# Pentacube Pair Pyramids

## Introduction

A pentacube is a solid formed by joining five equal cubes face to face. Here I show how copies of two pentacubes can be joined to make various pyramidal structures: quarter pyramids, half pyramids, full pyramids, and regular octahedra.

Click on a picture to show the structure of the tiling.

For the two-dimensional counterpart of this problem, see Erich Friedman's Math Magic for May 2006.

## The 29 Pentacubes

Here are the 29 pentacubes. Seventeen have mirror symmetry. The other 12 form six mirror pairs.

## Quarter Pyramids

5A + 5M5A + 5R5E + 5H5E + 5H′5E + 5R′5E + 5W
5F + 5H5F + 5W5H + 5M5H + 5R5H + 5R′5H + 5W
5H + 5Y5K + 5M5L + 5R5L + 5W5M + 5P5M + 5Q
5M + 5W5N + 5R5N + 5W5P + 5Y5Q + 5Y5R + 5V
5R + 5W5R + 5Y5W + 5Y

## Half Pyramids

5A + 5M5E + 5R5E + 5W5H + 5W5J + 5R5K + 5M
5L + 5W5M + 5N5M + 5Q5M + 5U5M + 5W5R + 5W
5R + 5Y5W + 5Y

## Full Pyramids

5A + 5M5H + 5M5M + 5N5M + 5P5M + 5Q5M + 5R
5M + 5W5R + 5X

## Octahedra

5I + 5M5M + 5W

Last revised 2019-02-11.

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Col. George Sicherman [ HOME | MAIL ]