Clipmarks | Kore7's 'math' clips

Ancient Blueprints of Calculus Uncovered in Archimedes Text

Sat, 06 Oct 2007 19:20:11 GMT

clipped by: Kore7
clipper's remarks: Details have been released from the nine-year-long reconstruction project to recover the Greek mathematician's writings from this one-of-a-kind find and the results are fascinating.

Buried beneath the surface of this gilded palimpsest, researchers discovered more extensive demonstrations of concepts such as infinite series, approximations, limits, and integral calculus than had been known to exist in ancient times.

Archimedes wrote The Method almost two thousand years before Isaac Newton and Gottfried Wilhelm von Leibniz developed calculus in the 1700s. Reviel Netz, an historian of mathematics at Stanford University who transcribed the text, says that the examination of Archimedes' work has revealed "a new twist on the entire trajectory of Western mathematics."

Clip Source: www.sciencenews.org

A long-lost text by the ancient Greek mathematician shows that he had begun to discover the principles of calculus.

The top layer of writing in this 700-year-old book describes Christian prayers. But underneath, almost obliterated, are the only surviving copies of many of the works of the ancient Greek mathematician Archimedes.

The picture on the left is an ordinary photograph, with the Archimedes text barely visible. The picture on the right is a multi-spectral image, and the Archimedes text and diagrams are mostly legible.

Sometime after Johan Heiberg examined the book in 1906, someone painted gold-leaf images over four of the pages (left). Multispectral imaging couldn't peer beneath the reflective metal paint, but x-ray fluorescence imaging revealed the underlying text (right).

Archimedes computed the area of the curved figure (left) by enclosing it in a bigger one with straight edges (right). He then examined random slices to compute the volume—using the concept of actual infinity.

Tags: math, science, archimedes, calculus, infinity, greek, ancient, history, x-ray, book

No Shortcuts to First-World Wealth

Sun, 02 Sep 2007 14:26:10 GMT

clipped by: Kore7
clipper's remarks: New cluster-analysis of the world's product export space reveals the differences in connectivity and diversity between nations' production capacities as well as the very sizable developmental gaps in this network that keep poorer countries on the industrial fringes.

The rich countries of the industrialized world tend to have broad portfolios of industries, and accordingly occupy large areas of the product space, usually including much of the network's core. Fast-growing developing countries such as China, Thailand, and Hungary are strong in some of those central, well-connected regions. The poorest countries, especially those in sub-Saharan Africa, tend to specialize in a few of the peripheral products—such as oil for Nigeria and copper for Zambia.

EDIT:My first title was too generic ("Mapping the Wealth of Nations.")

Clip Source: www.sciencenews.org

A country's competitive edge can spread industry to industry, like a disease

The economies of poor and developing countries often depend almost exclusively on a single product—perhaps timber or coffee—or on a handful of products at most.

HIDDEN LINKS. In the product space network above, nodes represent products. The more closely products are linked, the more likely they are to be produced and exported by the same countries. Each node's size represents the total world trade in that product, and the nodes' colors follow an older classification of products.
Hidalgo/Science

REGIONAL DIFFERENCES. In these illustrations, black squares mark products successfully exported. The industrialized countries' products (left) occupy the highly connected core of world trade. Goods from Southeast Asia and the Pacific region (center) cluster in the garment industry and in electronics, while sub-Saharan Africa's products (right) are mostly peripheral.
Hidalgo/Science

Tags: map, network, wealth, poverty, exports, industry, math, economics, countries, world

New Math Theory Explains Toddler's "Word Spurt"

Sun, 19 Aug 2007 18:36:01 GMT

clipped by: Kore7
clipper's remarks: A bell-shaped word distribution and a steady child learning rate turn out to be enough to bring about the extraordinary explosion seen in children's vocabularies around this age.

McMurray notes that languages have only a small number of very easy-to-learn words and many more intermediate words. So when a baby has been exposed to enough language to learn the easy words, she will acquire just a few words. As she is exposed to more language, she begins to learn the medium words. And because there are a lot of medium words, she is likely to pick up a lot of words at this stage. This, McMurray says, is the vocabulary explosion.

Clip Source: www.sciencenews.org

When children first begin to talk, they learn perhaps a word or two a week. Then suddenly, at around 14 to 18 months, they seem to soak up new words like a sponge, learning as many as ten a day.

Child psychologists have long been puzzled by this dramatic acceleration, and they have developed a variety of theories to explain it. Some researchers have argued that as the brain develops, it reorganizes to become much more efficient at storing new words. Others have argued that babies are able to use their knowledge of a few words to deduce the meanings of other words.

Bob McMurray, a child psychologist at the University of Iowa in Iowa City, now says that a bit of mathematics is enough to explain the phenomenon. Children's word absorption, he says, is the logical consequence of two basic principles: (1.) children work on learning many words at the same time, and (2.) some words are more difficult to learn than others.

Tags: toddler, children, child, word, words, vocabulary, learning, psychology, math, science

Math as a Civil Right

Sat, 21 Jul 2007 15:33:26 GMT

clipped by: Kore7
clipper's remarks:

The ubiquity of computers makes abstract, quantitative reasoning skills critical to a wide range of job opportunities. "Information age technology put math on the table as a literacy requirement in the same way that industrialism made reading literacy a requirement," says Moses. For that reason, he says, the country needs to raise math education standards for all students.

Clip Source: www.sciencenews.org

Voting rights advocate calls for mathematics literacy

Mathematics literacy is a new civil rights battleground, according to the renowned activist and political organizer Robert Parris Moses. Using the same ideas and methods that he once used to fight for voting rights in the South, Moses is working to increase access to quality mathematics education through the Algebra Project, a nationwide program that he founded.

Before the 1965 Voting Rights Act, many Americans were excluded from participation in their country's democracy through laws that made literacy a requirement to vote. Today, Moses says, many young people are excluded from full participation in the country's economy because they lack mathematical literacy.

Tags: math, mathematics, algebra, computers, technology, education, children, civil rights, usa, robert parris moses

New Math For Analyzing Evolutionary Trees

Sat, 12 May 2007 19:59:52 GMT

clipped by: Kore7
clipper's remarks:

"What this tells me is that you don't know what kind of mathematics is going to be useful to biology," Billera says. "It wasn't clear before this that geometry and topology would be useful to biology. Who would think they had anything to do with each other?"

Ernst Haeckel's classic hand-drawn diagram is just for fun—it's one of those wonderful diagrams that functions as both science and art.

Clip Source: www.sciencenews.org

Now, mathematicians have developed a new understanding of the mathematics of tree-shaped graphs, which makes possible the statistical analysis of evolutionary trees. The development will help biologists to make sense of the flood of newly available genetic information.

How similar are these three trees?
Billera, Holmes, and Vogtmann

This diagram has negative curvature. It shows a small portion of the space of trees with four nodes. The purple lines show the shortest distance between two pairs of points. Although each rectangular region is flat, the bends and folds create a rough curvature over a larger area.
Billera, Holmes, and Vogtmann

In 1866, biologist Ernst Haeckel drew this tree showing the evolutionary relationships among all living things, as they were understood at that time.

Tags: math, trees, evolution, genetics, biology, topology, dna, statistics, haeckel

The Mathematical Lives of Plants

Mon, 07 May 2007 00:43:06 GMT

clipped by: Kore7
clipper's remarks:

The seeds of a sunflower, the spines of a cactus, and the bracts of a pine cone all grow in whirling spiral patterns. Remarkable for their complexity and beauty, they also show consistent mathematical patterns that scientists have been striving to understand.
...
Scientists have puzzled over this pattern of plant growth for hundreds of years. Why would plants prefer the golden angle to any other? And how can plants possibly "know" anything about Fibonacci numbers?

For the first time, scientists have found convincing biochemical mechanisms responsible for the interlocking spiral growth patterns seen in many plants. (The Romanesco broccoli plant is a striking example.) The video of the experiment with magnetized liquid iron droplets demonstrates how the geometry of such growth could occur in nature.

Clip Source: www.sciencenews.org

Each yellow nub in the center of this daisy is actually its own miniature flower, complete with a full set of reproductive organs. The buds form interlocking clockwise and counterclockwise spirals.
Scott Hotton

The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio.
Wikipedia

This sunflower has 21 clockwise and 34 counterclockwise spirals.
Scott Hotton

Plants form new seeds or buds from the center. In this picture, the circle labeled 1 would be the most recent bud. The circle labeled 2 would have been formed just previously, and it forms the golden angle with bud 1

An image of the tip of a Norway spruce branch, viewed through an electron microscope, shows small buds that are primordial. In this case, they will eventually turn into needles. The primordia form at the tip and then move outward and downward.
R. Rutishauser

This cactus, a Mammilaria moellerana, has golden-angle spirals.
Eleanor Farrington

Tags: math, plants, biology, symmetry, sprial, science, nature, geometry, fibonacci, patterns

248-Dimensional Mathematical Map Calculated

Tue, 20 Mar 2007 16:04:59 GMT

clipped by: Kore7
clipper's remarks:

"The calculation was known to be possible in principle, but it was thought to be hopeless in practice," says Adams. "But four years ago a group of us said let's really try to do it. We're pretty sure we've got it right, but it's hard to be 100% sure."

"It's probably one of the most complicated pure mathematical calculations anyone's ever done," says Stewart. "Each entry is difficult to calculate — it's amazing they managed to do this."

Clip Source: www.nature.com

A map of one of the strangest and most complex entities in mathematics should be a powerful new tool for both mathematicians and physicists pursuing a unified theory of space, time and matter.

The strange 'thing' that has been mapped is a 'Lie group' called E8 — a set of maths that describes the symmetry of an (unimaginable to most) 57-dimensional object.

The creation of this map, which took 77 hours on a supercomputer, resulted in a matrix of 453,060 ? 453,060 cells, containing more than 205 billion entries — "all related in intricate and complex ways", says Jeffrey Adams, the project leader and a mathematician at the University of Maryland.

"It's perhaps the most beautiful structure in all of mathematics, but it's very complex," says physicist Hermann Nicolai of the Max Planck Institute for Gravitational Physics in Potsdam, Germany.

Tags: map, e8, lie group, math, mathematics, physics, science, topology, algebra, dimensions, symmetry, calculation, computation

Inventor of Fortran, John W. Backus, Dies

Tue, 20 Mar 2007 14:06:16 GMT

clipped by: Kore7
clipper's remarks: Not only did Backus give the world the first high-level (and highly-successful) programming language, but he had the genius in 1959 to develop Backus–Naur form, the meta-language used to define all possible programming languages, past, present and future. Our digital world wouldn't be the same without him.

Shortly before he graduated, Mr. Backus wandered by the I.B.M. headquarters on Madison Avenue in New York, where one of its room-size electronic calculators was on display.

When a tour guide inquired, Mr. Backus mentioned that he was a graduate student in math; he was whisked upstairs and asked a series of questions Mr. Backus described as math “brain teasers.” It was an informal oral exam, with no recorded score.

He was hired on the spot. As what? “As a programmer,” Mr. Backus replied, shrugging. “That was the way it was done in those days.”

Clip Source: www.nytimes.com

Fortran, released in 1957, was “the turning point” in computer software, much as the microprocessor was a giant step forward in hardware, according to J.A.N. Lee, a leading computer historian.

Fortran changed the terms of communication between humans and computers, moving up a level to a language that was more comprehensible by humans. So Fortran, in computing vernacular, is considered the first successful higher-level language.

Innovation, Mr. Backus said, was a constant process of trial and error.

“You need the willingness to fail all the time,” he said. “You have to generate many ideas and then you have to work very hard only to discover that they don’t work. And you keep doing that over and over until you find one that does work.”

Tags: backus, john backus, backus–naur form, fortran, programming, language, grammar, cfg, computer science, math, mathematics, computers, history

80-Year-Old Indian Math Mystery Solved

Sat, 10 Mar 2007 18:54:26 GMT

clipped by: Kore7
clipper's remarks: A few months into 2007 and already another long-standing mathematical mystery has been more-or-less put to rest. It will be hard to top Perelman's stunning proof of the legendary Poincaré Conjecture from last year, but in math and science, you never know when the next breakthrough will come.

(If you haven't already, read up on some of the incredible anecdotes about the life of the Indian genius, Ramanujan. He was truly one of a kind.)

Clip Source: www.sciencenews.org

A pair of mathematicians has solved a problem that had tantalized number-theory researchers for more than 8 decades. It is the so-called final problem of the legendary Indian mathematical genius Srinivasa Ramanujan.

"The mock theta functions are like beautiful butterflies that Ramanujan happened to find," says Freeman Dyson, an emeritus professor at the Institute for Advanced Study in Princeton, N.J. "But if you're a scientist, you want more—you want a theory of evolution, a framework of ideas to fit the butterflies in."

"I didn't really hope to see someone actually do this," says George Andrews of Pennsylvania State University in University Park, who had called the description of the mock theta functions one of the hardest math problems for the new millennium. Ono and Bringmann's accomplishment is "absolutely stunning," he concludes.

Tags: ramanujan, srinivasa ramanujan, 1729, math, mathematics, number theory, problem, mystery, mock theta, function, science, indian

Math Behind Ancient Islamic Tile Patterns Decoded

Sat, 24 Feb 2007 15:15:32 GMT

clipped by: Kore7
clipper's remarks:

When Peter J. Lu traveled to Uzbekistan, he had no idea of the mathematical journey that he was about to embark on as well.

See the full research article as published in Science. It's a wonderful example of original, multidisciplinary academic research bridging history and mathematics that happens to force us to re-think the sophistication of ancient geometrical knowledge.

When Lu looked at photographs of Islamic buildings, he found that he could break the patterns on their surfaces up into the same shapes, even though the shapes often weren't immediately visible. "I couldn't sleep for days," he said. "I skipped Christmas break to work on it."

Clip Source: www.sciencenews.org

Archway from the Darb-i Imam shrine in Isfahan, Iran, which was built in 1453 C.E. The larger pentagons outlined in pale blue were constructed using a large-scale girih tile pattern, and the small white pentagons were constructed using a small-scale girih tile pattern.
Image courtesy of K. Dudley and M. Elliff.

Kite and dart Penrose tiles.
Wikipedia

A Penrose tiling made up of fat and skinny diamonds.
Wikipedia

A 15th-century Timurid-Turkmen scroll now held by the Topkapi Palace Museum in Istanbul. The faint reddish lines outline the shapes of the underlying tiles. One example of each shape has been shaded in the picture.
Peter J. Lu

The five decorated shapes.
Peter J. Lu

An archway in the Sultan's Lodge in the Green Mosque in Bursa, Turkey from 1424.
Image courtesy of W.B. Denny

Although the underlying shapes aren't obvious in the final design, this diagram shows how the pattern from the Sultan's Lodge archway was constructed using girih tiles.
Image courtesy of W.B. Denny

Tags: art, tile, tiles, tiling, shape, islamic, islam, turkey, uzbekistan, patterns, pattern, symmetry, geometry, math, mathematics, penrose, girih, history, ancient, research, discovery, science

Fractal Food: Self-Similarity on the Supermarket Shelf

Fri, 09 Feb 2007 20:00:56 GMT

clipped by: Kore7
clipper's remarks: This great article on computational self-similarity in nature provided the author with an excuse to take a series of spectacular close-up photos of the incredible Romanesco broccoli plant. Fractals never looked so delicious!

(Click pictures for high-resolution images.)

Nearly exact self-similar fractal forms occur do in nature, but I'd never seen such a beautiful and perfect example until, some time after moving to Switzerland, I came across a chou Romanesco like the one above in a grocery store. This is so visually stunning an object that on first encounter it's hard to imagine you're looking at a garden vegetable rather than an alien artefact created with molecular nanotechnology. But of course, then you realise that vegetables are created with molecular nanotechnology, albeit the product of earthly evolution, not extraterrestrial engineering.

Clip Source: www.fourmilab.ch

Fractal Food
Self-Similarity on the Supermarket Shelf

by John Walker

Fractal forms--complex shapes which look more or less the same at a wide variety of scale factors, are everywhere in nature. From the fluctuations in the cosmic microwave background radiation to the coastlines of continents, courses of rivers, clouds in the sky, branches of plants and veins in their leaves, blood vessels in the lung, and the shape of seashells and snowflakes, these fractal or self-similar patterns abound. The self-similarity of most of these patterns is defined only in a statistical sense: while the general "roughness" is about the same at different scales, you can't extract a segment, blow it up, and find a larger scale segment which it matches precisely.

However, some of the most pleasing patterns in geometric art exhibit exact or almost exact self-similarity. These are patterns which are composed of smaller copies of themselves ad infinitum, or at least until some limit where the similarity breaks down due to the granularity of the underlying material.

Every self-similar pattern in nature breaks down at some scale--at the level of molecules and atoms if not before. The last photo shows the tiny structures near the top level spiral. As the spirals get smaller and smaller approaching the vertex, the spirals that make them up have less and less lower level detail, with the tiniest being little more than bumpy spheroids.

Tags: romanesco, broccoli, fractal, self-similar, computation, nature, math, mathematics, food, supermarket, photos, photography, zoom, vegetables, automata, complexity

Art of the Tetrahedron, Revisited

Sun, 14 Jan 2007 13:46:56 GMT

clipped by: Kore7
clipper's remarks: Wonderfully angular sculptures. And what an inspiring story!

Until age 50, Silverman had been a highly successful surgeon, practicing medicine with considerable enthusiasm and skill. Then he encountered an ailing colleague near death, who advised Silverman that if there were anything he might really want to do, then he ought to do it right away, before the chance slips away.

The encounter changed Silverman's life. He returned to interests that had captured his attention when he was a teenager. He had visited museums to gaze at statues, and he had tried his hand at carving wood. Later, when studying medicine at Tulane University, he had met a sculpture teacher who had invited him to classes and taught him to see, in the artistic sense.

Clip Source: www.sciencenews.org

Art of the Tetrahedron, Revisited

The tetrahedron is the simplest of all polyhedra. Any four points in space that are not all on the same plane mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron.

For more than 30 years, Arthur Silverman of New Orleans has created artworks arising out of explorations of this angular form. Many of his sculptures are on display in public spaces and various buildings in New Orleans and other cities from Florida to California.

"The tetrahedron is very exciting visually," Silverman says. "It's very difficult to anticipate what you are going to see. Every step around a piece gives you a different view."

We're accustomed to thinking about orientation in space in terms of three perpendicular axes. A tetrahedron has no right angles. So, a tetrahedral structure jars us out of spatial complacency. It has so few faces, its aspect changes abruptly as you move around to view it from different angles.

This colorful trio of tetrahedra can be found on Poydras Street in New Orleans. It's one of the few Silverman sculptures that's painted.

Located in front of City Hall in New Orleans, this sculpture was designed by removing tetrahedral forms from a rectangular block.

It's sometimes hard to tell that Silverman's sculptures are based on tetrahedra.

Arthur Silverman's "Echo" features a pair of elongated tetrahedra, each seemingly balanced on one edge. The sculpture is 60 feet tall and rests on a foundation that extends 20 feet into the ground.

Aluminum tiles based on cross sections of a tetrahedron create a striking wall design.

Silverman created an outdoor menorah for Temple Sinai in New Orleans.

Changing the orientation of this sculpture, about 20 inches high, gives observers startlingly different views.

Partially immersed in floodwaters for days, this sculpture displays the resulting waterline.

Silverman has turned his sculpture studio into a gallery, showcasing his many artworks.

Tags: math, mathematics, tetrahedron, triangles, geometry, art, artist, sculpture, arthur silverman, new orleans, katrina, flood, inspiration

What Did Descartes Really Know? (Book Review)

Sun, 19 Nov 2006 17:03:19 GMT

clipped by: Kore7
clipper's remarks: Great review of two new Descartes biographies that set some of the records straight on the great naturalist's works.

Despite his current reputation, the man himself seems to have been less interested in metaphysics than in applying algebra to geometry and delving into the innards of cows. He turned to philosophy relatively late in life, and out of fear that the Catholic Church would condemn his science. He would have been surprised at how he is remembered.

Most of all, he would have been aghast at the way in which “I think, therefore I am” has been ripped from its context, inflated into a one-sentence summary of his ideas, and turned into something absurd. The rot set in at the start of the nineteenth century, when Hegel made heavy weather of “I think, therefore I am” and took it to mean that thought and being are fundamentally the same thing.

Clip Source: www.newyorker.com

THINK AGAIN

What did Descartes really know?

by ANTHONY GOTTLIEB

Issue of 2006-11-20

Posted 2006-11-13

In “Descartes: A Biography” (Cambridge; $40), Desmond Clarke, a leading Descartes scholar at the National University of Ireland, Cork, argues that Descartes’s philosophy is distorted if it is not seen in the context of his wider scientific enterprise. Descartes certainly played down his own purely philosophical writings. In a letter to his friend Princess Elizabeth of Bohemia, in June, 1643, and again in a conversation with a theology student, Frans Burman, who interviewed him in April, 1648, Descartes warned against paying too much attention to his metaphysics. Read it once, he said, and move on. His “Discourse,” with the famous slogan, was published merely as a preface to a collection of treatises on optics, meteorology, and geometry. Clarke reminds us that Descartes’s philosophical works were intended to establish credentials for his system of nature, and to make it theologically acceptable. The “Meditations” originally bore the subtitle “in which are demonstrated the existence of God and the immortality of the soul.” Indeed, Clarke’s most distinctive claim is that Descartes’s account of the mind as an immaterial substance—his famous “dualism” of the mental and the physical, sometimes known as the doctrine of “the ghost in the machine”—is at best a provisional theory, aimed at providing support for the Catholic doctrine of the immortality of the soul, and sits uneasily with many other things that he wrote.

Tags: descartes, rene descartes, grayling, desmond clarke, philosophy, thinking, existence, metaphysics, science, nature, math, mathematics, history, biography, review, dualism, theology

St. Louis Arch Couldn't Exist According to Its Own Description

Sat, 04 Nov 2006 21:57:26 GMT

clipped by: Kore7
clipper's remarks: It turns out that for more than 40 years, there's been a blatant mathematical error etched into the plaque at the base of the Gateway Arch and apparently the mistake has only been caught now! The Arch is a giant inverted catenary, a curve with a precise mathematical equation. One of the geometric equations at the arch turns out to be meaningless. If the Arch were actually built to satisfy these equations, it wouldn't even exist!

This could be some kind of national embarrassment, given the prominence of this mistake's locale.

Clip Source: www.jonco48.com

Clip Source: www.geocities.com

Let me make one thing absolutely clear: The Gateway Arch is safe. There's a mistake at the base of the Gateway Arch, BUT it's a typo and the Gateway Arch is safe. Even though the Arch weighs 43,000 tons of stainless steel and it can sway as much as 18 inches with sufficiently strong winds, the Gateway Arch is safe. I don't want to panic residents of St. Louis, nor the National Park Service, nor the four million annual visitors to 11 North Fourth Street, St. Louis, Missouri, 63102.

When you enter the North Tram of the Gateway Arch, you see an exhibit on the science and engineering behind the Gateway Arch. In that exhibit, there's a white plaque which shows the mathematical equations used in designing the arch.

One of those equations is mistaken.

That's it. That's the mistake.

The Gateway Arch is itself an inverted catenary, whose exact formula somewhat resembles the catenary formalism above:

Y = A (cosh[X/L] C-1)

where L and A are values used in calculating parts of the Arch's base and top (diversion alert: click here for the story behind L and A). Do the algebra to find the value for X (go on, it's fun--I'll wait; when you're done you can check your answer here), and you get:

X = L/C[cosh^-1 (1 + [Y/A])]

These formulas for Y and X are listed on that plaque in the North Tram--except that the X formula is listed on that plaque as:

X = -[cos h (1+ Y/A)]

The Mistake at the Base of the Gateway Arch is actually three different mistakes on this single equation for X: there's no constant L/C preceding the inverse hyperbolic cosine, there's a minus sign preceding the inverse hyperbolic cosine, and there is no inverse hyperbolic cosine (denoted by a superscript ^-1 above the cosh). Matt told me that the missing inverse tipped him off to the discovery.

So what is drawn out by the formula for X that is listed on that plaque? Forget about trying to draw it. It's nothing, nonsense, a misteak. I have visited the Gateway Arch myself and confirmed the error firsthand.

Tags: st louis, missouri, landmark, arch, gateway, memorial, math, mathematics, catenary, equation, equations, error, mistake, geometry, cosine

The Exanding Mathematical Universe of Spidrons

Sun, 22 Oct 2006 16:45:48 GMT

clipped by: Kore7
clipper's remarks:

A field of triangles crumples and twists into a wavy crystalline sea. A crystal ball sprouts spiraling, labyrinthine passages. Faceted bricks stack snugly into a tidy, compact structure. Underlying each of these objects is a remarkable geometric shape made up of a sequence of triangles—a spiral polygon that resembles a seahorse's tail.

The result is beautiful to behold.

Clip Source: www.sciencenews.org

TWO ARMS, TWO ARMS. The spiral arms of a spidron consist of alternating sequences of equilateral and isosceles triangles.
Erdély

Erdély started his spidron tinkering while working in a Budapest printing house, where he noticed networks of lines and hexagons on rolls of paper prepared for printing. Subsequent doodling beginning with a hexagon led him to an intriguing pattern. He had connected every second vertex of the hexagon with a straight line, thus creating a six-pointed star. Inside the star was a smaller hexagon. He again connected every second vertex and continued the process until the figure in the center was too small to distinguish. The resulting inscribed pattern consisted of just two types of triangles—equilateral and isosceles—that got smaller as they neared the hexagon's center.

From his doodling, Erdély found that a hexagon contains six identical copies of a spiral sequence of triangles—a shape that he later called a spidron arm. His subsequent insight was to start with an array of inscribed hexagons drawn on a sheet of paper and laid as if they were bathroom tiles. By creasing the pattern in the right combination of mountains and valleys at the lines within each spidron arm and leaving a small hole at the center of each hexagon to allow movement, he crinkled the whole array into a dramatic three-dimensional relief.

SPIDRON CELEBRATION. This elaborate polyhedral structure is made up of 120 spidrons.
Pelletier, Erdély, van Ballegooijen

BY ANY OTHER NAME. This spidrohedron sculpture in a park near Leeuwarden in the Netherlands was assembled from 240 equilateral and 240 isosceles triangles. The object is a closed polyhedron made of 24 spidrons. Such spidrohedra can be stacked, each one with eight neighbors, to fill space.
Erdély, I. Sági

BUILDING ON MATH. Standard spidron-based units like the one shown below can be stacked to create a space-filling structure, like the one shown above.
Roelofs

Tags: dániel erdély, daniel erdely, erdely, spidron, spidrons, spiral, spirals, triangle, triangles, hexagon, math, mathematics, geometry, pattern, shape, beauty, art, sculpture, infinite, interesting

Clipmarks | Kore7's 'math' clips

Ancient Blueprints of Calculus Uncovered in Archimedes Text

A long-lost text by the ancient Greek mathematician shows that he had begun to discover the principles of calculus.

No Shortcuts to First-World Wealth

A country's competitive edge can spread industry to industry, like a disease

New Math Theory Explains Toddler's "Word Spurt"

Math as a Civil Right

Voting rights advocate calls for mathematics literacy

New Math For Analyzing Evolutionary Trees

The Mathematical Lives of Plants

248-Dimensional Mathematical Map Calculated

Inventor of Fortran, John W. Backus, Dies

80-Year-Old Indian Math Mystery Solved

Math Behind Ancient Islamic Tile Patterns Decoded

Fractal Food: Self-Similarity on the Supermarket Shelf

Fractal Food Self-Similarity on the Supermarket Shelf

Art of the Tetrahedron, Revisited

Art of the Tetrahedron, Revisited

What Did Descartes Really Know? (Book Review)

St. Louis Arch Couldn't Exist According to Its Own Description

The Exanding Mathematical Universe of Spidrons

Fractal Food
Self-Similarity on the Supermarket Shelf