## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## July 5, 2007

### Spiraling Triangles

Playing with triangles can lead to amazing patterns and three-dimensional structures. That's what Hungarian designer Dániel Erdély (below) found when he created an intriguing geometric form out of two spirals of triangles that get smaller and smaller.

Daniel Erdély holds a complex polyhedron constructed from spidrons. Photo by Regina Márkus.

He called the resulting S-shaped object a spidron. Each of its two arms looks a bit like a seahorse's tail.

The two spiral arms of a spidron consist of alternating sequences of equilateral and isosceles triangles (above). Erdély.

How to Draw a Spidron's Arm (above): Start with a regular hexagon, which has six corners. Connect every send corner with a straight line to make a six-pointed star. Inside the star is a smaller hexagon. Again connect every second corner. Continue the process until the shape in the center is so small that you can't put in any more lines. The resulting pattern contains six identical copies of a spidron arm. Erdély.

Even though spidrons are irregularly shaped, they can fit together without gaps or overlaps to cover a plane (above). For example, you could tile your bathroom floor with this pattern of spidrons. Erdély, Marc Pelletier, Amina Buhler Allen, Walt van Ballegooijen.

When spidrons are laid down like tiles on a flat surface, then creased in just the right way at the line within each spidron arm, the flat structure can be forced to fold accordion-style into a wavy surface (above). As the folds get steeper, the whole pattern twists and compacts. Erdély, Marc Pelletier, Amina Buhler Allen, Walt van Ballegooijen.

Creased and folded spidrons can be assembled into three-dimensional balls (above). This one is made of 120 spidrons. Erdély, Marc Pelletier, Amina Buhler Allen, Walt van Ballegooijen.

Are spidrons good for anything besides artwork and maybe bathroom floors? Erdély says spidron surfaces could be used for collapsible solar panels or shock absorbers. And spidron-based blocks might make an interesting toy. But mostly, he admits, they're just interesting for their own sake.