# 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-2^{2}),
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 C 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

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

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

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.

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

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 | — |

Last revised 2018-09-01.

Back to Polyform Tiling
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Polyform Curiosities

Col. George Sicherman
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