# Pieces at the Bridges Art-Exhibit, London, August 4-8, 2006

*"Klein Quartic Quilt" (Eveline Séquin) *

3D quilt, 12" diameter, 1993.

Tactile visualization of the group of 24 heptagons on a genus-3 surface,

exhibiting 24 of the original 168 symmetries of the Klein group.

Quilt based on a pattern obtained from Bill Thurston, MSRI.

* "Poincaré FishDish" (Carlo Séquin) *

2D print, 16" diameter, 2006.

A tiling with regular heptagons does not fit into the Euclidean plane,
since the 3 times

the dihedral angle of the heptagon exceeds 360 degrees.
However, if we introduce a

progressive scale factor, then the whole
hyperbolic plane can map into the Poincaré disc.

Here is a visualization of
a {7,3} tessellation where three heptagons join at every vertex,

using
a tiling motif inspired by the famous Dutch artist M.C. Escher. Each heptagon

is cut into seven identical pizza slices with irregular boundaries in the shape
of fish

that properly interlock with one another.

* "Poincaré Lace and Double Lace" (Carlo Séquin) *

2D print, 16" diameter, 2006.

A 7-lobed clover-leaf knot is the basic heptagonal tile for a
hyperbolic {7,3} tiling,

in which 3 heptagons join at every vertex.
Since 3 times the dihedral angle of the heptagon

(128.57 degrees) exceeds 360 degrees,
this tiling cannot be drawn in the Euclidean plane

without distortion.
Using the conformal mapping of the Poincaré hyperbolic disk,

which preserves angles between intersecting lines,
the entire hyperbolic plane

can be projected into a disk of finite radius.

The artistic motif of the mutually interlinking clover-leaf knots
may be seen as a symbol

for the various intertwined topics that
are covered by the annual BRIDGES conferences.

* "Ribbon Tetrus" (Carlo Séquin) *

Fused deposition model, 4" tall, 2006.

A single ribbon winds around the six arms of a Tetrus, thereby defining this shape.

This work was inspired by Escher's "Bond of Union" and by my own "Viae Globi."

The challenge was to achieve uniform coverage on all six tetrahedral arms and also

maintain the maximum possible symmetry. By using two different cork screws,

a single closed path could be formed and C-2 symmetry could be
maintained.

Carlo H. Séquin, Professor of Computer Science,

University of California, Berkeley, CA 94720-1776

EMAIL: sequin@cs.berkeley.edu

URL: http://www.cs.berkeley.edu/~sequin/