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Babylonians tracked Jupiter with sophisticated geometrical math

September 22, 2016 by Poly Hedra

Used geometry that hints at calculus 1,500 years before Europeans.

Trustees of the British Museum/Mathieu Ossendrijver

Trustees of the British Museum/Mathieu Ossendrijver

Even when a culture leaves behind extensive written records, it can be hard to understand their knowledge of technology and the natural world. Written records are often partial, and writers may have been unaware of some technology or simply considered it unremarkable. That’s why the ancient world can still offer up surprises like the Antikythera Mechanism, an ancient mechanical computer that highlighted the Greeks’ knowledge of math, astronomy, and the mechanical tech needed to tie them together.

It took several years after the discovery for the true nature of the Antikythera Mechanism to be understood. And now something similar has happened for the Babylonians. Clay tablets, sitting in the British Museum for decades, show that this culture was able to use sophisticated geometry to track the orbit of Jupiter, relying on methods that in some ways pre-figure the development of calculus centuries later.

We already knew that the Babylonians tracked the orbits of a variety of bodies. There are roughly 450 written tablets that describe the methods and calculations that we’re aware of, and they date from 400 to 50 BCE. Most of the ones that describe how to calculate orbital motion, in the words of Humboldt University’s Mathieu Ossendrijver, “can be represented as flow charts.” Depending on the situation, they describe a series of additions, subtractions, and multiplications that could tell you where a given body would be.
(Complicating matters, Babylonian astronomy worked in base-60, which leads to a very foreign-looking notation.)

The Babylonians did have a grasp of geometric concepts—Ossendrijver calls them “very common in the Babylonian mathematical corpus”—but none of them appeared in their known astronomical calculations.

In the British Museum, however, he located a tablet that hadn’t been formally described, and it contained parts of the procedure for tracking Jupiter. Combined with other tablets, it starts with Jupiter’s first morning rising, tracks it through its apparent retrograde motion, and finishes with its last visible setting at dusk. Again, it’s procedural. Different sections are used to predict the planet’s appearance at different segments of its orbit.

The trapezoid used to calculate the first 120 days of Jupiter's orbit. The red line divides the first shape into two equal areas. Picture by John Timmer

The trapezoid used to calculate the first 120 days of Jupiter’s orbit. The red line divides the first shape into two equal areas. Picture by John Timmer

Ossendrijver took the procedure for calculating the first 120 days and showed that calculating its daily displacement over time produces a trapezoid. In this case, the shape was largely a rectangle but with its top side angled downward over time in two distinct segments. A series of other tablets treated the calculations explicitly as producing a trapezoid.

Things get interesting in the next procedure, which is used to calculate when Jupiter reaches the midpoint in the first half of this stage of its motion. This procedure involved taking the left half of the trapezoid and dividing it into two pieces of equal area. The location of the dividing line (labelled vc above) then produces the answer. As Ossendrijver describes it, “They computed the time when Jupiter covers half this distance by partitioning the trapezoid into two smaller ones of ideally equal area.”

Figuring this out obviously required some sophisticated geometry. European scholars wouldn’t develop similar methods until the 14th century, when they became used at Oxford. The Greeks did use geometry for some astronomical work, but this involved calculations of actual space. The Babylonians here are working in an abstracted time-velocity space.

It’s also striking that this general approach is similar to some aspects of calculus. There, the area under a curve is calculated by mathematically creating an infinite number of small geometric figures and summing their areas. There’s no indication that the Babylonians were anywhere close to taking this intellectual leap given that they only divided this shape up a few times. But it does show that they recognised the value of the general approach.

By John Timmer via Ars Technica

Filed Under: History, Science Tagged With: Babylonians, British Museum, Clay tablets, Jupiter, Mathieu Ossendrijver, Orbit

A Sculptural Geometric Pop-Up Book

May 11, 2016 by Poly Hedra

Working somewhere between conceptual art and graphic design, California-born artist Tauba Auerbach creates compositions that exist somewhere in the state between two and three dimensions. For a colossal new book project, published in collaboration with New York-based independent bookstore Printed Matter and comprised of six die-cut paper sculptures that act as the book’s pages, director Sam Fleischner caught the aural experience of leafing through the weighty tome in the film premiered on Nowness.

You can see more of Auerbach’s designs on Instagram.

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via Eventhree, Nowness

 

Filed Under: Art, Design, Sculpture Tagged With: book, geometric, pop-up, sculpture, Tauba Auerbach

NASA Orbit Pavilion Lets Visitors Listen to the “Sounds of Space”

February 4, 2016 by Poly Hedra

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Based on the concept of listening to the sounds of the ocean inside a shell, STUDIOKCA, commissioned by NASA, has created the NASA Orbit Pavilion to immerse visitors in the sounds of satellites orbiting in outer space.

The traveling, nautilus-shaped pavilion provides a space in which to experience the trajectories of 19 satellites orbiting Earth. Made with 3,500 square feet of water-jet cut aluminum panels, the pavilion is “scribed with over 100 ‘orbital paths’ fitted together and bolted to a curved framework of aluminum tubes.”

 

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Speakers are arranged within the 30-foot diameter inner space, and programmed by artist and composer Shane Myrbeck to map, translate, and broadcast the sounds of the satellites in real time.

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“Surface perforations echo the orbital paths of the satellites, culminating around the oculus at the center of the sound chamber, while helping to mitigate exterior noise and decrease wind loads on the relatively light structure,” write the architects.

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The Pavilion made its debut at the World Science Festival in New York City this summer, and is slated to “wash ashore” at the Huntington Library Botanical Gardens in San Marino, California this spring.

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CREDITS:
Architects/Pavilion Design: Jason Klimoski and Lesley Chang, STUDIOKCA
Client: NASA
Sound Composition: Shane Myrbeck, ARUP
Creative Strategy: Dan Goods and David Delgado, NASA Jet Propulsion Laboratory
Structural Engineering: Ryan Miller, SILMAN

via ArchDaily, STUDIOKCA

 

Filed Under: Architecture, Design, Music, Science Tagged With: NASA, nautilus, Orbit, outer space, Pavilion, satellites, shell, sounds, STUDIOKCA

Gorgeous light installation

January 26, 2016 by Poly Hedra

SOFTlab, a design studio from New York City made a new installation for Melissa in Soho. It will be up for the several more months at 102 Greene St, New York, NY 10012. More great Photos by Alan Tansey after the jump.

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via SOFTlab

 

Filed Under: Architecture, Art, Design, Sculpture Tagged With: Installation, light, Melissa, sculpture, Softlab

3D Printed Mesmerizing Glowing Kinetic Sculpture

January 7, 2016 by Poly Hedra

Mesmerizing phosphorescent zoetrope designed by German designer Dieter Pilger in collaboration with Janno Ströcker and Frederik Scheve. This hypnotizing spherical zoetrope is printed with 3D printing and has been devised based on the mathematics of Fibonacci sequence.

Visual information is forwarded to the brain, where it is processed, interpreted and translated into sensory impressions. Generally speaking, visual perception is the product of filtering and reducing data, which enables us to depict our environment distinctly.
We envisioned a sculpture that displays an animation in the open physical space. The sphere is constructed according to the fibonacci sequence. It rotates in a certain speed and gets illuminated in a specific frequency.
The animation can be seen just by looking at it with your eyes. No external devices like a strobe or a camera are required. The fibonacci sequence thereby isn’t anything that only appeals to mathematicians, but is of great significance in the process of understanding aesthetics and harmony as a whole – as far as an impression can be expressed as visual perception. Thanks to John Edmark for the inspiration with his amazing project «Blooms».

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via Project Flux

Filed Under: Animation, Architecture, Art, Design, Sculpture Tagged With: 3D print, animation, fibonacci, kinetic sculpture, lamp, light, spherical, zoetrope

Brilliant Four-Story-Tall Star Lights Up Malaysian Skyline

December 17, 2015 by Poly Hedra

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A four-story star has landed in a run-down building in Butterworth, a city off mainland Penang, Malaysia. The massive LED construction, comprised of steel cables and over 500 meters of light, is artist Jun Hao Ong’s latest public installation. Constructed to raise awareness and illuminate the 2015 Urban Xchange art festival, the impressively large-scale sculpture pierces the building’s foundations, radiating outwards with a network of lights.

Appropriately titled The Star, this is just one of Ong’s many light sculptures. The Malaysian artist and architect enjoys experimenting with different light arrangements to couple “low-tech materials with hi-tech applications.” This piece is Ong’s submission to the festival, curated by the Hin Bus Depot, an art centre that aims to inspire the next generation of contemporary artists in Malaysia. The Xchange Festival itself is an event that works to bring together international and local artists and “encourage and challenge the artists and creative potential of the city’s urban environment.”

According to Ong, The Star is a “glitch in current political and cultural climate of the country, it is a manifestation of the sterile conditions of Butterworth, a once thriving industrial port and significant terminal between the mainland and island.” With its four-story tall radiance, the luminescent display certainly draws attention to the proceedings of the art festival in a big way.

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By Kristine Mitchell

via My Modern Met

Filed Under: Architecture, Art Tagged With: building, Jun Hao Ong, LED, light, Malaysia, sculpture, star

Man Ray – Human Equations

December 4, 2015 by Poly Hedra

There is a wonderful collection of Man Ray’s series Human Equations on Artsy, a fusion of two extremes: mathematical abstraction and human drama.

 

Julius Caesar, 1948 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Julius Caesar,
1948
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

As You Like It, 1948 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

As You Like It,
1948
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Hamlet 1949 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Hamlet
1949
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Mathematical Objects 1934-1935 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Mathematical Objects
1934-1935
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Mathematical Objects 1934-1935 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Mathematical Objects
1934-1935
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Real part of the function w=e1/z, 1900 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Real part of the function w=e1/z,
1900
“Man Ray – Human Equations” at Glyptoteket, Copenhagen


Surface generated by the normals of a rotational paraboloid 1900 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Surface generated by the normals of a rotational paraboloid
1900
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Surface of constant width 1911-1914 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Surface of constant width
1911-1914
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Torso 1936, "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Torso
1936,
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Twelfth Night, 1948 "Man Ray – Human Equations" at Glyptoteket, Copenhagen

Twelfth Night,
1948
“Man Ray – Human Equations” at Glyptoteket, Copenhagen

Mathematics – The Body – Shakespeare
In 1934, Man Ray was a frequent visitor at the Institut Henri Poincaré in Paris. His objective there was to photograph the Institute’s collection of three-dimensional mathematical models, which were used to illustrate the geometric properties of mathematical equations. The result was a series of iconic photographs which, by means of dramatic lighting and daring compositions, made the enigmatic mathematical models seem almost human.

In the 1940s he returned to this process, using his photographs as the basis for a series of 20 paintings. In some of these paintings he depicted the mathematical models on their own, in bright, vibrant colours; in others he would insert the models in complicated Surrealist tableaux. Augmented by titles from Shakespeare’s famous plays, these paintings added yet another ambitious layer to Man Ray’s artistic journey, which took him back and forth between two extremes: mathematical abstraction and human drama.

A drama in three acts
”Man Ray – Human Equations” is not arranged chronologically; instead, it follows a structure similar to that of three acts in a play. It shows the full artistic range and scope of Man Ray’s work, from his juvenilia to his late production, and three overarching themes focus on the material synergy between the works. 14 paintings from Shakespearean Equations series form the main axis of the show, providing a fundamental narrative about Man Ray’s fascination with universal enigmas. The paintings, photographs, and original models – on loan from Institut Henri Poincaré – offer a meeting between artistic practice and mathematical puzzles, human bodies, and drama. The three acts present the three themes as interconnected circuits: constantly overlapping, transforming, and returning to themselves. Only to enter into new circuits all over again.

Thought Passage and Mathematical Model Cabinet – be like Man Ray
The exhibition is directly linked to the Thought Passage. A space, where visitors can play chess or solve equations – with or without guidance from trained specialists.  You can also try the Mathematical Model Cabinet, a feature developed especially for this exhibition: a virtual, interactive equation transformer that allows visitors to change and vary the mathematical and formal parameters of geometric figures, playing around with the shapes and forms used by May Ray.

Text via Glyptoteket

Filed Under: Art, Math Tagged With: Artsy, Human Equations, Man Ray, mathematical models, photography, Shakespeare

Translating Classical Music Into Abstract Geometries

November 20, 2015 by Poly Hedra

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The project from visual artist and Creator Quayola is another audiovisual collaboration with Abstract Birds, who he teamed up with previously for Partitura 001, a real-time generative sound visualisation inspired by Kandinsky, Paul Klee, Oscar Fischinger, and Norman McLaren.

Using the same Partitura custom software (below), the work from Quayola and Abstract Birds is called Partitura-Ligeti and had been shown at the Nemo Festival in Paris 2012. The piece involves both a live performance—based on composer Ligeti’s sonata for viola solo and performed by Odile Auboin—and a triptych installation at MAC de Créteil.

Six Ligeti pieces were transformed into a visual score, translating the sounds into abstract shapes and geometric forms—and much like many of Quayola’s previous AV work, explores the concept of synesthesia in regards to sound and image.

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More about Quayola at his website and at Bitforms gallery

via Creators Project

Filed Under: Animation, Math, Music Tagged With: 3D, image, Ligeti, Quayola, sound

Spectacular Snowdon aviary in London Zoo

October 16, 2015 by Poly Hedra

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Also known as the Northern Aviary, this is the largest and most spectacular aviary in London Zoo and was built in 1962-65 as a result of Sir Hugh Casson’s 1958 redevelopment plan of the Zoo known as ‘The New Zoo’ as a replacement for the Great Aviary of 1888, the site of which is now occupied by the Michael Sobell Pavilions for Apes and Monkeys. It was Britain’s first walk-through aviary and was designed to allow the public close up views of birds in different number of habitats.

The aviary was pioneering on two fronts; it is a large tension structure and it is largely made of aluminium.

The aviary designs were by Antony Armstrong-Jones (Lord Snowdon) and Cedric Price, with some of the funding provided by Jack Cotton. The engineer was Frank Newby, of Felix Samuely and Partners, with landscape architects Margaret Maxwell and Peter Shepheard. The contractor was Leonard Fairclough Limited. It comprises an aluminium and steel frame with mesh cladding and concrete foundations, and measures 45 metres by 19 metres with a maximum height of 24 metres.

Public circulation within the interior was along the main axis following a zig zag path, the central section of which comprises a cantilevered reinforced concrete bridge. The interior has been landscaped to include waterfalls, ponds and pools. Birds ranging from birds of prey to waterfowl have been housed here.

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via Geograph

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Snowdon Aviary, London Zoo, Regent's Park

via Architecture.com

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The Photography of Eric de Maré via The Telegraph

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via Pinterest

Filed Under: Architecture, Engineering Tagged With: Architects, Aviary, Cedric Price, London Zoo, Lord Snowdon, Regent's Park

A Visual History of Human Knowledge

September 23, 2015 by Poly Hedra

How does knowledge grow? Sometimes it begins with one insight and grows into many branches. Infographics expert Manuel Lima explores the thousand-year history of mapping data — from languages to dynasties — using trees of information. It’s a fascinating history of visualizations, and a look into humanity’s urge to map what we know.

via TED

Filed Under: Design, Education, History, Psychology, Science Tagged With: data, History, Human knowledge, Infographics, information, Manuel Lima, TED, visualization

Attack on the pentagon results in discovery of new mathematical tile

September 1, 2015 by Poly Hedra

Joy as mathematicians discover a new type of pentagon that can cover the plane leaving no gaps and with no overlaps. It becomes only the 15th type of pentagon known that can do this, and the first discovered in 30 years

Five stars! The pentagon tiles are all identical. The colouring shows how they tile the plane when arranged in identical groups of three. Illustration: Casey Mann

Five stars! The pentagon tiles are all identical. The colouring shows how they tile the plane when arranged in identical groups of three. Illustration: Casey Mann

In the world of mathematical tiling, news doesn’t come bigger than this.

In the world of bathroom tiling – I bet they’re interested too.

If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to tile the plane.

Every triangle can tile the plane. Every four-sided shape can also tile the plane.

Things get interesting with pentagons. The regular pentagon cannot tile the plane. (A regular pentagon has equal side lengths and equal angles between sides, like, say, a cross section of okra, or, erm, the Pentagon). But some non-regular pentagons can.

The hunt to find and classify the pentagons that can tile the plane has been a century-long mathematical quest, begun by the German mathematician Karl Reinhardt, who in 1918 discovered five types of pentagon that do tile the plane.

(To clarify, he did not find five single pentagons. He discovered five classes of pentagon that can each be described by an equation. For the curious, the equations are here. And for further clarification, we are talking about convex pentagons, which are most people’s understanding of a pentagon in that every corner sticks out.)

Most people assumed Reinhardt had the complete list until half a century later in 1968 when R. B. Kershner found three more. Richard James brought the number of types of pentagonal tile up to nine in 1975.

That same year an unlikely mathematical pioneer entered the fray: Marjorie Rice, a San Diego housewife in her 50s, who had read about James’ discovery inScientific American. An amateur mathematician, Rice developed her own notation and method and over the next few years discovered another four types of pentagon that tile the plane. In 1985 Rolf Stein found a fourteenth. Way to go!

But then the hunt went cold. Until last month, when Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell announced last week that they had discovered this little beauty:

Illustration: Casey Mann

Illustration: Casey Mann

“We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey. “We were of course very excited and a bit surprised to find the new type of pentagon.

Pentagons remain the area of most mathematical interest when it comes to tilings since it is the only of the ‘-gons’ that is not yet totally understood.

As mentioned above, all triangles and quadrilaterals tile the plane. It was proved in 1963 that there are exactly three types of convex hexagon that tile the plane. And no convex heptagon, octagon, or anything else-gon tiles the plane. But full classification of the pentagons is still an open area of research.

“The problem of classifying convex pentagons that tile the plane is a beautiful mathematical problem that is simple enough to state so that children can understand it, yet the solution to the problem has eluded us for over 100 years,” said Casey. “The problem also has a rich history, connecting back to the 18th of David Hilbert’s famous 23 problems.”

Study of pentagonal tilings is interesting also because of its potential applications. “Many structures that we see in nature, from crystals to viruses, are comprised of building blocks that are forced by geometry and other dynamics to fit together to form the larger scale structure,” he added.

“I am too cautious to make predictions about whether or not more pentagon types will be found, but we have found no evidence preventing more from being found and are hopeful that we will see a few more. As we continue our computerized enumerations, we also hope to gather enough data to start making specific predictions that can be tested.”

For the time being, however, the choice of pentagonal tile types for your bathroom wall are these:

The 15 types of pentagonal tilings discovered so far. Photograph: Ed Pegg/Wikipedia

The 15 types of pentagonal tilings discovered so far. Photograph: Ed Pegg/Wikipedia

by Alex Bellos

via The Guardian

Filed Under: Architecture, Design, Math Tagged With: discovery, math, pentagon, plane, tiles

Everything about the way we teach math is wrong

August 26, 2015 by Poly Hedra

A 3-D representation of a 4-D shape called a 24-cell polychoron.

A 3-D representation of a 4-D shape called a 24-cell polychoron.

 

Mathematics is one of humanity’s most creative and poetic endeavors.

And it is a disaster that it isn’t taught this way to students.

“A Mathematician’s Lament” is a classic polemic (later expanded and published as a book) written by math teacher Paul Lockhart. The essay is a devastating and passionate assault on the mechanistic way mathematics is taught in most of our schools.

A Student’s Nightmare

Lockhart begins with a vivid parable in which a musician has a nightmare in which music is taught to children by rote memorization of sheet music and formal rules for manipulating notes. In the nightmare, students never actually listen to music, at least not until advanced college classes or graduate school.

The problem is that this abstract memorization and formal-method-based “music” education closely resembles the “math” education that most students receive. Formulas and algorithms are delivered with no context or motivation, with students made to simply memorize and apply them.

Part of why many students end up disliking math, or convincing themselves that they are bad at math, comes from this emphasis on formulas and notation and methods at the expense of actually deep understanding of the naturally fascinating things mathematicians explore. It’s understandable that many students (and adults) get frustrated at memorizing context-free strings of symbols and methods to manipulate them.

This goes against what math is really about. The essence of mathematics is recognizing interesting patterns in interesting abstractions of reality and finding properties of those patterns and abstractions. This is inherently a much more creative field than the dry symbol manipulation taught conventionally.

Playing With Triangles

Lockhart uses a geometry problem to illustrate. He draws a triangle inside a rectangle:

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How much of the rectangle does the triangle take up? Lockhart notes that mathematicians are interested in shapes in the abstract:

I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is — wondering, playing, amusing yourself with your imagination.

We’ve come up with this imaginary triangle, and now we want to better understand it. The way to do this is to try different things and see what they tell us about the triangle.

Lockhart presents one possibility that turns out to be useful in answering the question: drawing a vertical line from the top of the triangle to the base:

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This answers our question:

“If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box!”

Drawing a triangle and playing around with it and eventually realizing something about the relationship between the triangle and the rectangle is much closer to the spirit of mathematics than simply being told a formula.

In this example, we’ve actually figured out and proved the triangle area formula written in the front cover of any middle or high school geometry textbook: Area = (1/2) × (length of base) × (height). The length of the base times the height gives us the area of the rectangle, and we just observed that the area of the triangle is half of that.

Challenging students to think about shapes, numbers, symmetry, or motion are more fun than the standard practice of memorizing techniques and applying them over and over again. Allowing students to explore these concepts and figure things out for themselves also builds up the critical thinking and reasoning skills that we supposedly want our children to learn rather more effectively than applying a handful of memorized, unmotivated, and unexplained formulas dozens of times.

The Soul of Mathematics

At the heart of mathematics is a need to understand structures, real or imagined. This is a profoundly speculative and creative exercise: a strange type of higher dimensional shape might hint that it has some interesting properties; a data set describing Ebola infection rates could roughly fit the same pattern as uranium atoms undergoing atomic decay. The job of the mathematician is to find and, far more importantly, explain these kinds of properties and relationships.

While it is important for students to work through a few basic problems at every level of mathematics they encounter, we live in an era when, once an understanding of the underlying concepts is mastered, one can turn to calculators or computer programs to do the mindless symbolic manipulations needed to get an answer. Pedagogy needs to move away from finding the answer, and toward understanding why this is the answer and why we care about the answer.

Mathematics is unique among human endeavors because it combines our most “right brained” creative, abstract, imaginative instincts with our most “left brained” logical, evidence-based, focused instincts. Math is about making a poetry out of pure reason and about abstractions based on seeing patterns in our world, and it is very sad that so few people ever get to experience this.

Lockhart’s entire essay is a beautifully and passionately written plea for a better way of educating students to truly understand the wonderful world of mathematics. Anyone who has any interest in math and math education should read the whole thing.

By Andy Kiersz

Via Business Insider

Filed Under: Math Tagged With: Essay, Mathematician's Lament, Paul Lockhart, Teaching

After 400 years, mathematicians find a new class of solid shapes

December 16, 2014 by Poly Hedra

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The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature.

Since Plato’s work, two other classes of equilateral convex polyhedra, as the collective of these shapes are called, have been found: Archimedean solids (including truncated icosahedron) and Kepler solids (including rhombic polyhedra). Nearly 400 years after the last class was described, researchers claim that they may have now invented a new, fourth class, which they call Goldberg polyhedra. Also, they believe that their rules show that an infinite number of such classes could exist.

Platonic love for geometry

Equilateral convex polyhedra need to have certain characteristics. First, each of the sides of the polyhedra needs to be of the same length. Second, the shape must be completely solid: that is, it must have a well-defined inside and outside that is separated by the shape itself. Third, any point on a line that connects two points in a shape must never fall outside the shape.

Platonic solids, the first class of such shapes, are well known. They consist of five different shapes: tetrahedron, cube, octahedron, dodecahedron and icosahedron. They have four, six, eight, twelve and twenty faces, respectively.

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Platonic solids in ascending order of number of faces. nasablueshift

These highly regular structures are commonly found in nature. For instance, the carbon atoms in a diamond are arranged in a tetrahedral shape. Common salt and fool’s gold (iron sulfide) form cubic crystals, and calcium fluoride forms octahedral crystals.

The new discovery comes from researchers who were inspired by finding such interesting polyhedra in their own work that involved the human eye. Stan Schein at the University of California in Los Angeles was studying the retina of the eye when he became interested in the structure of protein called clathrin. Clathrin is involved in moving resources inside and outside cells, and in that process it forms only a handful number of shapes. These shapes intrigued Schein, who ended up coming up with a mathematical explanation for the phenomenon.

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Goldberg polyhedron.

During this work, Schein came across the work of 20th century mathematician Michael Goldberg who described a set of new shapes, which have been named after him, as Goldberg polyhedra. The easiest Goldberg polyhedron to imagine looks like a blown-up football, as the shape is made of many pentagons and hexagons connected to each other in a symmetrical manner (see image to the left).

However, Schein believes that Goldberg’s shapes – or cages, as geometers call them – are not polyhedra. “It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces,” Schein said.

Instead, in a new paper in the Proceedings of the National Academy of Sciences, Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others.

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Blown up dodecahedron. stblaize

A crude way to describe Schein and Gayed’s work, according to David Craven at the University of Birmingham, “is to take a cube and blow it up like a balloon” – which would make its faces bulge (see image to the right). The point at which the new shapes breaks the third rule – which is, any point on a line that connects two points in that shape falls outside the shape – is what Schein and Gayed care about most.

Craven said, “There are two problems: the bulging of the faces, whether it creates a shape like a saddle, and how you turn those bulging faces into multi-faceted shapes. The first is relatively easy to solve. The second is the main problem. Here one can draw hexagons on the side of the bulge, but these hexagons won’t be flat. The question is whether you can push and pull all these hexagons around to make each and everyone of them flat.”

During the imagined bulging process, even one that involves replacing the bulge with multiple hexagons, as Craven points out, there will be formation of internal angles. These angles formed between lines of the same faces – referred to as dihedral angle discrepancies – means that, according to Schein and Gayed, the shape is no longer a polyhedron. Instead they claimed to have found a way of making those angles zero, which makes all the faces flat, and what is left is a true convex polyhedron (see image below).

Their rules, they claim, can be applied to develop other classes of convex polyhedra. These shapes will be with more and more faces, and in that sense there should be an infinite variety of them.

Playing with shapes

Such mathematical discoveries don’t have immediate applications, but often many are found. For example, dome-shaped buildings are never circular in shape. Instead they are built like half-cut Goldberg polyhedra, consisting of many regular shapes that give more strength to the structure than using round-shaped construction material.

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Only the one in the right bottom corner is a convex polyhedra. Stan Schein/PNAS

However, there may be some immediate applications. The new rules create polyhedra that have structures similar to viruses or fullerenes, a carbon allotrope. The fact that there has been no “cure” against influenza, or common flu, shows that stopping viruses is hard. But if we are able to describe the structure of a virus accurately, we get a step closer to finding a way of fighting them.

If nothing else, Schein’s work will invoke mathematicians to find other interesting geometric shapes, now that equilateral convex polyhedra may have been done with.

via The Conversation

Filed Under: Math, Science Tagged With: geometric forms, geometry, Goldberg polyhedra, new class, Plato, solid shapes

Mesmerizing Gifs

October 9, 2014 by Poly Hedra

When Ireland-based PhD student and digital artist Dave Whyte isn’t studying the physics of foam, he’s creating transfixing GIFs based on mathematical data. Whyte’s animated dots, lines, cubes, and spheres move and shift in endlessly mesmerizing geometrical patterns that we could watch all day.

Whyte, who creates the gifs using an open-source coding language called Processing, updates his blog, Bees & Bombs, constantly, so be sure to check back there for more amazing gifs almost daily. He also takes requests!

Check out some of our favorites gifs that Whyte has made below. While his posts are minimal on descriptions, the Gifs are mesmerizing:

 

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By Christian Storm via Business Insider

Filed Under: Animation, Art, Math Tagged With: animation, Dave Whyte, Gifs, patterns, Processing

Fascinating Resonance Experiment

July 28, 2014 by Poly Hedra

frequency on metal plate01

Salt and sound form mesmerising geometric shapes on a metal plate. Amazing power of frequency!

Filed Under: Music, Nature, Science Tagged With: experiment, frequency, metal plate, music, resonance, salt, science, sound, vibrations

Sparkling Table

June 13, 2014 by Poly Hedra

Beautiful and creative table made out of glass by talented artist John Foster.

When the light shines through the “Sparkle Palace Cocktail Table”, various shapes of different colors get projected onto the walls, floor, and ceiling.
Sparkling table will transform any room into unique art installation.

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via toxel

Filed Under: Architecture, Art, Craft Tagged With: glass, interior design, light, table

Four Plates from the Book “Vielecke und Vielflache”, (1900)

May 26, 2014 by Poly Hedra

Prof. Dr, Max Bruckner, Four Plates from the Book “Vielecke und Vielflache”, (1900)

Regular convex polyhedra, frequently referenced as “Platonic” solids, are featured prominently in the philosophy of Plato, who spoke about them, rather intuitively, in association to the four classical elements (earth, wind, fire, water… plus ether). However, it was Euclid who actually provided a mathematical description of each solid and found the ratio of the diameter of the circumscribed sphere to the length of the edge and argued that there are no further convex polyhedra than those 5: tetrahedron, hexahedron (also known as the cube), octahedron, dodecahedron and icosahedron.

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Via RUDY/GODINEZ

Filed Under: Art, Philosophy, Science Tagged With: “Platonic” solids, euclid, Max Bruckner, Plato, polyhedra

Uncoiling the spiral: Maths and hallucinations

May 7, 2014 by Poly Hedra

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Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. But it’s not just hallucinogenic drugs like LSD, cannabis or mescaline that conjure up these geometric structures. People have reported seeing them in near-death experiences, as a result of disorders like epilepsy and schizophrenia, following sensory deprivation, or even just after applying pressure to the eyeballs. So common are these geometric hallucinations, that in the last century scientists began asking themselves if they couldn’t tell us something fundamental about how our brains are wired up. And it seems that they can.

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Computer generated representations of form constants. The top two images represent a funnel and a spiral as seen after taking LSD, the bottom left image is a honeycomb generated by marijuana, and the bottom right image is a cobweb. Image from [1], used by permission.

Geometric hallucinations were first studied systematically in the 1920s by the German-American psychologist Heinrich Klüver. Klüver’s interest in visual perception had led him to experiment with peyote, that cactus made famous by Carlos Castaneda, whose psychoactive ingredient mescaline played an important role in the shamanistic rituals of many central American tribes. Mescaline was well-known for inducing striking visual hallucinations. Popping peyote buttons with his assistant in the laboratory, Klüver noticed the repeating geometric shapes in mescaline-induced hallucinations and classified them into four types, which he called form constants: tunnels and funnels, spirals, lattices including honeycombs and triangles, and cobwebs.

In the 1970s the mathematicians Jack D. Cowan and G. Bard Ermentrout used Klüver’s classification to build a theory describing what is going on in our brain when it tricks us into believing that we are seeing geometric patterns. Their theory has since been elaborated by other scientists, including Paul Bressloff, Professor of Mathematical and Computational Neuroscience at the newly established Oxford Centre for Collaborative Applied Mathematics.

How the cortex got its stripes…

In humans and mammals the first area of the visual cortex to process visual information is known as V1. Experimental evidence, for example from fMRI scans, suggests that Klüver’s patterns, too, originate largely in V1, rather than later on in the visual system. Like the rest of the brain, V1 has a complex, crinkly, folded-up structure, but there’s a surprisingly straight-forward way of translating what we see in our visual field to neural activity in V1. “If you imagine unfolding [V1],” says Bressloff, “You can think of it as neural tissue a few millimetres thick with various layers of neurons. To a first approximation, the neurons through the depth of the cortex behave in a similar way, so if you compress those neurons together, you can think of V1 as a two-dimensional sheet.”

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The visual cortex: the area V1 is shown in red. Image:Washington irving 

An object or scene in the visual world is projected as a two-dimensional image on the retina of each eye, so what we see can also be treated as flat sheet: the visual field. Every point on this sheet can be pin-pointed by two coordinates, just like a point on a map, or a point on the flat model of V1. The alternating regions of light and dark that make up a geometric hallucination are caused by alternating regions of high and low neural activity in V1 — regions where the neurons are firing very rapidly and regions where they are not firing rapidly.

To translate visual patterns to neural activity, what is needed is acoordinate map, a rule which links each point in the visual field to a point on the flat model of V1. In the 1970s scientists including Cowan came up with just such a map, based on anatomical knowledge of how neurons in the retina communicate with neurons in V1 (see the box on the right for more detail). For each light or dark region in the visual field, the map identifies a region of high or low neural activity in V1.

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So how does this retino-cortical map transform Klüver’s geometric patterns? It turns out that hallucinations comprising spirals, circles, and rays that emanate from the centre correspond to stripes of neural activity in V1 that are inclined at given angles. Lattices like honeycombs or chequer-boards correspond to hexagonal activity patterns in V1. This in itself might not have appeared particularly exciting, but there was a precedent: stripes and hexagons are exactly what scientists had seen when modelling other instances of pattern formation, for example convection in fluids, or, more strikingly, the emergence of spots and stripes in animal coats. The mathematics that drives this pattern formation was well known, and it now suggested a mechanism for modelling the workings of the visual cortex too.

…and how the leopard got its spots

The first model of pattern formation in animal coats goes back to Alan Turing, better known as the father of modern computer science and Bletchley Park code breaker. Turing was interested in how a spatially homogeneous system, such as a uniform ball of cells making up an animal embryo, can generate a spatially inhomogeneous but static pattern, such as the stripes of a zebra.

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Turing hypothesised that these animal patterns are a result of a reaction-diffusion process. Imagine an animal embryo which has two chemicals living in its skin. One of the two chemicals is aninhibitor, which suppresses the production of both itself and the other chemical. The other, an activator, promotes the production of both.

Initially, at time zero in Turing’s model, the two chemicals exactly balance each other — they are in equilibrium, and their concentrations at the various points on the embryo do not change over time. But now imagine that, for some reason or other, the concentration of activator increases slightly at one point. This small perturbation sets the system into action. The higher local concentration of activator means that more activator and inhibitor are produced there — this is a reaction. But both chemicals also diffuse through the embryo skin, inhibiting or activating production elsewhere.

For example, if the inhibitor diffuses faster than the activator, then it quickly spreads around the point of perturbation and decreases the concentration of activator there. So you end up with a region of high activator concentration bordered by high inhibitor concentration — in other words, you have a spot of activator on a background of inhibitor. Depending on the rates at which the two chemicals diffuse, it is possible that such a spotty pattern arises all over the skin of the embryo, and eventually stabilises. If the activator also promotes the generation of a pigment in the skin of the animal, then this pattern can be made visible. (See the Plus article How the leopard got its spots for more detail.)

Turing wrote down a set of differential equations which describe the competition between the two chemicals (see the box on the right), and which you can let evolve over time, to see if any patterns emerge. The equations depend on parameters capturing the rate at which the two chemicals diffuse: if you choose them correctly, the system will eventually stabilise on a particular pattern, and you can vary the pattern by varying the parameters. Here is an applet (kindly provided by Chris Jennings) which allows you to play with different parameters and see the patterns form.

Patterns in the brain

Neural activity in the brain isn’t a reaction-diffusion process, but there are analogies to Turing’s model. “Neurons send signals to each other via their output lines called axons,” says Bressloff. Neurons respond to each other’s signals, so we have a reaction. “[The signals] propagate so quickly relative to the process of pattern formation, that you can think of them as instantaneous interactions.” So rather than diffusion, which is a local process, we have instantaneous interaction at a distance in this case. The roles of activator and inhibitor are played by two different classes of neurons. “There are neurons which are excitatory — they make other neurons more likely to become active — and there are inhibitory neurons, which make other neurons less likely to become active,” says Bressloff. “The competition between the two classes of neurons is the analogue of the activator-inhibitor mechanism in Turing’s model.”

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Stripy, hexagonal and square patterns of neural activity in V1 generated by a mathematical model. Image from [1], used by permission.

Inspired by the analogies to Turing’s process, Cowan and Ermentrout constructed a model of neural activity in V1, using a set of equations that had been formulated by Cowan and Hugh Wilson. Although the equations are more complicated than Turing’s, you can still play the same game, letting the system evolve over time and see if patterns in neural activity evolve. “You find that, under certain circumstances, if you turn up a parameter which represents, for example, the effect of a drug on the cortex, then this leads to a growth of periodic patterns,” says Bressloff.

Cowan and Ermentrout’s model suggests that geometric hallucinations are a result of an instability in V1: something, for example the presence of a drug, throws the neural network off its equilibrium, kicking into action a snowballing interaction between excitatory and inhibitory neurons, which then stabilises in a stripy or hexagonal pattern of neural activity in V1. In the visual field we then “see” this pattern in the shape of the geometric structures described by Klüver.

Symmetries in the brain

In reality, things aren’t quite as simple as in Cowan and Ermentrout’s model, because neurons don’t only respond to light and dark images. Through the thickness of V1, neurons are arranged in collections of columns, known as hypercolumns, with each hypercolumn roughly responding to a small region of the visual field. But the neurons in a hypercolumn aren’t all the same: apart from detecting light and dark regions, each neuron specialises in detecting local edges — the separation lines between light and dark regions in a part of an image — of a particular orientation. Some detect horizontal edges, others detect vertical edges, others edges that are inclined at a 45° angle, and so on. Each hypercolumn contains columns of neurons of all orientation preferences, so that a hypercolumn can respond to edges of all orientations from a particular region of the visual field. It is the lay-out of hypercolumns and orientation preferences that enables us to detect contours, surfaces and textures in the visual world.

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Connections in V1: Neurons interact with most other neurons within a hypercolumn. But they only interact with neurons in other hypercolumns, if the columns lie in the direction of their orientation, and if the neurons have the same preference. Image from [2], used by permission.

Over recent years, much anatomical evidence has accumulated showing just how neurons with various orientation preferences interact. Within their own hypercolumn, neurons tend to interact with most other neurons, regardless of their orientation preference. But when it comes to neurons in other hypercolumns they are more selective, only interacting with those of similar orientations and in a way which ensures that we can detect continuous contours in the visual world.

Bressloff, in collaboration with Cowan, the mathematician Martin Golubitsky and others, has generalised Cowan and Ermentrout’s original model to take account of this new anatomical evidence. They again used the plane as the basis for a model of V1: each hypercolumn is represented by a point $(x,y)$ on the plane, while each point$(x,y)$ in turn corresponds to a hypercolumn. Neurons with a given orientation preference $\theta $ (where $\theta $ is an angle between 0 and $\pi $) are represented by the location $(x,y)$ of the hypercolumn they’re in, together with the angle $\theta $, that is, they are represented by three bits of information, $(x,y,\theta )$. So in this model V1 is not just a plane, but a plane together with a full set of orientations for each point.

In keeping with anatomical evidence, Bressloff and his colleagues assumed that a neuron represented by $(x_0,y_0,\theta _0)$ interacts with all other neurons in the same hypercolumn $(x_0,y_0).$ But it only interacts with neurons in other hypercolumns, if these hypercolumns lie in their preferred direction $\theta _0$: on the plane, draw a line through $(x_0,y_0)$ of inclination $\theta _0.$ Then the neurons represented by $(x_0,y_0,\theta _0)$ interact only with neurons in hypercolumns that lie on this line, and which have the same preferred orientation $\theta _0$.

This interaction pattern is highly symmetric. For example, the pattern doesn’t appear any different if you shift the plane along in a given direction by a given distance: if two elements $(x_0,y_0,\theta _0)$ and $(s_0,t_0,\phi _0)$ interact, then the elements you get to by shifting along, that is $(x_0+a,y_0+b,\theta _0)$ and $(s_0+a,t_0+b,\phi _0)$ for some $a$ and $b$, interact in the same way. In a similar way, the pattern is also invariant under rotations and reflections of the plane.

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If two elements (x,y,θ) and (s,t,θ) interact, then so do the elements of the same orientation at (x+a,y+b) and (s+a,t+b), and the elements of orientation -θ at (x,-y) and (s,-t).

Bressloff and his colleagues used a generalised version of the equations from the original model to let the system evolve. The result was a model that is not only more accurate in terms of the anatomy of V1, but can also generate geometric patterns in the visual field that the original model was unable to produce. These include lattice tunnels, honeycombs and cobwebs that are better characterised in terms of the orientation of contours within them, than in terms of contrasting regions of light and dark.

What’s more, the model is sensitive to the symmetries in the interaction patterns between neurons: the mathematics shows that it is these symmetries that drive the formation of periodic patterns of neural activity. So the model suggests that it is the lay-out of hypercolumns and orientation preferences, in other words the mechanisms that enable us to detect edges, contours, surfaces and textures in the visual world, that generate the hallucinations. It is when these mechanism become unstable, for example due to the influence of a drug, that patterns of neural activity arise, which in turn translate to the visual hallucinations.

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A lattice tunnel hallucination generated by the mathematical model. It strongly resembles hallucinations seen after taking marijuana. Image from [1], used by permission.

Beyond hallucinations

Bressloff’s model does not only provide insight into the mechanisms that drive visual hallucinations, but also gives clues about brain architecture in a wider sense. In collaboration with his wife, an experimental neuroscientist, Bressloff has looked at the connection circuits between hypercolumns in normal vision, to see how visual images are processed. “People used to think that neurons in V1 just detect local edges, and that you have to go to higher levels in the brain to put these edges together to detect more complicated features like contours and surfaces. But the work I have done with my wife shows that these structures in V1 actually allow the earlier visual cortex to detect contours and do more global processing. It used to be thought that you process more and more complex aspects of an image as you go higher up in the brain. But now it’s realised that there is a huge amount of feedback between higher and lower cortical areas. It’s not a simple hierarchical process, but an incredibly complicated and active system it will take many years to understand.”

Practical applications of this work include computer vision — computer scientists are already building the inter-connectivity structures that Bressloff and his colleagues played around with into their models, with the aim of teaching computers to detect contours and textures. On a more speculative note, Bressloff’s research may also one day help to restore vision to visually impaired people. “The question here is if you can somehow stimulate part of the visual cortex, [bypassing the eye], and use that to guide a blind person,” says Bressloff. “If one can understand how the cortex is wired up and responds to stimulation, perhaps one would then have a better way of stimulating it in the right way.”

There are even applications that have nothing at all to do with the brain. Bressloff applied the insights from this work to other situations in which objects are located in space and also have an orientation, for example fibroblast cells found in human and animal tissue. He showed that under certain circumstances these interacting cells and molecules can line up and form patterns analogous to those that arise in V1.

People have reported seeing visual hallucinations since the dawn of time and in almost all human cultures — hallucinatory images have even been found in petroglyphs and cave paintings. In shamanistic traditions around the world they have been regarded as messages from the spirit world. Few neuroscientists today would agree that spirits have anything to do with it, but as messengers from a hidden world — this time the hidden world of our brain — these hallucinations seem to have lost none of their potency.

by Marianne Freiberger

via Plus Magazine

Further reading

Bressloff’s work on visual hallucinations is summarised in the paper What geometric visual hallucinations tell us about the visual cortex ([1]). A more detailed description can be found in the paper Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex ([2]).

Filed Under: Nature, Psychology, Science Tagged With: brain, drugs, geometric structures, geometry, hallucinations

Can’t Miss Golden Ratio In Masterpieces!

April 7, 2014 by Poly Hedra

“This is not art,” designers Hadi Alaeddin and Mothanna Hussein write of their series dubbed, not surprisingly, “NOT ART.”

The Jordan-based designers, who go by the collective name Warsheh, strip art history’s greatest hits down to their leading characters, splicing up the remaining imagery based on the Golden Ratio. For anyone who needs a refresher, the golden ratio (also known as the divine proportion) is a number that can be found when a line or shape is divided into two parts so that the longer part divided by the smaller part is also equal to the whole length or shape divided by the longer part. Still with us?

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“We know nothing about these paintings, we even used google image search to look up the names,” the designers continue. It’s this utterly non-pretentious attitude that makes Alaeddin and Mothanna’s endeavor so refreshing and endearing. Not to mention the fact that, removed from their backgrounds and entwined in the geometry that enabled their creation, the best of Jacques-Louis David, Johannes Vermeer and company gain a hypnotic, contemporary appeal.

If you’re looking for some profound message behind the stunning series, don’t get your hopes up. “We’re proud of being designers and not artists, the main difference being that we will sometimes happily admit that we worked on something just for the heck of it and not to claim any deeper meanings and hidden philosophies,” they say. “Sometimes we do make posters just because they look cool. And sometimes those posters, the way we imagine them, are liked by so many people, people that we are very much grateful are actually out there.”

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via Huffington Post

Filed Under: Art, Design Tagged With: art, geometry, golden ratio

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