Salt and sound form mesmerising geometric shapes on a metal plate. Amazing power of frequency!
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
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).
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.
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.
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.
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.
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 (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.
Figure 5: The reverse of a two pound coin.
Figure 6: Triple helix.
Watch the triple helix move:
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.
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.
Figure 8: The “round” Möbius strip.
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.
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
Celebrating the perfect imperfection of circles by Thomas Brown and Lightning + Kinglyface
If you’ve ever whiled away time (in a coffee shop or solicitor’s office say) by trying to draw the perfect circle you’ll know how ruddy frustrating it can be (damn you last minute bump!). But hang on, it turns out there’s a theory that it’s an impossible task to even create the perfect circle in this world as some outside influence will always distort it however minutely.
Superb photographer Thomas Brown and brilliant set design duo Lightning + Kinglyface have teamed up to explore this idea, “creating the most perfect, imperfect circles, embracing improbability, chaos and letting the world around us disturb mathematical perfection to make beauty.”
The project, called 2-(y-b)2 = r2 (the equation of a circle) has been released gradually over a six-week period and as you’d expect from such talented creative minds the results have been breathtaking, using smoke, sugar, fabric and bubbles to make their point that the pursuit of perfection is a Sisyphean endeavour.
As the last circle is released today we are delighted to present the entire series in all its barnstorming glory.
By Rob Alderson
via its nice that
Physicists have discovered a jewel-like geometric object that dramatically simplifies calculations of particle interactions and challenges the notion that space and time are fundamental components of reality.
“This is completely new and very much simpler than anything that has been done before,” said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work.
The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.
“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at Harvard University and one of the researchers who developed the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.”
The new geometric version of quantum field theory could also facilitate the search for a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe. Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. The amplituhedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: locality and unitarity.
“Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the new work, which he is presenting in talks and in a forthcoming paper. “Both are suspect.”
Locality is the notion that particles can interact only from adjoining positions in space and time. And unitarity holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one. The concepts are the central pillars of quantum field theory in its original form, but in certain situations involving gravity, both break down, suggesting neither is a fundamental aspect of nature.
In keeping with this idea, the new geometric approach to particle interactions removes locality and unitarity from its starting assumptions. The amplituhedron is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.
“It’s a better formulation that makes you think about everything in a completely different way,” said David Skinner, a theoretical physicist at Cambridge University.
The amplituhedron itself does not describe gravity. But Arkani-Hamed and his collaborators think there might be a related geometric object that does. Its properties would make it clear why particles appear to exist, and why they appear to move in three dimensions of space and to change over time.
Because “we know that ultimately, we need to find a theory that doesn’t have” unitarity and locality, Bourjaily said, “it’s a starting point to ultimately describing a quantum theory of gravity.”
The amplituhedron looks like an intricate, multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated, “scattering amplitudes,” which represent the likelihood that a certain set of particles will turn into certain other particles upon colliding. These numbers are what particle physicists calculate and test to high precision at particle accelerators like the Large Hadron Collider in Switzerland.
The 60-year-old method for calculating scattering amplitudes — a major innovation at the time — was pioneered by the Nobel Prize-winning physicist Richard Feynman. He sketched line drawings of all the ways a scattering process could occur and then summed the likelihoods of the different drawings. The simplest Feynman diagrams look like trees: The particles involved in a collision come together like roots, and the particles that result shoot out like branches. More complicated diagrams have loops, where colliding particles turn into unobservable “virtual particles” that interact with each other before branching out as real final products. There are diagrams with one loop, two loops, three loops and so on — increasingly baroque iterations of the scattering process that contribute progressively less to its total amplitude. Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity — the requirement that probabilities sum to one.
“The number of Feynman diagrams is so explosively large that even computations of really simple processes weren’t done until the age of computers,” Bourjaily said. A seemingly simple event, such as two subatomic particles called gluons colliding to produce four less energetic gluons (which happens billions of times a second during collisions at the Large Hadron Collider), involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the scattering amplitude.
In 1986, it became apparent that Feynman’s apparatus was a Rube Goldberg machine.
To prepare for the construction of the Superconducting Super Collider in Texas (a project that was later canceled), theorists wanted to calculate the scattering amplitudes of known particle interactions to establish a background against which interesting or exotic signals would stand out. But even 2-gluon to 4-gluon processes were so complex, a group of physicists had written two years earlier, “that they may not be evaluated in the foreseeable future.”
Stephen Parke and Tommy Taylor, theorists at Fermi National Accelerator Laboratory in Illinois, took that statement as a challenge. Using a few mathematical tricks, they managed to simplify the 2-gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula, which a 1980s supercomputer could handle. Then, based on a pattern they observed in the scattering amplitudes of other gluon interactions, Parke and Taylor guessed a simple one-term expression for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of quantum field theory, involving hundreds of Feynman diagrams worth thousands of mathematical terms, was obfuscating something much simpler. As Bourjaily put it: “Why are you summing up millions of things when the answer is just one function?”
“We knew at the time that we had an important result,” Parke said. “We knew it instantly. But what to do with it?”
The message of Parke and Taylor’s single-term result took decades to interpret. “That one-term, beautiful little function was like a beacon for the next 30 years,” Bourjaily said. It “really started this revolution.”
In the mid-2000s, more patterns emerged in the scattering amplitudes of particle interactions, repeatedly hinting at an underlying, coherent mathematical structure behind quantum field theory. Most important was a set of formulas called the BCFW recursion relations, named for Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten. Instead of describing scattering processes in terms of familiar variables like position and time and depicting them in thousands of Feynman diagrams, the BCFW relations are best couched in terms of strange variables called “twistors,” and particle interactions can be captured in a handful of associated twistor diagrams. The relations gained rapid adoption as tools for computing scattering amplitudes relevant to experiments, such as collisions at the Large Hadron Collider. But their simplicity was mysterious.
“The terms in these BCFW relations were coming from a different world, and we wanted to understand what that world was,” Arkani-Hamed said. “That’s what drew me into the subject five years ago.”
With the help of leading mathematicians such as Pierre Deligne, Arkani-Hamed and his collaborators discovered that the recursion relations and associated twistor diagrams corresponded to a well-known geometric object. In fact, as detailed in a paper posted to arXiv.org in December by Arkani-Hamed, Bourjaily, Cachazo, Alexander Goncharov, Alexander Postnikov and Jaroslav Trnka, the twistor diagrams gave instructions for calculating the volume of pieces of this object, called the positive Grassmannian.
Named for Hermann Grassmann, a 19th-century German linguist and mathematician who studied its properties, “the positive Grassmannian is the slightly more grown-up cousin of the inside of a triangle,” Arkani-Hamed explained. Just as the inside of a triangle is a region in a two-dimensional space bounded by intersecting lines, the simplest case of the positive Grassmannian is a region in an N-dimensional space bounded by intersecting planes. (N is the number of particles involved in a scattering process.)
It was a geometric representation of real particle data, such as the likelihood that two colliding gluons will turn into four gluons. But something was still missing.
The physicists hoped that the amplitude of a scattering process would emerge purely and inevitably from geometry, but locality and unitarity were dictating which pieces of the positive Grassmannian to add together to get it. They wondered whether the amplitude was “the answer to some particular mathematical question,” said Trnka, a post-doctoral researcher at the California Institute of Technology. “And it is,” he said.
Arkani-Hamed and Trnka discovered that the scattering amplitude equals the volume of a brand-new mathematical object — the amplituhedron. The details of a particular scattering process dictate the dimensionality and facets of the corresponding amplituhedron. The pieces of the positive Grassmannian that were being calculated with twistor diagrams and then added together by hand were building blocks that fit together inside this jewel, just as triangles fit together to form a polygon.
Like the twistor diagrams, the Feynman diagrams are another way of computing the volume of the amplituhedron piece by piece, but they are much less efficient. “They are local and unitary in space-time, but they are not necessarily very convenient or well-adapted to the shape of this jewel itself,” Skinner said. “Using Feynman diagrams is like taking a Ming vase and smashing it on the floor.”
Arkani-Hamed and Trnka have been able to calculate the volume of the amplituhedron directly in some cases, without using twistor diagrams to compute the volumes of its pieces. They have also found a “master amplituhedron” with an infinite number of facets, analogous to a circle in 2-D, which has an infinite number of sides. Its volume represents, in theory, the total amplitude of all physical processes. Lower-dimensional amplituhedra, which correspond to interactions between finite numbers of particles, live on the faces of this master structure.
“They are very powerful calculational techniques, but they are also incredibly suggestive,” Skinner said. “They suggest that thinking in terms of space-time was not the right way of going about this.”
Quest for Quantum Gravity
The seemingly irreconcilable conflict between gravity and quantum field theory enters crisis mode in black holes. Black holes pack a huge amount of mass into an extremely small space, making gravity a major player at the quantum scale, where it can usually be ignored. Inevitably, either locality or unitarity is the source of the conflict.
“We have indications that both ideas have got to go,” Arkani-Hamed said. “They can’t be fundamental features of the next description,” such as a theory of quantum gravity.
String theory, a framework that treats particles as invisibly small, vibrating strings, is one candidate for a theory of quantum gravity that seems to hold up in black hole situations, but its relationship to reality is unproven — or at least confusing. Recently, a strange duality has been found between string theory and quantum field theory, indicating that the former (which includes gravity) is mathematically equivalent to the latter (which does not) when the two theories describe the same event as if it is taking place in different numbers of dimensions. No one knows quite what to make of this discovery. But the new amplituhedron research suggests space-time, and therefore dimensions, may be illusory anyway.
“We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics,” Arkani-Hamed said. “We have to learn new ways of talking about it. This work is a baby step in that direction.”
Even without unitarity and locality, the amplituhedron formulation of quantum field theory does not yet incorporate gravity. But researchers are working on it. They say scattering processes that include gravity particles may be possible to describe with the amplituhedron, or with a similar geometric object. “It might be closely related but slightly different and harder to find,” Skinner said.
Physicists must also prove that the new geometric formulation applies to the exact particles that are known to exist in the universe, rather than to the idealized quantum field theory they used to develop it, called maximally supersymmetric Yang-Mills theory. This model, which includes a “superpartner” particle for every known particle and treats space-time as flat, “just happens to be the simplest test case for these new tools,” Bourjaily said. “The way to generalize these new tools to [other] theories is understood.”
Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery of the amplituhedron could cause an even more profound shift, Arkani-Hamed said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of the universe arose out of pure geometry.
“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless.”
While more work is needed, many theoretical physicists are paying close attention to the new ideas.
The work is “very unexpected from several points of view,” said Witten, a theoretical physicist at the Institute for Advanced Study. “The field is still developing very fast, and it is difficult to guess what will happen or what the lessons will turn out to be.”
By: Natalie Wolchover
via QUANTA Magazine
Diagrams from Geometrical psychology, or, The science of representation: an abstract of the theories and diagrams of B. W. Betts (1887) by Louisa S. Cook, which details New Zealander Benjamin Bett’s remarkable attempts to mathematically model the evolution of human consciousness through geometric forms. From the Introduction:
The symbolic forms which Mr. Betts has evolved through his system of Representation resemble, when developed in two dimensions, conventionalised but very scientifically and beautifully conventionalised leaf-outlines. When in more than two dimensions they approximate to the forms of flowers and crystals…
The fact that he has accidentally portrayed plant-forms when he was studying human evolution is an assurance to Mr. Betts of the fitness of the symbols he has developed, as it affords presumptive evidence that the laws he is studying intuitively admit of universal application.