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A Fight to Fix Geometry’s Foundations

February 14, 2017 by neogeo

When two mathematicians raised pointed questions about a classic proof that no one really understood, they ignited a years-long debate about how much could be trusted in a new kind of geometry.

Daan Botlek for Quanta Magazine

Daan Botlek for Quanta Magazine

In the 1830s, the Irish mathematician William Rowan Hamilton reformulated Newton’s laws of motion, finding deep mathematical symmetries between an object’s position and its momentum. Then in the mid-1980s the mathematician Mikhail Gromov developed a set of techniques that transformed Hamilton’s idea into a full-blown area of mathematical research. Within a decade, mathematicians from a broad range of backgrounds had converged to explore the possibilities in a field that came to be known as “symplectic geometry.”

The result was something like the opening of a gold-rush town. People from many different areas of mathematics hurried to establish the field and lay claim to its fruits. Research developed rapidly, but without the shared background knowledge typically found in mature areas of mathematics. This made it hard for mathematicians to tell when new results were completely correct. By the start of the 21st century it was evident to close observers that significant errors had been built into the foundations of symplectic geometry.

The field continued to grow, even as the errors went largely unaddressed. Symplectic geometers simply tried to cordon off the errors and prove what they could without addressing the foundational flaws. Yet the situation eventually became untenable. This was partly because symplectic geometry began to run out of problems that could be solved independently of the foundational issues, but also because, in 2012, a pair of researchers — Dusa McDuff, a prominent symplectic geometer at Barnard College and author of a pair of canonical textbooks in the field, and Katrin Wehrheim, a mathematician now at the University of California, Berkeley — began publishing papers that called attention to the problems, including some in McDuff’s own previous work. Most notably, they raised pointed questions about the accuracy of a difficult, important paper by Kenji Fukaya, a mathematician now at Stony Brook University, and his co-author, Kaoru Ono of Kyoto University, that was first posted in 1996.

This critique of Fukaya’s work — and the attention McDuff and Wehrheim have drawn to symplectic geometry’s shaky foundations in general — has created significant controversy in the field. Tensions arose between McDuff and Wehrheim on one side and Fukaya on the other about the seriousness of the errors in his work, and who should get credit for fixing them.

More broadly, the controversy highlights the uncomfortable nature of pointing out problems that many mathematicians preferred to ignore. “A lot of people sort of knew things weren’t right,” McDuff said, referring to errors in a number of important papers. “They can say, ‘It doesn’t really matter, things will work out, enough [of the foundation] is right, surely something is right.’ But when you got down to it, we couldn’t find anything that was absolutely right.”

The Orbit Counters

The field of symplectic geometry begins with the movement of particles in space. In flat, Euclidean space, that motion can be described in a straightforward way by Newton’s equations of motion. No further wrangling is required. In curved space like a sphere, a torus or the space-time we actually inhabit, the situation is more mathematically complicated.

This is the situation William Rowan Hamilton found himself considering as he studied classical mechanics in the early 19th century. If you think of a planet orbiting a star, there are several things you might want to know about its motion at a given point in time. One might be its position — where exactly it is in space. Another might be its momentum — how fast it’s moving and in what direction. The classical Newtonian approach considers these two values separately. But Hamilton realized that there is a way to write down equations that are equivalent to Newton’s laws of motion that put position and momentum on equal footing.

To see how that recasting works, think of the planet as moving along the curved surface of a sphere (which is not so different from the curved space-time along which the planet actually moves). Its position at any point in time can be described by two coordinate points equivalent to its longitude and latitude. Its momentum can be described as a vector, which is a line that is tangent to the sphere at a given position. If you consider all possible momentum vectors, you have a two-dimensional plane, which you can picture as balancing on top of the sphere and touching it precisely at the point of the planet’s location.

You could perform that same construction for every possible position on the surface of the sphere. So now you’d have a board balancing on each point of the sphere, which is a lot to keep track of. But there’s a simpler way to imagine this: You could combine all those boards (or “tangent spaces”) into a new geometric space. While each point on the original sphere had two coordinate values associated to it — its longitude and latitude — each point on this new geometric space has four coordinate values associated to it: the two coordinates for position plus two more coordinates that describe the planet’s momentum. In mathematical terms, this new shape, or manifold, is known as the “tangent bundle” of the original sphere. For technical reasons, it is more convenient to consider instead a nearly equivalent object called the “cotangent bundle.” This cotangent bundle can be thought of as the first symplectic manifold.

To understand Hamilton’s perspective on Newton’s laws, imagine, again, the planet whose position and momentum are represented by a point in this new geometric space. Hamilton developed a function, the Hamiltonian function, that takes in the position and momentum associated to the point and spits out another number, the object’s energy. This information can be used to create a “Hamiltonian vector field,” which tells you how the planet’s position and momentum evolve or “flow” over time.

Symplectic manifolds and Hamiltonian functions arose from physics, but beginning in the mid-1980s they took on a mathematical life of their own as abstract objects with no particular correspondence to anything in the world. Instead of the cotangent bundle of a two-dimensional sphere, you might have an eight-dimensional manifold. And instead of thinking about how physical characteristics like position and momentum change, you might just study how points in a symplectic manifold evolve over time while flowing along vector fields associated to any Hamiltonian function (not just those that correspond to some physical value like energy).

Lucy Reading-Ikkanda/Quanta Magazine

Lucy Reading-Ikkanda/Quanta Magazine

Once they were redefined as mathematical objects, it became possible to ask all sorts of interesting questions about the properties of symplectic manifolds and, in particular, the dynamics of Hamiltonian vector fields. For example, imagine a particle (or planet) that flows along the vector field and returns to where it started. Mathematicians call this a “closed orbit.”

You can get an intuitive sense of the significance of these closed orbits by imagining the surface of a badly warped table. You might learn something interesting about the nature of the table by counting the number of positions from which a marble, rolled from that position, circles back to its starting location. By asking questions about closed orbits, mathematicians can investigate the properties of a space more generally.

A closed orbit can also be thought of as a “fixed point” of a special kind of function called a symplectomorphism. In the 1980s the Russian mathematician Vladimir Arnold formalized the study of these fixed points in what is now called the Arnold conjecture. The conjecture predicts that these special functions have more fixed points than the broader class of functions studied in traditional topology. In this way, the Arnold conjecture called attention to the first, most fundamental difference between topological manifolds and symplectic manifolds: They have a more rigid structure.

The Arnold conjecture served as a major motivating problem in symplectic geometry — and proving it became the new field’s first major goal. Any successful proof would need to include a technique for counting fixed points. And that technique would also likely serve as a foundational tool in the field — one that future research would rely upon. Thus, the intense pursuit of a proof of the Arnold conjecture was entwined with the more workaday tasks of establishing the foundations of a new field of research. That entanglement created an uneasy combination of incentives — to work fast to claim a proof, but also to go slow to make sure the foundation was stable — that was to catch up with symplectic geometry years later.

How to Count to Infinity

In the 1990s the most promising strategy for counting fixed points on symplectic manifolds came from Kenji Fukaya, who was at Kyoto University at the time, and his collaborator, Kaoru Ono. At the time they released their approach, Fukaya was already an acclaimed mathematician: He’d given a prestigious invited talk at the 1990 International Congress of Mathematicians and had received a number of other awards for his fundamental contributions to different areas of geometry. He also had a reputation for publishing visionary approaches to mathematics before he’d worked out all the details.

“He would write a 120-page-long thing in the mid-1990s where he would explain a lot of very beautiful ideas, and in the end he would say, ‘We don’t quite have a complete proof for this fact,’” said Mohammed Abouzaid, a symplectic geometer at Columbia University. “This is very unusual for mathematicians, who tend to hoard their ideas and don’t want to show something which is not yet a polished gem.”

Kenji Fukaya, a mathematician at Stony Brook University, argues that his work has always been both complete and correct.

Kenji Fukaya, a mathematician at Stony Brook University, argues that his work has always been both complete and correct.

Fukaya and Ono saw the Arnold Conjecture as essentially a counting problem: What’s the best way to tally fixed points of symplectomorphisms on symplectic manifolds?

One method for tallying comes from work by the pioneering mathematician Andreas Floer and involves counting another complicated type of object called a “pseudo-holomorphic curve.” Counting these objects amounts to solving a geometry problem about the intersection points of two very complicated spaces. This method only works if all the intersections are clean cuts. To see the importance of having clean cuts for counting points of intersection, imagine you have the graph of a function and you want to count the number of points at which it intersects the x-axis. If the function passes through the x-axis cleanly at each intersection, the counting is easy. But if the function runs exactly along the x-axis for a stretch, the function and the x-axis now share an infinite number of intersection points. The intersection points of the two become literally impossible to count.

In situations where this happens, mathematicians fix the overlap by perturbing the function — adjusting it slightly. This has the effect of wiggling the graph of the function so that lines cross at a single point, achieving what mathematicians call “transversality.”

Fukaya and Ono were dealing with complicated functions on spaces that are far more tangled than the x-y plane, but the principle was the same. Achieving transversality under these conditions turned out to be a difficult task with a lot of technical nuance. “It became increasingly clear with Fukaya trying to prove the Arnold conjecture’s most general setup that it’s not always possible to achieve transversality by simple, naive methods,” said Yakov Eliashberg, a prominent symplectic geometer at Stanford University.

Lucy Reading-Ikkanda/Quanta Magazine

Lucy Reading-Ikkanda/Quanta Magazine

The main obstacle to making all intersections transversal was that it wasn’t possible to wiggle the entire graph of the function at once. So symplectic geometers had to find a way to cut the function space into many “local” regions, wiggle each region, and then add the intersections from each region to get an overall count.

“You have some horrible space and you want to perturb it a little so that you can get a finite number of things to count,” McDuff said. “You can perturb it locally, but somehow you have to fit together those perturbations in some consistent way. That’s a delicate problem, and I think the delicacy of that problem was not appreciated.”

In their 1996 paper, Fukaya and Ono stated that they used Floer’s method to solve this problem, and that they had achieved a complete proof of the Arnold conjecture. To obtain the proof — and overcome the obstacles around counting and transversality — they introduced a new mathematical object called Kuranishi structures. If Kuranishi structures worked, they belonged among the foundational techniques in symplectic geometry and would open up huge new areas of research.

But that’s not what happened. Instead the technique languished amid uncertainty in the mathematical community about whether Fukaya’s approach worked as completely as he said it did.

The End of the Low-Hanging Fruit

In mathematics, it takes a community to read a paper. At the time that Fukaya and Ono published their work on Kuranishi structures, symplectic geometry was still a loosely assembled collection of researchers from different mathematical backgrounds — algebraists, topologists, analysts — all interested in the same problems, but without a common language for discussing them.

In this environment, concepts that might have been clear and obvious to one mathematician weren’t necessarily so to others. Fukaya’s paper included an important reference to a paper from 1986. The reference was brief, but consequential for his argument, and hard to follow for anyone who didn’t already know that work.

“When you write a proof, it is implicitly checkable by somebody who has the same background as the person who wrote it, or at least sufficiently similar so that when they say, ‘You can easily see such a thing,’ well, you can easily see such a thing,” Abouzaid said. “But when you have a new subject, it’s difficult to figure out what is easy to see.”

Fukaya’s paper proved difficult to read. Rather than guiding future research, it got ignored. “There were people who tried to read it and they couldn’t, they had problems, so the adoption was actually extremely slow; it didn’t happen,” said Helmut Hofer, a mathematician at the Institute for Advanced Study in Princeton, New Jersey, who has been developing foundational techniques for symplectic geometry since the 1990s. “A lot of people just listened to other people and said, ‘If they have difficulties, I don’t even want to try.’”

Fukaya explains that in the years after he published his paper on Kuranishi structures, he did what he could to make his work intelligible. “We tried many things. I talked in many conferences, wrote many papers, abstract and expository, but none of it worked. We tried so many things.”

During the years that Fukaya’s work languished, no other techniques emerged for solving the basic problem of creating transversality and counting fixed points. Given the lack of tools they could trust and understand, most symplectic geometers retreated from this area, focusing on the limited class of problems they could address without recourse to Fukaya’s work. For individual mathematicians building their careers, the tactic made sense, but the field suffered for it. Abouzaid describes the situation as a collective action problem.

“It’s completely reasonable for one person to do this, it’s completely reasonable for a small number of people to do that, but if you end up in a situation where 90 percent of the people are working in small generality from a small number of cases in order to avoid the technical things that are done by the 10 percent minority, then I’d say that’s not very good for the subject,” he said.

By the late 2000s, symplectic geometers had worked through most of the problems they could address independently of the foundational questions involved in Fukaya’s work.

“Usually people go for the low-hanging fruits, and then the fruits hang a little higher,” Hofer said. “At some point, a certain pressure builds up and people ask what happens in the general case. That discussion took a while, it sort of built up, then more people got interested in looking into the foundations.”

Then in 2012 a pair of mathematicians broke the silence on Fukaya’s work. They gave his proof a thorough examination and concluded that, while his general approach was correct, the 1996 paper contained important errors in the way Fukaya implemented Kuranishi structures.

A Break in the Field

In 2009 Dusa McDuff attended a lecture at the Mathematical Sciences Research Institute in Berkeley, California. The speaker was Katrin Wehrheim, who was an assistant professor at the Massachusetts Institute of Technology at the time. In her talk, Wehrheim challenged the symplectic geometry community to face up to errors in foundational techniques that had been developed more than a decade earlier. “She said these are incorrect things; what are you going to do about it?” recalled McDuff, who had been one of Wehrheim’s doctoral thesis examiners.

For McDuff, the challenge was personal. In 1999 she’d written a survey article that had relied on problematic foundational techniques by another pair of mathematicians, Gang Liu and Gang Tian. Now, 10 years later, Wehrheim was pointing out that McDuff’s paper — like a number of early papers in symplectic geometry, including Fukaya’s — contained errors, particularly about how to move from local to global counts of fixed points. After hearing Wehrheim’s talk, McDuff decided she’d try to correct any mistakes.

Dusa McDuff, a mathematician at Barnard College, struggled for years to fix what she saw as gaps in the foundations of symplectic geometry.

Dusa McDuff, a mathematician at Barnard College, struggled for years to fix what she saw as gaps in the foundations of symplectic geometry.

“I had a bad conscience about what I’d written because I knew somehow it was not completely right,” she said. “I make mistakes, I understand people make mistakes, but if I do make a mistake, I try to correct it if I can and say it’s wrong if I can’t.”

McDuff and Wehrheim began work on a series of papers that pointed out and fixed what they described as mistakes in Fukaya’s handling of transversality. In 2012 McDuff and Wehrheim contacted Fukaya with their concerns. After 16 years in which the mathematical community had ignored his work, he was glad they were interested.

“It was around that time a group of people started to question the rigor of our work rather than ignoring it,” he wrote in an email. “In 2012 we got explicit objection from K. Wehrheim. We were very happy to get it since it was the first serious mathematical reaction we got to our work.”

To discuss the objections, the mathematicians formed a Google group in early 2012 that included McDuff, Wehrheim, Fukaya and Ono, as well as two of Fukaya’s more recent collaborators, Yong-Geun Oh and Hiroshi Ohta. The discussion generally followed this form: Wehrheim and McDuff would raise questions about Fukaya’s work. Fukaya and his collaborators would then write long, detailed answers.

Whether those answers were satisfying depended on who was reading them. From Fukaya’s perspective, his work on Kuranishi structures was complete and correct from the start. “In a math paper you cannot write everything, and in my opinion this 1996 paper contained a usual amount of detail. I don’t think there was anything missing,” he said.

Others disagree. After the Google group discussion concluded, Fukaya and his collaborators posted several papers on Kuranishi structures that together ran to more than 400 pages. Hofer thinks the length of Fukaya’s replies is evidence that McDuff and Wehrheim’s prodding was necessary.

“Overall, [Fukaya’s approach] worked, but it needed much more explanation than was originally given,” he said. “I think the original paper of Fukaya and Ono was a little more than 100 pages, and as a result of this discussion on the Google group they produced a 270-page manuscript and there were a few hundred pages produced explaining the original results. So there was definitely a need for the explanation.”

Abouzaid agrees that there was a mistake in Fukaya’s original work. “It is a paper that claimed to resolve a long-standing problem, and it’s a paper in which this error is a gap in the definition,” he said. At the same time, he thinks Kuranishi structures are, generally speaking, the right way to deal with transversality issues. He sees the errors in the 1996 paper as having occurred because the symplectic geometry community wasn’t developed enough at the time to properly review new work.

“The paper should have been refereed much more carefully. My opinion is that with two or three rounds of good referee reports that paper would have been impeccable and there would have been no problem whatsoever,” Abouzaid said.

In August 2012, following the Google group discussion, McDuff and Wehrheim posted an article they’d begun to write before the discussion that detailed ways to fix Fukaya’s approach. They later refined and published that paper, along with two others, and plan to write a fourth paper on the subject. In September 2012, Fukaya and his co-authors posted some of their own responses to the issues McDuff and Wehrheim had raised. In Fukaya’s mind, McDuff and Wehrheim’s papers did not significantly move the field forward.

“It is my opinion that the papers they wrote do not contain new and significant ideas. There is of course some difference from earlier papers of us and other people. However, the difference is only on a minor technicality,” Fukaya said in an email.

Hofer thinks that this interpretation sells McDuff and Wehrheim’s contributions short. As he sees it, the pair did more than just fix small technical details in Fukaya’s work — they resolved higher-level problems with Fukaya’s approach.

“They understood very well the different pieces and how they worked together, so you couldn’t just say: ‘Here, if that’s problematic, I fixed it locally,’” he said. “You could also know then more or less where a possible other problem would arise. They understood it on an extremely high level.”

The difference in how mathematicians evaluate the significance of the errors in Fukaya’s 1996 paper and the contributions Wehrheim and McDuff made in fixing them reflects a dichotomy in ways of thinking about the practice of mathematics.

“There are two conceptions of mathematics,” Abouzaid said. “There’s mathematics as: The currency of mathematics is ideas. And there’s mathematics as: The currency of mathematics is proofs. It’s hard for me to say on which side people stand. My personal attitude is: The most important thing in mathematics is ideas, and the proofs are there to make sure the ideas don’t go astray.”

Fukaya is a geometer with an instinct to think in broad strokes. Wehrheim, by contrast, is trained in analysis, a field known for its rigorous attention to technical detail. In a profile for the MIT website Women in Mathematics, she lamented that in mathematics, “we don’t write good papers anymore,” and likened mathematicians who doesn’t spell out the details of their work to climbers who reach the top of a mountain without leaving hooks along the way. “Someone with less training will have no way of following it without having to find the route for themselves,” she said.

These different expectations for what counts as an adequate amount of detail in a proof created a lot of tension in the symplectic geometry community around McDuff and Wehrheim’s objections. Abouzaid argues that it’s important to be tactful when pointing out mistakes in another mathematician’s work, and in this case Wehrheim might not have been diplomatic enough. “If you present it as: ‘Everything that has appeared before us is wrong and now we will give the correct answer,’ that’s likely to trigger some kinds of issues of claims of priority,” he said.

Wehrheim declined multiple requests to be interviewed for this article, saying she wanted to “avoid further politicization of the topic.” However, McDuff thinks that she and Wehrheim had no choice but to be forceful in pointing out errors in Fukaya’s work: It was the only way to get the field’s attention.

“It’s sort of like being a whistleblower,” she said. “If you point [errors] out correctly and politely, people need not pay attention, but if you point them out and just say, ‘It’s wrong,’ then people get upset with you rather than with the people who might be wrong.”

Regardless of who gets credit for fixing the issues with Fukaya’s paper, they have been fixed. Over the last few years, the dispute surrounding his work has settled down, at least as a matter of mathematics.

“I would say it was a somewhat healthy process. These problems were realized and eventually fixed,” Eliashberg said. “Maybe this unnecessarily caused too many passions on some sides, but overall I think everything was handled and things will go on.”

New Approaches

A developing field does not have many standard results that everyone understands. This means each new result has to be built from the ground up. When Hofer thinks about what characterizes a mature field of mathematics, he thinks about brevity — the ability to write an easily understood proof that takes up a small amount of space. He doesn’t think symplectic geometry is there yet.

“The fact is still true that if you write a paper today in symplectic geometry and give all the details, it can very well be that you have to write several hundred pages,” he said.

For the last 15 years Hofer has been working on an approach called polyfolds, a general framework that can be used as an alternative to Kuranishi structures to address transversality issues. The work is nearing completion, and Hofer explains that his intention is to break symplectic geometry into modular pieces, so that it’s easier for mathematicians to identify which pieces of knowledge they can rely on in their own work, and easier for the field as a whole to evaluate the correctness of new research.

“Ideally it’s like a Lego piece. It has a certain function and you can plug it together with other things,” he said.

Polyfolds are one of three approaches to the foundational issues that have vexed symplectic geometry since the 1990s. The second is the Kuranishi structures, and the third was produced by John Pardon, a young mathematician at Princeton University who has developed a technique based on Wehrheim and McDuff’s work, but written more in the language of advanced algebra. All three approaches do the same kind of thing — count fixed points — but one approach might be better suited to solving a particular problem than another, depending on the mathematical situation.

In Abouzaid’s opinion, the multiple approaches are a sign of the strength of the field. “We are moving away from these questions of what’s wrong, because we’ve gotten to the point where we have different ways of approaching the same question,” he said. He adds that Pardon’s work in particular is succinct and clear, resulting in a tool that’s easy for others to wield in their own research. “It would have been fantastic if he’d done this 10 years before,” he said.

Abouzaid thinks symplectic geometry is doing well along other measures as well: New graduate students are coming into the field, senior researchers are staying, and there’s a steady stream of new ideas. (Though Fukaya, after his experiences, holds a different view: “It is hard for me to recommend my students go to that area because it’s dangerous,” he said.)

For Eliashberg, the main attraction of symplectic geometry remains, in a sense, the uncertainty in the field. In many other areas of mathematics, he says, there is often a consensus about whether particular conjectures are true or not, and it’s just a question of proving them. In symplectic geometry, however, there’s less in the way of conventional wisdom, which invites contention, but also creates exciting possibilities.

“For me personally, what was exciting in symplectic geometry is that whatever problem you look at, it’s completely unclear from the beginning what would be the answer,” he said. “It could be one answer or completely the opposite.”

Update and correction: On February 10 this article was updated to include the work of Andreas Floer and to clarify the timing of the various papers that were posted following the 2012 Google group discussions.

By Kevin Hartnett via Quanta magazine

Filed Under: Math, Science

The Hive – Kew Gardens’ spectacular new bee-inspired sculpture

October 7, 2016 by neogeo

The critically acclaimed structure The Hive encapsulates the story of the honey bee and the important role of pollination, through an immersive sound and visual experience.

What is The Hive?

The Hive is a unique structure, inspired by scientific research into the health of bees. Designed by UK based artist Wolfgang Buttress, it was originally created as the centrepiece of the UK Pavilion at the 2015 Milan Expo.

The installation is made from thousands of pieces of aluminium which create a lattice effect and is fitted with hundreds of LED lights that glow and fade as a unique soundtrack hums and buzzes around you.

These multi-sensory elements of the Hive are in fact responding to the real-time activity of bees in a beehive behind the scenes at Kew. The sound and light intensity within the space changes as the energy levels in the real beehive surge, giving visitors an insight into life inside a bee colony.

The award-winning honeycomb structure features lighting and music which fluctuate in harmony with a real beehive.


It may look like an alien spacecraft has landed at Kew Gardens , but this shimmering whorl of aluminium is actually an ode to an every day visitor – the humble bee.
The Hive looms amid the greenery like a vast swarm but as you approach, the intricate structure modelled upon the architecture of a honeycomb is revealed.
What’s more, Wolfgang Buttress’s spectacular sculpture is hooked up a real beehive in the gardens and contains nearly 1,000 LED lights which flicker in time to vibrations as the winged insects chatter among themselves.

The Hive's creator, artist Wolfgang Buttress, inside his bee-inspired sculpture at Kew Gardens

The Hive’s creator, artist Wolfgang Buttress, inside his bee-inspired sculpture at Kew Gardens

To complete the mood, and immerse you even more completely into the inner world of the pollinator, the sonorous hum of a specially composed soundtrack envelops visitors as they enter the 17 metre tall mesh – growing louder as activity levels rise within the colony.

‘Conversation between visitors and the bees’

As The Hive’s creator puts it, the towering installation is an attempt to engender a “conversation” between humans and the unpaid workers responsible for pollinating the vast majority of crops we consume.
“The bee is a barometer of the earth’s health – a kind of sentinel of the planet – and by connecting (the sculpture) to a real beehive I hoped to give visitors a sense of connection to bees and to nature,” he says.

Looking up through the lattice from within The Hive at Kew Gardens

Looking up through the lattice from within The Hive at Kew Gardens

“It’s like a conversation between you, the bees, the landscape and the sculpture.”
The Hive – all 170,000 pieces and 40 tonnes of it – was imported from the Milan Expo 2015, where it was the centrepiece of the UK Pavilion.
A wildflower meadow was specially created to accommodate it, meaning visitors ascending the spiral staircase can imagine themselves as bees returning to the colony.

Insight into bees’ vital role in crop production

Kew Gardens’ resident bee expert Phil Stevenston says he hopes the installation will give visitors an insight into the insects’ lives and the importance of protecting the diverse array of plants they rely on for food.
“Diversity is incredibly important for bees and other pollinators, and it doesn’t take much effort, just a little knowledge, to contribute,” he says.
“Anyone with a garden can help by growing plants that provide forage for pollinators and are not just there to look attractive.

The transparent floor of The Hive at Kew Gardens

The transparent floor of The Hive at Kew Gardens

“With some of the multi-petalled flowers (specially bred for their looks), bees are unable to access the nectar or pollen.”
As well as soaking up the sights and sounds as they harmonise with nature, visitors can learn more about bees and their role in the ecosystem.

Listening posts enable you to eavesdrop on bee communication

On the ground floor, for example, a series of listening posts allow you to eavesdrop on bees’ communication via vibration.

The basement of The Hive at Kew Gardens, where listening posts enable you to feel the vibrations bees make as they communicate

The basement of The Hive at Kew Gardens, where listening posts enable you to feel the vibrations bees make as they communicate

Just stick a lollipop into one of the slots, and you can feel the buzz of the insects giving directions, begging for food or squaring up to one another.
Each day, from 11am-5pm a group of “hive explainers” will be on hand to lead you around the structure and explain the importance of bees.

Visitors can also follow a “pollination trail” around the gardens, and you can also meet the experts and get the lowdown on growing wild flowers at a series of special events.
Kew will also host three “Hive Lates” this September, giving you the chance to take in the sculpture’s beauty while sipping on honey-infused cocktails as dusk descends.

The Hive at Kew Gardens will host a series of "lates", enabling visitors to experience the interactive sculpture at dusk

The Hive at Kew Gardens will host a series of “lates”, enabling visitors to experience the interactive sculpture at dusk

THE HIVE FACTS

• The sculpture stands 17 metres tall and weighs 40 tonnes
• It consists of nearly 170,000 pieces of aluminium
• There are nearly 1,000 LED lights
• Bees tend to be more active just before a storm, as they return to the hive, meaning the changing light and music levels within the sculpture act as a handy barometer
• The soundscape of bees, cello and vocals was composed by space-rock band Spiritualized, along with Sigur Ros’s regular string section Amiina, Buttress’s singer daughter Camille and Dr Bencsik’s cellist wide Deirdre
• The music, which also features unusual bee noises, was made after discovering bees hum in the key of C. It is available as an album BE.ONE
• The meadow around the hive is planted with 34 different native UK plant species that provide important forage for bees
• The global value of pollination is estimated to be between £162bn and £399bn a year. In the UK alone it is £600m
• Of the 100 crop species which provide 90% of food worldwide, 70 are pollinated by bees
• There are 270 species of bees, of which 90% are solitary creatures
• More than a quarter of bee species are “cuckoo bees”, meaning they lay their eggs in other bees’ nests
• The nectar of the rhododendron is toxic to honey bees, but not bumblebees

The Hive contains nearly 1,000 LED lights which pulse in time to activity levels within a nearby beehive

The Hive contains nearly 1,000 LED lights which pulse in time to activity levels within a nearby beehive

HOW CAN YOU HELP BEES?

We asked horticulturists at Kew which were the best plants gardeners could grow to support bees, and this is what they recommended
• White clover – the meadow around the hive is oversown with white clover, which although common and viewed by some gardeners as a weed, is a great food source for bees
• Prunella vulgaris
• Cowslip
• Wild carrot

 

The Hive, BE.ONE performance 29 September 2016

The Hive, BE.ONE performance 29 September 2016

The Hive, BE.ONE performance 29 September 2016

The Hive, BE.ONE performance 29 September 2016

The Hive, BE.ONE performance 29 September 2016

The Hive, BE.ONE performance 29 September 2016, Photographs: Jeff Eden

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Photographs: Mark Haddon

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Inside The Hive at Milan Expo 2015 – image: Hufton & Crow

Via Kew, Get West London and Wolfgang Buttress

Filed Under: Architecture, Nature, Sculpture Tagged With: Bees, honeycomb, Kew Gardens, Milan Expo, Wolfgang Buttress

Geometrical Auditorium in Poland

July 7, 2016 by neogeo

Stunning Concert Hall “Jordanki” of the city of Torun in Poland features remarkable geometrical play of walls and ceilings. Their shapes allow creation of various interesting staging and lightning effects during performances. The author, Spanish architect Fernando Menis imagined this monumental auditorium as a “solution for the fusion between old and new and that is achieved also through the materials which are going to be used for its construction. For the interior, bricks are going to be used as a recall of the façades of the old town. For the exterior very clear, almost white concrete is going to be used and in some place it is “broken” by some cracks which reveal the interior coating. The façade reinterprets the traditional handicraft of the bricks and establishes a parallel between the tectonics of the city and the strategic situation of the plot. The 2 colours -red and white – are also a way for outlining the relation between the traditional use of bricks and the technology and modernity of new urban developments.”

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via Fubiz

Filed Under: Architecture, Design Tagged With: architecture, Concert Hall, Fernando Menis, Jordanki, Poland, Torun

ECOsystem GEOmetry

March 28, 2016 by neogeo

Captivating video by Noble Sweat for international garden expo 2013 Suncheon bay – Korea.

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via Noble Sweat

Filed Under: Animation, Design, Nature

Brilliant array of precisely aligned architecture

February 11, 2016 by neogeo

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Geometry Club is a collaboration of precisely aligned architecture photographs from around the world.

It began just over a year ago as an Instagram account. The purpose was to curate a gallery of identically composed architecture photographs. Instagrammers are encouraged to participate and submit their photographs to be featured using the hashtag #geometryclub.

Geometry Club currently has over 50 contributors in more than 20 countries, featuring shots from Paris, London, LA, Moscow, Sydney, Tokyo, Berlin etc.

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Visit geometryclub.org
Follow @geometryclub

Filed Under: Architecture, Photography, Technology

Taxonomy Of Shapes – Reuleaux Polygon meets Golygon

November 17, 2015 by neogeo

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As any compulsive list-maker knows, there’s something immensely satisfying about creating a list, even—no, especially—if you already know its contents by heart. Pop Chart Lab, the team behind brilliantly simple infographic posters like Pie Charts of Pies and The Grand Taxonomy of Rapper Names (they also do some complex ones), have made a business out of it.
This week, the Brooklyn studio released its latest chart, A Synoptic Scheme of Shapes, is about as simplistic as they come. The poster is a taxonomy of shapes, ranging from the humble ellipse, to the majestic golygon, to the sexy-looking cardioid.

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Patrick Mulligan, Pop Chart Lab’s founder, tells Co.Design that after years of dreaming up high-concept charts, his team decided to make the simplest list they could think of. “We were trying to think of the most minimalist thing we could chart, and after tossing around a few odd ideas like “a chart… of charts!” we decided to investigate doing one based on simple shapes,” he explains. “The research process felt like finally covering all the stuff we slept through in high-school geometry class.”

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Sure, this poster is a simplistic exercise in visual classification. But then again, I had no idea what a Reuleaux polygon was before today.
You can buy A Synoptic Scheme of Shapes here.

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via Fast Company

Filed Under: Art, Education, Math Tagged With: Infographics, Pop Chart Lab, shapes

Surreal Geometry Constructs a World of Loops [Music Video]

November 3, 2015 by neogeo

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The endless loop: what are its possibilities and limitations? As part of a final project for Berliner Technische Kunsthochschule, university student Felix Neumann, who previously co-created a cascading portal of light and geometry, built a 2D music video full of looping, animated isometric (cube-based) shapes. Titled “Beauty of the Loop,” it resembles mobile game Monument Valley with its cascading geometry.

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“The animation tries to fathom the possibilities and restrictions of exclusively working with loops,” Neumann notes. “The combination of these loops and the song takes the viewer on an expedition through a surreal geometric world—over and over again.”

A wide variety of isometric shapes can be drawn in two dimensions on isometric paper, which is often used for sketching out engineering designs. Neumann applies this technique for his music video, with pretty striking results. Together with music by Bluestaeb, Neumann creates a wonderful little exploration of mathematical loops in the form of sound and vision.

BEAUTY OF THE LOOP /// Animation from Felix von Líska on Vimeo.

Click here to see how Felix Neumann’s Beauty of the Loop took shape.

 

By DJ Pangburn via The Creators Project

 

Filed Under: Animation, Art, Math, Music Tagged With: Felix Neumann, isometric, loops, music, shapes, video

Math Generated Art: Leonardoworx’ The Iterative Method

October 5, 2015 by neogeo

 

Case studies about iterative method.
How can it be connected to generative life.
It’s about how math and physic can be the key for explaining human life and perception, even the feelings we can not quantify, like love, that goes beyond time and space dimensions.

NOTE:
This project started developing code functions in Max/Msp and Processing using Arrays and the “for” instruction.
After i decided to connect all research i made (studying physic books, TED talks, M.I.T. documents) in one project, trying to enhance the conncetion between these studies and my life experience.

ABOUT ITERATIVE METHOD:
In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common.

In the problems of finding the root of an equation (or a solution of a system of equations), an iterative method uses an initial guess to generate successive approximations to a solution. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution.
Iterative methods are often the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.

via Leonardoworx

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Working side by side with multimedia artist Leonardoworx is not a merely professional duty, it’s an incredible experience. His latest project, aptly called The Iterative Method, sees Leonardoworx raising the bar of his comprehensive approach to visual art and music, this time using a combination of maths, physics and coding to generate art.

Every shape, line or movement of The Iterative Method hasn’t been put together in a software suite but has been originated by an array of numbers (variables) set into formulas created by artist himself and that followed the paradigm of Quantic Physics. That is to say that Leonardo didn’t know the visual outcome of the artworks he was creating, just that if the math was correct they were going to appear as an orderly shape—the incredible beauty of the project lies both in the final result and in the creative process, which required an astonishing amount of skills and knowledge.

But if you are like us and mathematical functions are just a fading high school memory, what is The Iterative Method? We asked Leonardoworx to give us an insight for Dummies on this incredibly complex yet fascinating project that expands the realm of aesthetics into math, physics and coding and ultimately to philosophy.

M: The artwork we see in the Iterative Method was completely created by mathematical functions: how did this idea came about?

LWX: It all started when I was working on the House of Peroni installation, as it was the first time I used iterative functions translated in coding with Processing. I was immediately intrigued by the idea of creating an infinite number of auto-generative artworks that evolve from a simple structure into more and more complex creations that I call module or DNA — for its similarity to genetic code. The Iterative Method gives an infinite number of options to create these complex modules, however I intentionally decided to keep the function simple in order to unveil its beauty.

M: The Iterative Method project encompassed maths, physics and coding: how did those three elements worked together?

LWX: The goal of physics is to study all natural events, events that can described and quantified mathematically. This is the reason why the connection between mathematics and physics (especially with the evolution of the modules movements) was quite natural. The coding was simply the language I’ve used to communicate to the computer the formula that governed the generation of the modules.

M: How did you create these modules? How did you choose them?

LWX:  In order to have more control on both the shape and the movements of the modules I’ve used just a limited amount of variables. The process was quite straight forward: once I inserted variables into the functions, the Mac would generate an endless series of modules. I then chose a range of variables in a set moment of time that would create the modules I thought were more interesting and emotionally engaging.

M: Were all the modules you’ve created used for the project or was there one that didn’t make the final cut?

LWX:  I discarded one and that’s because it kind of looked like a flying saucer.

M: That’s interesting: most of the modules were good from the very start: is it fair to say that it wasn’t a tentative creative process?

LWX: I’m so used to write coding that whenever I write strings I can already visualise what it is going to appear on the screen and to be honest, if the math is right there is no chaos.

M: How did this project effect your creative process?

LWX: There was definitely a shift in my approach, especially as this time I wasn’t using a 3D or interface-based software: I was working directly with the computer, forcing myself to create a fully functioning basic language able to run complex creative process.

M: You’ve also made a series of artworks to complement the video the project: how are they linked together?

LWX: Each complex module has been originated by running the function of “simplier” modules (respectively the orange, cyan, violet and yellow) in the iterative function.

M: What was your inspiration for the music?
LWX: I composed everything on a OP-1 and a Maschine. Mirroring the visual part, I wrote the score with a pattern form on Max Msp and then reused them. I didn’t use any sound design as I wanted the sound to be very minimal and focusing mostly on the voice of Alison (the computer who introduces the modules). Speaking of Alison, I think that I worked so much on this project that I might had a chat or two with her in my coffee break!

M: Although The Iterative Method is based on math and physics the result is not cold at all, quite the contrary, is incredibly powerful from the emotional point of view.

LWX: When I started this project I felt there was a strong connection between the iterative method and some of my personal experiences, although I didn’t clearly understand why. I then started researching online and went through a lot of M.I.T. papers, thesis, TED talks and all those sources shared one idea: that all our feelings, like the love for someone, even someone who is not with us anymore, go beyond a determined space and time and follow an iterative pattern that can be quantified with one “value” that we already know. This is the reason why I did The Iterative Method.

via MACHAS

Filed Under: Art, Math Tagged With: Digital Art, Motion Graphics, music

College Kites 1957

April 14, 2015 by neogeo

These images come from the University of Illinois “Kite Derby Day”. All the entries shown here are from the sophomores in the department of Industrial Design.

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via MONDOBLOGO

Filed Under: Craft, Design, Education Tagged With: 1957, college, Industrial Design, kite, Kite Derby Day, students, University of Illinois

Wearable Structures by Matija Čop

February 25, 2015 by neogeo

An Interview with Croatian Fashion Designer Matija Čop

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Working on the cutting edge of contemporary fashion, not to mention unafraid of dealing with some difficult questions that sit at the very heart of his practice (fashion is, after all not just about the clothes), young fashion designer Matija Čop is definitely going to prove to be a name to remember in the coming years. Born and raised in Croatia, before turning to the world of fashion, he was a national champion in athletics and a student of Croatian language and literature.Currently a graduate student of fashion design of the Faculty of Textile Technology in Zagreb, Čop combines a conceptual approach with a more intuitive hands-on working method, depending on the nature of each project and what he wants to achieve each time. Less focused on how groundbreaking or provocative his garments could be, instead, he turns his attention to the role fashion (including fashion makers) plays in its wider socio-cultural context.

Today, after spending a few months in Sweden researching an approach to fashion design that seeks to create ”fashion without clothing,” Čop is presently involved in a new project leading him in new directions. Though reluctant to reveal any more detail about his current undertakings, the 27-year-old designer was very generous in answering the following questions for us, thereby offering us a broader look into his work and design philosophy.

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Object 12-1, photo © Zvonimir Ferina.

All of your past projects are in one way or another related to architecture. What is it exactly in architecture that you find so interesting as a fashion designer?
A garment is in itself an architectural object, and vice versa – but not because of its appearance, its structure or monumentality, but because of the idea behind it. Both disciplines deal with similar problems, such as the relation between body and space – it’s only the scale that actually changes. As a fashion designer I have been interested in space in my past projects. Since space is a distinctive feature of architecture, it just so happened that my explorations often leaned on the language of architecture, which is why the outcome of my projects is so often reminiscent of architecture.

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Object 12-1, photo © Vanja Šolin.

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Object 12-1, photo © Zvonimir Ferina.

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Object 12-1, photo © Zvonimir Ferina.

Another ‘fashion week month’ is about to start, with everyone excited to see what’s new in the fashion world. What is your opinion on ‘high fashion’? Do you aspire to become a very popular ‘high-end’ fashion designer in the future?
High fashion is a form of fashion that emphasizes, above all, a specific fashion idea. These ideas change and are reinterpreted from fashion show to fashion show. I consider that valuable. Fashion design changes just as man does. In the past, these changes came about slowly. Nowadays it’s faster. These changes that are so closely related to fashion, are a reflection of the individual responsible for a collection, and consequently a reflection of society (the local one more than the global one). I like to communicate with other designers in that way. I strive to show my own version of the moment – but I don’t think about whether that will suffice in order to become a high-end fashion designer.

Apart from being an entrepreneur and a good dress-maker, what do you believe is the role (or even, the responsibility) of the contemporary fashion designer in society today?
Even though being a good dress maker and entrepreneur is difficult enough as it is, the contemporary fashion designer should be a loud interpreter of the moment. No matter where their interests lie (be it to explore sustainability in fashion, custom-made clothing or something else), they should be brave in what they’re doing and be free of the anxiety of influence because of their predecessors.

Fashion always happens now, in the moment. Its authenticity arises from the courage of the individual, no matter what direction that may go in.

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Object 12-1, photo © Vanja Šolin.

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Object 12-1, photo © Vanja Šolin.

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Object 12-1, photo © Vanja Šolin.

You mentioned that you have recently spent some time in Sweden working on a ‘‘theoretical approach’’ to fashion design. Can you tell us more about this approach?
I spent a semester in Sweden at the University of Borås where I dealt with the subject of abstracting the body – fashion without clothes. If man is permanently connected to a garment/textile object and is a part of it – what is man without it? That was one of the questions I asked myself and subsequently went about looking for the answers – doing experiments by covering the body with non-clothing/non-textile objects, in turn abstracting it and turning it into the object itself.

You were involved in Lady Gaga‘s G.U.Y. video. Can you tell us more about your participation in that project?
Garments from my ‘City Lace’ and ‘Object 12-1’ collections were used in the music video for ‘G.U.Y.’; I consider Lady Gaga a relevant figure in modern design/art, so it has been a great honour to be a small part of that moment.

2-object-12-1Matija-Cop-yatzer

Object 12-1, photo © Zvonimir Ferina.

What do you consider as a ‘successful’ project?
I consider a successful project as one which achieves some sort of communication with the public (and not necessarily a public great in number). The fact that it triggers a reaction is enough for me – to be able to receive feedback (be it good or bad, it doesn’t matter). Another, perhaps, more intimate criterion would be the feeling I get when the project is finished. It is then that I sense the project is a success in itself because it answered the questions it was initially faced with.

Matija Čop’s “Homeless in Heaven” project was one of the winning entries in the Young Balkan Designers competition 2014. His ”Object 12-1” collection will be exhibited at the MAR museum in Ravenna, Italy, next October.

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Object 12-1, photo © Zvonimir Ferina.

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Object 12-1, photo © Nives Milješić.

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Object 12-1, photo © Zvonimir Ferina.

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Object 12-1, photo © Zvonimir Ferina.

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Object 12-1, photo © Zvonimir Ferina.

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City Lace, photo © Zvonimir Ferina.

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City Lace, photo © Zvonimir Ferina.

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City Lace, photo © Zvonimir Ferina.

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Homeless in heaven, photo © Matija Čop.

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Homeless in heaven, photo © Matija Čop.

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Homeless in heaven, photo © Matija Čop.

via yatzer

Filed Under: Art, Design, Fashion Tagged With: clothes, fashion, fashion design, Matija Čop

“Godel, Escher Bach” More Than Thirty Years Later…

September 5, 2014 by neogeo

geb first

Remember the book “Godel, Escher Bach: An Eternal Golden Braid?” and Douglas Hofstadter and Artificial Intelligence? What they are doing now:

This book was a sensation back in 1980.

It was an absolutely, amazingly brilliant work from a totally unknown first-time professor/author.
So much so that Scientific American’s Martin Gardner praised it to the skies, rightly so, pushing it to the best-seller book lists, and not because was yet another detective novel or a political rant or a ghostwritten memoir by someone rich and famous.

No, it was an well-written, highly entertaining book about the connections among mathematics, computer languages,  English and other ancient and modern human languages, DNA code, artificial intelligence, science, history, music,  what it means to be human and to think and do stuff. Brilliant, original ideas and clear, sparkling language on every beautifully-written page — with lots of illustrations and diagrams, too!!

The fact that the author’s father (Robert) shared the Nobel Prize in nuclear Physics in 1961 considerably upped the odds that Hofstadter would grow up in intellectual atmosphere that valued independent thinking rather than mindless obedience. According to this Atlantic review, his parents more than tolerated young Douglas’ tendency to go off on various tangents and delve into them deeply and thoroughly and even obsessively for some period of time, until he felt he had another hunch or tangent, which he would again jump into with both feet and all his weight. And all of it carefully and brilliantly documented.
Those documents, I discovered in reading this essay, became the book GEB.

He and the rest of the Artificial Intelligence community agree that they have gone in different directions since then.
AI today no longer tries to imitate the actions of the human brain, but they are doing some pretty amazing stuff with sheer computational speed and power.
Hofstadter thinks that may all be very nice, but that approach does not really help understand how humans think — how we make all those connections in our head in which we strip off 99% of the details about one thing and find one or two ways in which it relates to another thing, constantly and unexpectedly

[I gave some copies to some of my students; I wish I could have afforded to give away more. Instead, I developed lists of books on math and science and math field trips and tessellations and had kids read some of the books and do various projects that I though would illustrate some topic and develop pride and character and a belief that math of whatever sort I was teaching to them was actually worth something and useful in real life as well as pretty cool as an abstract creation of humanity…]

Douglas Hofsadter, the author of GEB is not working for Google or Apple or any other such company helping to develop complex computer  programs that do complex things either very well at least some of the time — because DH thinks they won’t lead to more understanding of human or animal intelligence. According to this review, DH has the greatest job in the world — he doesn’t have to teach classes. or  attend any meetings at all, or perform experiments. or write grant applications. For a number of years,. he took over the Mathematical Games that Martin Gardner used to write for SciAm, and renamed it “Metamagical Themas” – an anagram of the original name.
A few interesting quotes from the article: (The man who would teach machines to think…)
“Correct speech isn’t very interesting; it’s like a well-executed magic trick—effective because it obscures how it works. What Hofstadter is looking for is “a tip of the rabbit’s ear … a hint of a trap door.”

Now, some quotes from Hofstadter himself, which I got from a collection of his quotes, and which remind me why I thopught his work was so brilliant in the first place:

Meaning lies as much
in the mind of the reader
as in the Haiku.

“How gullible are you? Is your gullibility located in some “gullibility center” in your brain? Could a neurosurgeon reach in and perform some delicate operation to lower your gullibility, otherwise leaving you alone? If you believe this, you are pretty gullible, and should perhaps consider such an operation.”
“Hofstadter’s Law: It always takes longer than you expect, even when you take into account Hofstadter’s Law”
“Sometimes it seems as though each new step towards AI, rather than producing something which everyone agrees is real intelligence, merely reveals what real intelligence is not. ”
“In the end, we self-perceiving, self-inventing, locked-in mirages are little miracles of self-reference.”
“I would like to understand things better, but I don’t want to understand them perfectly.”
“This idea that there is generality in the specific is of far-reaching importance.”

via GFBRANDENBURG’S BLOG

If you haven’t fried your brain yet HERE IS GEB IN PDF

MIT lectures:

And some random comments from Amazon, Goodreads and Reddit:

“Expand your mind! Not for the faint of heart & yet by no means dry. 
Hofstadter makes some fascinating observations about emergent properties (such as intelligence) and diverts us into the extremely heavy mathematics of Godel via the self referencing systems that are Bach’s fugues and Escher’s ‘optical illusion’ style artwork.”

“I could not with a clear conscience recommend this book to everyone, because I’m simply not that cruel. It would be like recommending large doses of LSD to everyone: some small minority will find the experience invaluably enlightening, but for most people it’s just going to melt their brain.”

“GEB is an astonishing achievement in popularizing mathematical philosophy (!), and among the few truly life-changing books I’ve read.”

“You don’t need any formal philosophy to get something out. But if you are not the type that enjoys pondering on the abstract side of things to reach the concrete, you might end up getting bored. With that under your belt, if you are interested in cognition, sense of self, consciousness, thinking, thinking about thinking, thinking about thinking about thinking… and so on, you are in for a ride.”

“It seems surprising to me that Hofstadter would constrain his own book to having only one central message–surely he should understand that a book of this complexity will mean many things to many different people, and that indeed is the reason for its popularity.
So, I still highly recommend this book, but I’m left just a little disappointed that Hofstadter seems somewhat at war with his readers and as a result, won’t attempt to update the book or try to help us reconcile the many events of the last 20 years with the themes of his book.”

Here is excerpt from the Twentieth-Anniversary Edition Preface:

On what GEB is really all about (twenty years later)

GEB-XX

So what is this book, Gödel, Escher, Bach: an Eternal Golden Braid — usually known by its acronym, “GEB” — really all about?

That question has hounded me ever since I was scribbling its first drafts in pen, way back in 1973. Friends would inquire, of course, what I was so gripped by, but I was hard pressed to explain it concisely. A few years later, in 1980, when GEB found itself for a while on the bestseller list of The New York Times, the obligatory one-sentence summary printed underneath the title said the following, for several weeks running: “A scientist argues that reality is a system of interconnected braids.” After I protested vehemently about this utter hogwash, they finally substituted something a little better, just barely accurate enough to keep me from howling again.

Many people think the title tells it all: a book about a mathematician, an artist, and a musician. But the most casual look will show that these three individuals per se, august though they undeniably are, play but tiny roles in the book’s content. There’s no way the book is about these three people!

Well, then, how about describing GEB as “a book that shows how math, art, and music are really all the same thing at their core”? Again, this is a million miles off — and yet I’ve heard it over and over again, not only from nonreaders but also from readers, even very ardent readers, of the book.

And in bookstores, I have run across GEB gracing the shelves of many diverse sections, including not only math, general science, philosophy, and cognitive science (which are all fine), but also religion, the occult, and God knows what else. Why is it so hard to figure out what this book is about? Certainly it’s not just its length. No, it must be in part that GEB delves, and not just superficially, into so many motley topics — fugues and canons, logic and truth, geometry, recursion, syntactic structures, the nature of meaning, Zen Buddhism, paradoxes, brain and mind, reductionism and holism, ant colonies, concepts and mental representations, translation, computers and their languages, DNA, proteins, the genetic code, artificial intelligence, creativity, consciousness and free will — sometimes even art and music, of all things! — that many people find it impossible to locate the core focus.

The Key Images and Ideas that Lie at the Core of GEB

Needless to say, this widespread confusion has been quite frustrating to me over the years, since I felt sure I had spelled out my aims over and over in the text itself. Clearly, however, I didn’t do it sufficiently often, or sufficiently clearly. But since now I’ve got the chance to do it once more — and in a prominent spot in the book, to boot — let me try one last time to say why I wrote this book, what it is about, and what its principal thesis is.

In a word, GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle? What is an “I” and why are such things found (at least so far) only in association with, as poet Russell Edson once wonderfully phrased it, “teetering bulbs of dread and dream” — that is, only in association with certain kinds of gooey lumps encased in hard protective shells mounted atop mobile pedestals that roam the world on pairs of slightly fuzzy, jointed stilts?

GEB approaches these questions by slowly building up an analogy that likens inanimate molecules to meaningless symbols, and further likens selves (or “I”’s or “souls” if you prefer — whatever it is that distinguishes animate from inanimate matter) to certain special swirly, twisty, vortex-like, and meaningful patterns that arise only in particular types of systems of meaningless symbols. It is these strange, twisty patterns that the book spends so much time on, because they are little known, little appreciated, counterintuitive, and quite filled with mystery. And for reasons that should not be too difficult to fathom, I call such strange, loopy patterns “strange loops” throughout the book, although in later chapters, I also use the phrase “tangled hierarchies” to describe basically the same idea.

This is in many ways why M. C. Escher — or more precisely, his art — is prominent in the “golden braid”: because Escher, in his own special way, was just as fascinated as I am by strange loops, and in fact he drew them in a variety of contexts, and wonderfully disorienting and fascinating.

[…] GEB was inspired by my long-held conviction that the “strange loop” notion holds the key to unraveling the mystery that we conscious beings call “being” or “consciousness.”

(GEB: Twentieth-Anniversary Edition, Preface, pp. P1-P2)

via Stanford

 

Filed Under: Art, Math, Music, Philosophy, Science Tagged With: Artificial Intelligence, Doublas Hofstadter, EGB, GEB

DIY wax candles or concrete vases in trending geometric forms

August 23, 2014 by neogeo

Faceted geometric forms are trending nowadays. While the simple, colorful, paper shape DIYs that appear all over Pinterest are fun and easy to make, the results aren’t lasting. Homemade Modern created a series of free, downloadable templates that could be used as forms and/or molds to make more sustainable objects. Download and print Bloktagon series on cardstock paper from Homemade Modern to create forms for wax candles or concrete coat hooks and vases. 

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Filed Under: Art, Craft, Design Tagged With: concrete, design, Faceted geometric forms, Homemade Modern, sustainable objects, templates, wax

Technological Mandalas

August 19, 2014 by neogeo

Artist Leonardo Ulian offers another interpretation of  the mandala with his assemblages of electronic components, copper wire, and more. The intricate, finely detailed works radiate the innards of what makes technology tick. Ulian crafts smaller geometric patterns within a larger, more general shape that become more impressive once you see close up shots of his handiwork.

The mandala is typically a spiritual symbol that is often destroyed after its created (like the ones created from sand). This is a practice that establishes a sacred space, which is Ulian’s technological collage can be a metaphor for. Circuit boards, computer chips, and wire connectors have not only transformed the way we live, but the way in which we see the world. The artist could be saying that our dependency on it is akin to the worshiping of a larger being. (Via The Inspiration Provider)

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via beautiful/decay

Filed Under: Art, Technology Tagged With: assemblage, collage, Leonardo Ulian, mandala

SoundSelf – A Virtual Reality Experience

August 18, 2014 by neogeo

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Genuinely new experiences are rare; even new works of art, intentionally or otherwise, reference past artistic creations. But when I entered the latest version of SoundSelf, Robin Arnott’s multisensory Oculus Rift experience, which fuses sound, vision, and vibration into one sublime biofeedback loop, I had to discard everything I’d seen before to describe what I’d experienced. The closest comparison could be a combination of six hours spent in an isolation chamber, a visit to La Monte Young’s Dream House, and an eye-straining optical illusion— as far as personal experiences go, the term sui generis doesn’t even cut it.

As Arnott lowered the new Oculus Rift onto my head, fixed headphones onto my ears, attached two microphones around my voice box, then positioned my back against a flat, soft subwoofer, I didn’t know what to expect. This wasn’t helped by the facts that SoundSelf began with a black void, and that the ambient sounds of the room I was in that could barely be heard through the headphones.

Arnott told me to hum monotones and hold them until I wanted to try out some other notes. If I modulated my voice too much, he cautioned, the experience wouldn’t work as well. So, I hummed— and suddenly, a brand new, fifteen-minute cavalcade of kaleidoscopic, geometric shapes and colors burst and drifted into my field of vision, merging with my voice, and the vibrations coming out of the subwoofer.

Some people might say that SoundSelf is a “mind-melting experience.” This, in my opinion, isn’t quite accurate. The game, which Arnott describes as something like a trance or virtual meditation, doesn’t melt the mind so much as activate and elevate consciousness to a new type of reality. Melting implies mental overload or breakdown. SoundSelf, on the other hand, pulls the user into a beautiful and dynamic multi-sensory reality, one that creates a sense of calm peacefulness.

The word “psychedelic,” in Arnott’s opinion, is a horrible aesthetic crutch that’s overly used when talking about non-chemical trips. SoundSelf, comprised of original software written in C++ (one that builds the system, the other that visualizes sound), clearly isn’t meant to be a virtual substitute for mind-altering chemicals. Since the game is the result of arranged computer code, its new realities are fundamentally different. Arnott’s hope is that in a few years developers and users will create a new vocabulary for trance-inducing virtual reality experiences.

“With SoundSelf, it’s such a difficult thing to describe, and one of the only things I can try to leverage a little is the ‘star gate scene’ in 2001: A Space Odyssey,” said Arnott. “So, the way I try to pull that into the description is I describe it as ‘an odyssey of light and sound’ and try to make people make the connection.”

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Arnott, who created sound design for the mind-bending experimental games Antichamber and The Stanley Parable, originally became interested in gaming during his time at New York University’s film school, Tisch. In his last year at NYU, the school founded the NYU Game Center, which he describes as the current equivalent of the university’s film school in the 1980s. This game center helped hurdle Arnott down his current course in experimental gaming experiences.

Before working on the above games, Arnott created his own, Deep Sea, where the player, underwater and blinded with a gas mask, fights a sea monster that he or she can only hear. Arnott used two little microphones embedded in breath-in/breath-out ports to allow the computer to sense players’ breathing.The player aims and fires a weapon, hoping to hear the creature cry out in pain. More often, the only sound the player hears is “their shot disappearing uselessly into the void.”

Deep Sea is ultimately, as Arnott told me, about being “vulnerable.” Though it was very much a game (as well as an installation piece), its concept and design would have a big impact on the SoundSelf development process.

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Arnott also explained that his game design background allowed him to think about SoundSelf’s user interaction as “flow,” which he said game designers have down to a discipline. Flow can be thought of as a player interacting in the moment with the game system. Side effects of these systems’ interactivity, Arnott said, are trance states. So, what he is trying to do with SoundSelf is combine gaming’s interactivity discipline with what monks are trying to do with singing bowls or mandalas. A clash of two different technologies, as it were.

Originally, Arnott programmed SoundSelf’s visuals himself. Never too confident when it came to designing visuals or deploying color, it took about a year before Arnott felt he was proficient at programming dynamic shapes and movement. Eventually Arnott enlisted programmer Evan Balster to help broaden SoundSelf’s current visual palette through C++ programming, which had to visualize the game’s music as geometry in a generative way.

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When Arnott finally started offering demos at gaming conferences like E3’s Indiecade booth, the input he got was invaluable. It also proved that SoundSelf challenged gamers in ways they’d never been challenged before.

“There is a lot of interesting things to explore other than what it feels like to be powerful, such as what it feels like to be weak,” said Arnott. “With SoundSelf, when gamers come into it they realize that it’s an interactive system responding to their voice, and they’ll try to take control of it. They’ll try to explore it and understand what it does, playing it like an instrument or using it like a gun in a shooter game, and it doesn’t work like that.”

Arnott said that this type of gamer will then give up. But when they finally do let go, they find they’re able to fully fall into SoundSelf’s virtual reality.

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Maybe because I am not a rabid gamer, my entry into SoundSelf’s latest virtual reality experience was smooth. That I immediately let go might have also had something to do with the fact that I knew the game had close analogues with non-gaming, mind-bending experiences.

In SoundSelf’s design, Arnott attempted to reach the six qualities of a spiritual experience: a sense of unity, an intuitive sense of deep truth, sacredness, positive mood, transcendence of time and space, and ineffability. Arnott was interested in how these qualities can be experienced across cultures in a number of different ways, and wondered if it could be done in a virtual environment. While he said SoundSelf doesn’t achieve an intuitive sense of deep truth or sacredness, he believes the game hits the other qualities pretty well.

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Having experienced SoundSelf, I would agree. While the fusion of beautifully rendered, three-dimensional visuals, monotone singing, and the subwoofer’s vibrations was indescribable, it didn’t produce feelings of deep truth or sacredness. But this might have more to do with VR being in its infancy than any real faults with SoundSelf’s design or concept. In the future, it could be possible for Arnott and other virtual reality developers to reach those lofty goals.

What is encouraging is that there were moments in SoundSelf, especially when the visuals resembled soft or silky holographic shapes, where I felt a sense of unity. In these moments, SoundSelf’s synchronized input and output—the humming, the virtual shapes flying around, and the vibrations—combined to produce a feeling that I was in another space, but still simply existing in or seeing just another aspect of reality within the cosmos.

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Equally as important, the game induced a beautiful and seamless trance that made me forgot that I was in virtual reality. Though fleeting, the moments were powerful; something for Arnott and his team to build on in future SoundSelf iterations. The only other way to try and describe SoundSelf is by comparing it to a synaesthetic experience, where sight, sound, and other sensations blur into one.

“Synaesthesia is one of the dragons I’m chasing, which puts you off balance in the way you perceive things,” said Arnott. “With SoundSelf, the hope is that you experience a oneness of self while you’re also experiencing a oneness of sensation.”

While SoundSelf is not pure synaesthesia at this point, technological advances in devices that create virtual sensations—from sound to touch, though perhaps not smell—should help Arnott further refine his virtual odyssey. And when the Oculus Rift consumer model comes in the near future, then we can all trip the light fantastic in SoundSelf’s ever-evolving world.

For more info and to stay updated on SoundSelf, head over to the game’s homepage.

By DJ Pangburn

via The Creators Project

Filed Under: Animation, Art, Music, Psychology Tagged With: Gaming, Immersive, Oculus Rift, Robin Arnott, sound visualizer, Soundself, Virtual Reality

This Is What Math Equations Look Like in 3-D

July 26, 2014 by neogeo

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In this model of a mathematical surface, every aspect of every swoop, dip and pinch is encoded in a single equation. That equation has a singularity where the plaster would be drawn infinitely thinly. In a concession to physics, the final gap is bridged by a tiny wire.  UIUC ALTGELD COLLECTION

 

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Knowledge of curves in the plane can be bootstrapped to build surfaces by first making metal arcs then connecting them with string. This model was constructed by Arnold Emch, the only major American model maker. UIUC ALTGELD COLLECTION

 

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By changing the parameters in the equation defining the Clebsch cubic (discussed below), you can morph the shape of the surface. Here three throats have pinched off into singular points. This process is called degeneration. UIUC ALTGELD COLLECTION

 

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This surface is the graph of the real part of a complex function (i.e., the square root of -1 is involved). Each peak, called a pole, comes with two towers and two pits. UIUC ALTGELD COLLECTION

 

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After a century of tension, many of this model’s strings have frayed and snapped. The Smithsonian’s National Museum of American History has a large collection of deteriorating models. None of them are on display. UIUC ALTGELD COLLECTION

 

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A single equation cuts out the two separate pieces of this model. This is similar to the way that the equation x2 = 1 has two separate solutions, x=1 and x=-1. UIUC ALTGELD COLLECTION

 

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A glimmering curve is suspended at the center of this model, where two surfaces made of string intersect. UIUC ALTGELD COLLECTION

 

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This plaster model shows the potential energy of an electric dipole. The negative charge generates an infinite well and the positive charge an infinite peak. Far away the charges approximately cancel and the surface is almost flat. UIUC ALTGELD COLLECTION

 

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The bends and bulges in this bone-white ball are defined by a single equation of degree four in x, y, and z. The label, in German, hints at the size of the 1911 Schilling model catalog: This model is number 4 in series 9 of 40. UIUC ALTGELD COLLECTION

 

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This surface is described by the equation z = xy/(x2 + y4). A severe singularity forms at the origin where the equation yields z = 0/0. There a vertical sheet drops from the high parabolic ridge to the low parabolic valley. A perpendicular mid-height ridge runs through the sheet. UIUC ALTGELD COLLECTION

 

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These six bulges (two are in back) exhibit a dihedral symmetry and a confident attitude. The creases where they meet correspond to singularities in the defining equation. UIUC ALTGELD COLLECTION

 

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The spiral of this helicoid is self-similar: Each turn has identical curvature. To verify this, you can slide the brass band around the shaft. UIUC ALTGELD COLLECTION

 

The doors to MIT are always unlocked. If you slip in at night and take a long walk down the fluorescent hallway called the Infinite Corridor, you will pass flatscreen monitors displaying friendly robots, gleaming lab equipment behind large plate glass, and advertisements for the bitcoin club. Turn off the main drag into an alcove in the building numbered 2, and you’ll find something that seems out of place: a locked display case stuffed with strange forms made of plaster and string.*Were they not dulled by age and covered with dust, they might pass for products of a modern fab lab or the nearby school of design. But those mysterious surfaces were made more than a century ago by mathematicians to answer a simple question: What does an equation look like?

The philosopher Descartes realized, more than 400 years ago, that the shapes the ancient Greeks drew with ruler and compass could be described with algebraic equations written with x’s and y’s. The circle, that primal and mystical form, is just the set of points (x,y) in the (Cartesian!) plane satisfying x2 + y2 = 1. Shapes and equations became intertwined forever, and the appropriately named field of algebraic geometry was born.

As the centuries rolled by, mathematicians began to understand the possibilities of curves on paper, circles and parabolas and cubics, and by the end of the 19th century, German mathematicians were leading the way from the flat plane into three-dimensional space.

In 1893, a prominent mathematician named Felix Klein brought a boatload of models from his laboratory in Göttingen to the World’s Fair in Chicago. These perfect plasters stood out in the pavilion showcasing Germany’s technical achievements. The scientists who walked by took note. Soon major American universities had ordered hundreds of surface models from thick catalogs, and had them shipped thousands of miles over the Atlantic. Large collections remain at MIT, the University of Arizona, Harvard, and the University of Illinois at Urbana-Champaign, whose models feature prominently in this gallery.

The Clebsch diagonal surface is one of the most exacting and beautiful models. Governed by a highly symmetric cubic equation, the Clebsch was selected for a remarkable property: There are supposed to be exactly 27 straight lines that lie flush to its surface. Most surfaces contain no lines (like the sphere), and a few of them are made entirely of lines (like the string figures in the gallery). But this one bends through space in such a way that just 27 lines lie nestled in its curves. Clebsch had proved this mathematically, substituting one equation into another. Klein wanted to see lines for himself.

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The Clebsch diagonal surface, c. 1880, from the University of Göttingen.  Jonathan Chertok

To build the model, even to figure out what it should look like, workers in Klein’s laboratory painstakingly drew the horizontal sections solving a planar version of the equation. Each cross-section was cast separately, in a plaster made from powdered chalk, bone glue, double varnish, essence of lavender and essence of clove. Then the layers were carefully stacked, glued together, and sanded smooth. To check for correctness, the mathematicians traced the vertical cross-sections on cardboard, cut out the inside, and fit the remainder over the model. And lo! The surface has three swooping branches that join in a thin central column and again in a flaring base. The 27 lines, etched in black, dash through three gaping holes and extend toward infinity.

The models were part of a program to make algebra palpable. It’s one thing to check that the derivatives of a function are zero and another to feel the plaster taper to a sharp point. It’s one thing to prove that a surface contains a bunch of straight lines. It’s another to build the surface out of those lines, sweeping it out with taut strings. It’s one thing to know that a function blows up at a point like the electric potential produced by a perfect charge, and another to see negative infinity carved as a pit in plaster.

By making models you can hold in your hands, Klein hoped to keep mathematics anchored to the physical world. “Collections of mathematical models and courses in drawing are calculated to disarm, in part at least, the hostility directed against the excessive abstractness of the university instruction,” Klein said at the 1893 Evanston colloquium. An image or an object does more than ease fear of the unseen, it makes the equation real. It puts a face to the name, so you know who you’re talking about.

Mathematician George Francis, who made a detailed study of the collection at the University of Illinois, feels the same way today. “The physical activity of tracing a curve or of holding a surface fixes the mind. I still believe in the right and left brain, which I know is deprecated now but nevertheless. These engage the other half of the brain,” Francis told WIRED. It was the involvement of that other half of the brain, of the intuitive aspect, that proved the models’ downfall.

In the early 1900′s, there was a growing realization that arguments made from geometric intuition, from drawing pictures and making models, might not be airtight logically. A theorem by the German Max Dehn on the untangling of twisted disks provided a cautionary tale. His elaborately illustrated argument was found to have a crucial gap 20 years after publication, and took another 20 years to repair. A hyper-rational French school of mathematics emerged in response. Led by the collective pseudonym Nicholas Bourbaki, they hoped to unify all of mathematics in a maximally abstract and precise language. An advanced calculus textbook from the era, affectionately called Baby Rudin, featured zero pictures or illustrations. By the 1940′s, Klein’s beautiful models were consigned to university basements and dusty out-of-the-way shelves to protect students from the dangers of intuition.

On occasion, the models would catch the eye of an artist or architect who happened to walk through a math department. The painter Man Ray did a series of impressionistic portraits of the plaster models called Shakespearean Equations, including a tragic singular surface he named King Lear.

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The Clebsch diagonal surface, 2001. 3D printed by Jonathan Chertok.

In the late 1990′s, the curves of the Clebsch cubic entranced and perplexed designer Jonathan Chertok. He wanted to understand their “sublime rationality” and “elusive logic.” Computer-aided architecture was growing in popularity, led by Frank Gehry’s undulating facades, and there were a few software packages for simulating surfaces using meshes of tiny triangles. So Chertok set himself a challenge: to reproduce the classical models using modern rapid prototyping techniques.

The new methods were strikingly parallel to the old. Software computed the cross-sections numerically, then ZCorp hardware stacked thin layers of plaster powder and glue to build up a 3D shape. Now computer rendering software is fast enough and 3D printing cheap enough that a generation of designers is incorporating mathematical methods into their work. Whether making algorithmic jewelry or curvaceous skyscrapers, designers are able to generate and modify shapes by tuning the numbers in formulas. The sublime forms once secreted inside old equations are beginning to adorn our necklines and our skylines.

* A warning to intrepid travelers: Building 2 at MIT is closed for renovations and will open again in 2016.

via WIRED

Filed Under: Science Tagged With: 3-D, Math Equations

Bizarre Organic Quasicrystal Accidentally Created in Lab

March 6, 2014 by neogeo

quasicrystal

Quasicrystals have teased and intrigued scientists for three decades. Now, this already strange group of materials has a bizarre new member: a two-dimensional quasicrystal made from self-assembling organic molecules.

This odd quasicrystal is flat, made from a single layer of molecules with five-sided rings. The molecules form groups within the layer as weak hydrogen bonds link them together. These molecular groups are assembled in a way that forces other molecules in the layer into shapes including pentagons, stars, boats, and rhombi. If this were a regular old crystal, you’d expect to see these groups and shapes repeated over and over throughout the layer in a predictable way. But in this quasicrystal, you’ll see the same shapes over and over in the layer, but not in any organized pattern.

The things that set these quasicrystals apart from all the others, scientists say, are its organic materials and self-assembling parts.

“They’re markedly different from just about everything else out there,” said physical chemist Alex Kandel, whose lab at the University of Notre Dame described the material today in Nature. Previously known quasicrystals are mostly metallic, and tied together by strong ionic bonds rather than the weaker hydrogen bonds that can be found in complex organic molecules like DNA.

As their name suggests, quasicrystals have a structure that’s part crystalline, part disorganized. In other words, they are something in between a structure with repeating, symmetric units, and one with completely random building blocks. Their atomic units are locally symmetric, but are not regularly repeated over longer distances. Because of these arrangements, quasicrystals are slippery and have been used in things like non-stick frying pans.

The first quasicrystal of any sort was also accidentally made in the lab, in 1982, by materials scientist Daniel Schechtman who won a Nobel Prize for the discovery in 2011. Up until that point, scientists thought the semi-organized structure of quasicrystals was an impossibility. Now, we know that’s not true. Not only can quasicrystals be grown in the lab, they can also grow in nature. In 2012, Princeton University physicist Paul Steinhardt showed that quasicrystals found in eastern Russia had fallen to Earth in a meteorite.

Kandel’s group discovered the organic quasicrystal accidentally. Instead of trying to make the thing, they were actually hoping to study how electrons are distributed in ferrocenecarboxylic acid, the molecule the quasicrystal is built from. To do that, the team needed to build a stable, linear group of molecules. But when the scientists tried, they produced a two-dimensional quasicrystal instead.

“The first images were quite a shock,” Kandel said. “Certainly, 2-D quasicrystals aren’t easy to make, which is why we’re only seeing very recent reports of them now, some 30-odd years after the first quasicrystalline materials were discovered.”

Wolf Widdra of Germany’s Martin Luther University, who made the first 2-D quasicrystal, reported in October 2013, is a bit skeptical of the new research. He doesn’t think there’s enough evidence yet to prove quasicrystal structure over a large enough area.

There is also disagreement among scientists about what it means to be self-assembling. Widdra thinks the term could be applied to all quasicrystal structures, not just this new one. Kandel argues that structures assembled by way of strong chemical bonds — like the other quasicrystals — aren’t actually self-assembled. Those strong chemical bonds, he says, overwhelm the forces holding individual building blocks together and leave the material no choice but to form. In this new quasicrystal, those building blocks are joined by weak hydrogen bonds.

“Self-assembly is interesting precisely because the forces that drive organization are weaker than the forces responsible for the individual structure,” Kandel said.

By Nadia Drake

via WIRED

Filed Under: Nature, Science Tagged With: bizarre, geometry, nature, pattern, quasicrystal, science

Rocks Your Geometry Teacher Would Love

March 5, 2014 by neogeo

If you saw a perfectly shaped six-sided rock with straight, smooth edges, what would you think? It had to be man-made, carved for some specific purpose, right? Wrong! Your geometry teacher would be proud to know that the Earth actually forms its own hexagonal rocks. Known as Giant’s Causeway, this area in the northeast coast of Ireland has become a popular tourist destination.

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Most of these rock formations are in the form of hexagons (a six-sided image), but there are also some with as little as four and as many as eight sides. The tops of them are flat—perfect to use as stepping-stones. And you may just need to use them as such in order to see all the sights at Giant’s Causeway, since some of the pencil-like pillars reach up to 39 feet high.

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Where did it get the name Giant’s Causeway? Legend has it that an Irish giant by the name of Finn MacCool (Sounds to me like a friend of The Fonz!) built a causeway to cross the North Channel. The reason? To put the beat down on another giant in Scotland. At least he was picking on someone his own size, I suppose. Adding credence to the myth, identical rocks can also be found at Fingal’s Cave, which is across the sea on the Scottish Isle of Staffa.

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If for some reason the legend of Finn MacCool isn’t true, you likely wonder how these rocks were formed in such an amazing configuration. They are the result of lava, which was forced upwards from below the Earth’s crust long ago. As the molten rock met with the air, it cooled and shrank. These hexagonal shapes were then formed by the cracks. But why did all 40,000 of them form into interlocking geometric shapes? That could only happen because of the amazing chemical composition of basalt, of which they are all formed.

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via Our Amazing Earth

Filed Under: Nature Tagged With: geometric shapes, geometry, Giant’s Causeway, hexagons, Ireland, lava, nature, rocks

Beautiful Geometry in Architecture by Nick Frank

January 18, 2014 by neogeo

The 38-year-old autodidact, who has taught himself how to take pictures since 2010, manages to confront objects with an unusual view, giving them a wholly new perspective.

[Read more…]

Filed Under: Architecture, Art, Photography Tagged With: architecture, art, geometry, photography

3D Printing Mathematics

January 15, 2014 by neogeo

When learning about existing mathematics, and especially when trying to produce new mathematics, we spend a lot of time thinking about examples. How do parts of the example interact with each other? What are the regularities and symmetries? Does it come in a family of examples, or does it live on its own? In many cases, the first thing to do is to try and draw a picture. We are both geometric topologists, working mostly with two and three-dimensional objects. As such, two-dimensional pictures are important currency in our field. These pictures are typically drawn on blackboards, on pieces of paper, and even on tablecloths and napkins, as famously depicted in Douglas Adams’ discussion of Bistromathics.

A natural extension of drawing in two dimensions is drawing in three dimensions. In this direction, we have been using 3D printing as an aid to visualising mathematical objects. We design sculptures that help us and others to understand the mathematics better. Also, these sculptures are beautiful in their own right!

Here are a few favourite examples.

Half of a 120-cell

Half of a 120-cell depicts a projection of the 120-cell, one of the four-dimensional regular polytopes (from the Greek — “poly” for many, “topos” for place).

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Figure 1: Half of a 120-cell, with views showing the 2-, 3- and 5-fold symmetries.

The familiar pentagon is a two-dimensional polytope having five facets, all of which are edges. The dodecahedron is a three-dimensional polytope having 12 pentagonal facets. Finally the 120-cell is a four-dimensional polytope having 120 dodecahedral facets. In each case the facets are polytopes of one dimension lower.

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Figure 2: Polytopes of dimensions zero through four: point, interval, pentagon, dodecahedron, 120-cell.

To understand how to project this four-dimensional object into three-dimensional space, we need to develop some intuition from lower dimensions.

In dimension two the corners of a square sit on a circle which contains the whole shape. If we place a light at the center of the circle, the edges of the square cast shadows on the circle.

In dimension three, a cube sits inside a sphere. We arrange the cube so that one of its square facets is horizontal, with the North pole directly above the facet’s center. Now we place a light at the centre of the sphere. The edges of the cube cast shadows onto the sphere, making a “beach ball cube”. We delete the cube, and concentrate on the beach ball version. Move the light to the North pole. The edges of the beach ball cube cast their shadows on the horizontal plane: the plane the sphere is sitting on. This last step is called stereographic projection, from the sphere to the plane. See figure 3 below for pictures of this process.

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Figure 3: Projecting a cube onto the plane.

Finally, in dimension four, the 120-cell sits inside a three-sphere (the unit sphere in four-dimensional space). By casting shadows out from the center of the three-sphere we make a “beach ball 120-cell”. We use stereographic projection to get the beach ball 120-cell into our ordinary three-dimensional space. This gives the rightmost image in figure 2.

Note the massive complexity near the center. This cannot be printed using current technology — at least not within our budget! We came up with the following idea: we cut the projection of the 120-cell in half along a sphere, and we threw away the outside. The result is shown in figure 1. You can find out more in this movie:

The internal structure is now visible. In addition, the ratio between the diameter and the smallest features is much more reasonable, making the sculpture printable.

 

Quintessence

After playing with Half of a 120-cell for quite a while, we were inspired to design a family of interlocking puzzles which we call Quintessence; here chains of dodecahedra living in the Half of a 120-cell are combined to build various structures.

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Figure 4: Quintessence; copies of the six “rib” pieces shown in the top left can make all of these puzzles.

See how the puzzles work in this movie:

Triple helix

Triple Helix (figure 6) is a mechanism with three helical gears, meshing in pairs, all at right angles to each other. There is an amusing mistake in graphic design, where three (or any odd number) of planar gears are arranged in a circle. Perhaps the best known example is the reverse of the two-pound coin (see figure 5), designed by Bruce Rushin, which shows 19 gears that symbolise the Industrial Revolution. The mistake comes from the fact that neighbouring gears must rotate in opposite directions. Thus any circle with an odd number of planar gears is frozen! Triple helix is one solution to this paradox.

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Figure 5: The reverse of a two pound coin.

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Figure 6: Triple helix.

Watch the triple helix move:

Triple gear

Another, more complicated, solution titled Triple gear (figure 7) involves using three toothed-rings, all pairwise linked. Unlike our other work, these sculptures move. For much more on Triple gear and related mechanisms, check out our paper on the subject, which appeared in the Proceedings of the 2013 Bridges conference on mathematics, music, art, architecture and culture.

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Figure 7: Triple gear.

The triple gear in action:

Round Möbius strip

Round Möbius strip answers another puzzle — it is a Möbius strip with a circular boundary! Famously, the Möbius strip can be made by taking a long strip of paper, giving it a half-twist (twist by 180 degrees), and gluing the two short edges together (see this Plus article for more information). The resulting object has a single edge and one side. Unlike an ordinary piece of paper, you do not need to cross an edge to get from one side to the other side. The boundary of the paper Möbius strip describes a curve in space; this curve is not geometrically a circle, but it can be deformed into one. This movie illustrates the idea:

Our sculpture shows what happens when you drag the surface of the Möbius strip along as you “straighten out” the circle. We have also dragged one point of the strip to infinity to bring out the symmetries of the Möbius strip, but you can see the round circle at the centre — this is the boundary of the original Möbius strip.

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Figure 8: The “round” Möbius strip.

3D printing

3D printing is a term that covers a number of closely related technologies, also known as additive manufacturing. In all of these the idea is to build a physical object up, layer by layer, starting from nothing. This contrasts with more traditionalsubtractive manufacturing, such as lathing or carving. Complicated internal structures are difficult to make in subtractive manufacturing, but are easy in additive manufacturing. When subtracting, the object can get in the way of the carving tool. When adding, the print head always works from above, and at each moment the printed layers are below. Intricate details are now just a matter of persistence on the part of the designer; at the manufacturing stage it is as easy to print a block as it is to print a delicate filigree occupying the same volume.

The printing process is almost entirely automated, which means that the printed object very closely approximates the computer design. For us, this means that our prints are very close to the mathematical ideal. Many of our sculptures are entirely or almost entirely generated by (Python) code that directly expresses the desired geometry. Thus the mathematics described by our programs gets translated into physical objects with very few choices or possibilities of error.

However, there are limits to what 3D printers can do. There is a basic tension between the minimum feature size and the overall size of a sculpture. As we learned from the 120-cell, if some features are too small then the sculpture will have parts that are fragile or perhaps just unprintable. The easiest solution to this problem is to scale the design up — however if the resulting volume is too large the sculpture will be too expensive and perhaps again unprintable (if it does not fit inside the printer).

Getting into 3D printing is becoming easier and easier. These days many schools and universities, and even hobbyists have 3D printers. There are also many 3D printing services that let you upload a model, which they then 3D print and send to you. In addition one of us (Segerman) has given a workshop on 3D printing, using the programs Mathematica and Rhinoceros. The workshop materials are available here.

These include a preprint of the paper 3D printing for mathematical visualisation that appeared in the Mathematical Intelligencer.

Ideas for the future

We are currently thinking about objects that move, or that can be taken apart and played with. Such sculptures play to the strengths of physical objects as opposed to pictures or even computer animations. As an example of cool stuff that wants to be 3D printed we mention planar linkages — a fascinating piece of engineering along these lines is the Chaos machine of Robert MacKay. We are also thinking of sculptures based on hyperbolic rather than spherical geometry, and about the favourite knot of hyperbolic geometers, the figure-eight knot.

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About the authors

Saul Schleimer is a geometric topologist, working at the University of Warwick. His other interests include combinatorial group theory and computation. He is especially interested in the interplay between these fields and additionally in visualisation of ideas from these fields.

Henry Segerman is an assistant professor in the Department of Mathematics at Oklahoma State University. His mathematical research is in 3-dimensional geometry and topology. He also makes mathematical artwork, often about geometry and topology, but also involving procedural generation, self-reference, ambigrams and puzzles.

via Plus Magazine

Filed Under: Design, Science Tagged With: 3D printing, geometry, math, Möbius strip, polyhedra, science, triple gear

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