# Tiling a Scaled Pentacube with a Pentacube

## Introduction

A pentacube is a solid made of five equal cubes joined face to face. There are 23 such figures, not distinguishing reflections and rotations:

The six blue tiles have distinct mirror images. Kate Jones's names are shown in green. Donald Knuth's names are shown in red.

Here I show which pentacubes can tile a pentacube scaled up by a factor of two or three. If the tile is the same pentacube as that being scaled, the tiling is called a reptile. See Polycube Reptiles.

To see an example of a tiling, click on the # in the appropriate cell of the table.

## Disallowing Reflection

These tables show results when the tile may not be reflected.

### Scale 2

SCALE 2 A B E F G H I J K L M N P Q R S T U V W X Y # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #
SCALE 3 A B E F G H I J K L M N P Q R S T U V W X Y # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

## Allowing Reflection

These tables show results when the tile may be reflected; that is, the tile and its mirror image may both be used in the tiling.

### Scale 2

SCALE 2 A B E F G H I J K L M N P Q R S T U V W X Y # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

### Scale 3

SCALE 3 A B E F G H I J K L M N P Q R S T U V W X Y # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

Last revised 2020-04-23.

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Col. George Sicherman [ HOME | MAIL ]