# The story of gyrangle

The Story of Gyrangle

Reza Sarhangi

(pictures of gyrangle:

http://www.ams.org/mathimagery/albums/userpics/10002/zcorp-model-3-fold-epostcard.jpg

http://www.georgehart.com/DC/tetrahedron-rotation.gif )

The first USA Science and Engineering Festival was

a call by President Obama to invite the nation to

the National Mall in Washington, D.C. for a celebration

of science and engineering achievements and

their contributions to our society. For two beautiful

sunny days, Saturday and Sunday, October 23–24,

2010, thousands of people attended the festival at

the National Mall and enjoyed talking to scientists

and engineers to learn about their new inventions

and discoveries.

People of all ages played with scientific toys

and machines, and participated in interesting and

exciting activities presented in hundreds of booths

located all over the Mall.

Mathematicians appeared in this national celebration

by presenting mathematical ideas and

activities at several booths, including ones for the

AMS and the Mathematical Association of America

(MAA). The award-winning mathematics writer

Ivars Peterson, director of publications and communications

at the MAA, was among the hosts in

the MAA booth. One gift from the MAA booth was

a folded one-sheet booklet, “A Field Guide to Math

on the National Mall”, highlighting mathematical

aspects of some interesting sites in the Washington,

D.C., area.

The sculptor, mathematician, and computer

scientist George Hart led a public sculpture barnraising

of his latest work, Gyrangle (Figure 1), at

the AMS booth. The 38-inch-high sculpture consists

of hundreds of laser-cut steel units bolted

together in a novel way. To be exact, there are 490

flat or folded hollow equilateral triangles in four

colors. It illustrates a discrete version of the gyroid

surface, made entirely from equilateral triangles.

The gyroid is a smooth, infinite, triply periodic,

minimal surface discovered by Alan Schoen in

1970. Channels run through it in many directions

and connect at an angle to other channels. The direction

of connection travels in a spiral along each

tunnel, giving rise to the name “gyroid”. Figure 2,

downloaded from Wikipedia, illustrates a cubical

portion of the infinite gyroid surface.

The gyroid is a smooth surface, but Hart discovered

a way to triangulate it entirely with equilateral

triangles. Both the faces and vertices in the construction

exhibit a property known as uniformity.

Although the gyroid itself contains no straight

lines, Hart’s triangular discretization contains

infinite straight lines embedded in various directions.

It divides all space into two congruent but

mirror-image volumes. Another interesting property

is that the faces do not meet edge-to-edge, but

instead each triangle shares six half-edges with six

neighboring triangles. The construction is previously

unpublished, so the sculpture serves as the

first presentation of this discovery. As it illustrates

a gyroid made of triangles, Hart coined the term

Gyrangle for it.

The full infinite surface would be visually repetitive

(and infinitely large) so is not itself suitable

for a sculpture. Instead, Hart chose a tetrahedral

portion of space as the outer form of the sculpture

and inhabited it with the Gyrangle surface as a kind

of three-dimensional texture. To seal off the edges

he introduced a folded triangle unit that closes off

the boundary. For visual and mathematical interest,

the 100 pounds of laser-cut steel triangles are