Polycube Reptiles

Introduction

A reptile (or rep-tile) is a polyform that can tile an enlarged copy of itself. For example, four copies of the L tetromino can make a double-scale L tetromino:

This tiling is said to be rep-4 (or rep-2²), because it uses four tiles.

Polycubes are polyforms made by joining equal cubes face to face. Little is known about polycube rep-tilings, because computing polycube tilings is lengthy and complex.

Any polycube that can tile a rectangular box is a reptile. Rectangular boxes can always be joined to form a cube, and copies of this cube can form an enlarged copy of the original polycube. For example, the U pentacube can form boxes with dimensions 2×3×5 and 4×4×5, and such boxes can tile a cube of side 10. So the U pentacube has a rep-1000 tiling. I show below that it also has a rep-27 tiling.

The smallest polycubes that cannot reptile themselves (nor tile any box) are the G and X pentacubes:

For the smallest box tilings of the other pentacubes, see Pentacubes in a Box. On this page I show only the smallest known rep-tiling for each polycube. If the rep-tiling is assembled from big cubes, I give only a description of the tiling. It would be interesting to know all possible rep-tiling values for each polycube.

Polycubes of orders 4 and up may be chiral, which means that they have distinct mirror images. Some chiral polyforms have known rep-tilings only if reflection is allowed. If a polyform can tile an enlarged copy of its mirror image without reflection, it can also tile an enlarged copy of itself without reflection, by iterating the mirror-image tiling.

If you find a small solution for a case not shown here, please write!

Links

Karl Scherer has a Wolfram Demonstration of finite and infinite polycube rep-tilings. Andrew Clarke's Poly Pages include several pages of Polycube Reptiles found by Mike Reid, Patrick Hamlyn, and Clarke. Torsten Sillke has an extensive catalogue of box tilings. Many of his results are used here.

Navigation

Monocube, Dicube, Tricubes

Tetracubes

Pentacubes

Achiral

Chiral, Disallowing Reflection

Chiral, Allowing Reflection

Monocube, Dicube, Tricubes

Except for the L-tricube, polycubes of order 1–3 are rectangular boxes and have regular rep-tilings with 8 tiles.

Tetracubes

Every tetracube can be rep-tiled. In the illustration below, the K and S tetracube rep-tilings are assembled from 2×2×2 boxes.

Pentacubes

Achiral

All tilings in the picture are minimal. The tilings shown for the I, L, N, P, U, and Y pentacubes have mirror symmetry.

Mike Reid first solved pentacube U at scale 3 and pentacube K at scale 4. Andrew Clarke first solved pentacubes Q and Y at scale 4.

Pentacube Boxes Cube Tiles Minimal?

A 6×10×10
8×10×10 20×20×20 8000 —

M 10×10×20 20×20×20 8000 —

T 10×10×10 10×10×10 1000 —

W 5×10×10 10×10×10 1000 —

Z 10×10×10 10×10×10 1000 —

Pentacube	Cube	Tiles	Minimal?
A	6×10×10 8×10×10	20×20×20	8000	—
M	10×10×20	20×20×20	8000	—
T	10×10×10	10×10×10	1000	—
W	5×10×10	10×10×10	1000	—
Z	10×10×10	10×10×10	1000	—

Chiral, Disallowing Reflection

The tilings shown in the pictures are minimal. Mike Reid found the solutions for the E and H pentacubes at scale 4.

Pentacube E

Pentacube H

Pentacube J

Pentacube Boxes Cube Tiles Minimal?

R 5×10×10 10×10×10 1000 —

S 4×10×10
6×10×10 10×10×10 1000 —

Pentacube	Boxes	Cube	Tiles	Minimal?
R		5×10×10	10×10×10	1000	—
S		4×10×10 6×10×10	10×10×10	1000	—

Chiral, Allowing Reflection

All tilings in the picture are minimal. The tiling shown for the S pentacube has diagonal rotary symmetry.

Pentacube Boxes Cube Tiles Minimal?

R 2×10×10 10×10×10 1000 —

Pentacube	Boxes	Cube	Tiles	Minimal?
R		2×10×10	10×10×10	1000	—

Last revised 2018-09-01.

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Col. George Sicherman [ HOME | MAIL ]