Swirling seas, crystal balls: spirals of triangles crinkle into intricate structures

Science News,  Oct 21, 2006  by Ivars Peterson

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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.

Hungarian industrial designer and graphic artist Daniel Erdely called the form a spidron when he discovered it in the early 1970s. In so doing, he evoked the figure's two spiral arms and the polygonal structures that can result when spidrons are joined.

A standard spidron consists of two alternating, adjoining sequences of equilateral and isosceles triangles. Start with an equilateral triangle. Draw lines from the three corners of the triangle to a spot at its center, creating three identical isosceles triangles, each with angles of 30[degrees], 120[degrees], and 30[degrees]. Then, draw a reflection of one of these isosceles triangles so that it projects from the side of the original triangle.

Next, make a new equilateral triangle, using one of the two short sides of the jutting isosceles triangle as a base. Repeat the procedure again and again, producing a spiraling sequence of ever-smaller triangles. Erase the original equilateral triangle, and join two of these structures along the long side of the largest isosceles triangle to create the s-shape of a spidron.

Erdely observed that within each arm, the area of any equilateral triangle equals the sum of the areas of all the triangles with areas smaller than the given quilateral triangle. In other words, all the smaller triangles would fit together to fill the larger one.

Far more startling, however, is what happens when spidrons, laid down like tiles on a flat surface, are creased in just the right way and the flat, tiled structure is forced to fold accordion-style. The transformation from two to three dimensions creates mountain and valley folds that steepen. Each section of the pattern rotates as the configuration tightens. At its limit, it's a wavy, three-dimensional surface made up of triangles at a few set angles to each other.

Sets of these creased and folded spidrons can themselves be assembled into a wide variety of intricate forms that resemble exotic crystal geodes.

"There's a massive potential for sculpture here," says artist Marc Pelletier, co-founder of the geometric construction-kit company Zometool in Denver. "It's really beautiful."

Recent collaborations between Erdely and several artists and mathematicians have vastly increased the potential applications of spidrons, not only for creating intriguing art objects but also for engineering finely adjustable dynamic structures. For example, spidron reliefs could be used as shock absorbers or crumple zones in vehicles, Erdely says. Spidron surfaces could serve as flexible acoustic walls or solar panels. Spidron-based structures could also be used as blocks for builders--or construction toys.

Pelletier, his colleague Amina Buhler Allen, and math enthusiast Walt van Ballegooijen of the Netherlands have been working with Erdely for the past year, coming up with many new spidron-based designs. They presented their work in August in London at Bridges, a conference on mathematical connections among math, music, and art.

HEXAGONAL TWIST Erdely 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, Erdely 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.

In Erdely's words, the folding pattern "mobilizes" the hexagonal array, permitting a flat surface to take on a range of three-dimensional forms. The surface area of the hexagon remains unchanged, and the constituent triangles neither bend nor distort.

In 1979, Erdely presented this swirling, moveable relief to his teacher, Erno Rubik at the Hungarian University of Applied Art. According to Erdely, Rubik--known for the invention of Rubik's cube--said that he'd never seen anything like it. That interest encouraged Erdely to continue experimenting with spidron structures.