The story of gyrangle

The Story of Gyrangle
Reza Sarhangi
(pictures of gyrangle: )
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