# 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 systematic names are shown in green. In both nomenclatures pentacubes that lie all in one plane are named for the corresponding pentominoes.

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 C E F H I J K L M N P Q R S T U V W X Y # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

### Scale 3

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

### Scale 3

SCALE 3 A B C E F 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 ]