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SCIENCE AT WILLIAMS

1995-1996

FACULTY AND STUDENTS IN THE NATURAL SCIENCES

1996

A Knotted Tiling of 3-Dimensional Space

Mathematicians have been interested in tiling for many years. Traditionally, one tries to find an interesting shape for tiling the plane. That is, a single shape for a tile such that carbon copies of that tile can be fitted together to cover the entire plane, leaving no gaps. In recent years, mathematicians have investigated tilings of 3-dimensional space as well. Such tilings come up in crystalline structures in nature, where a particular 3-dimensional molecular structure repeats again and again, forming a crystal. Until relatively recently, the only 3-dimensional tiles considered were essentially deformed versions of a ball. That is, they had no holes or handles, as a doughnut or pretzel does.

The cover depicts a shape for a tile discovered by Colin Adams, Professor of Mathematics. It is "knotted", in the sense that at its core is a knot with a variety of protrusions attached, none of which affect its knottiness. Copies of this tile can be fit together to fill up all of 3-dimensional space, yielding a "knotted" tiling of 3-space. The "divining stick" on the left-hand side of the picture fits into the knot that would be in the adjacent tile on the left, filling its crossings. The front and back half-molds are each filled with a knot from a tile in front and a tile behind. Half-molds protruding from each of these tiles encase the knot from the original tile. Continuing in this manner, we can fill up all of 3-space with these tiles.

This idea was used by Adams to prove that any shape in 3-space with one
boundary can, after deformation, be used as the shape of a tile, copies of
which can then be used to tile 3-space. The result generalizes to n-space.
Additional information on these concepts can be found in "Tiling Three-Space
with Knotted Tiles," by Colin Adams in the *Mathematical Intelligencer*,
Vol. 17, No. 2, 41-51 (1995) or the article by Ian Stewart in *Scientific
American*, Nov. 1995, p.100-101 about Adams' work. The drawing is by Pier
Gustafson.

The Science Executive Committee wishes to express its gratitude to the
extensive efforts of all the science department secretaries in preparing
contributions for this publication, and to Alice J. Seeley for assembling this
material in its final form.

Editor: Dr. Bryce Babcock

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