# Pentacube Nomenclature

• Introduction
• Serena Sutton Besley Tollefson's Nomenclature
• Kate Jones's Nomenclature
• C. J. Bouwkamp's Nomenclature
• Stewart Coffin's Nomenclature
• Ekkehard Künzell's Nomenclature
• Abaroth's Nomenclature
• Donald Knuth's Nomenclature
• My Nomenclature
• Reconciliation
• Acknowledgments
• ## Introduction

A pentacube is a solid made of five cubes joined face to face. It is the 3-dimensional counterpart of a pentomino.

In 1939 the Fairy Chess Review published as Problem 3930 an inquiry by W. D. Rawlings of Pointe Claire, P.Q., Canada: in how many ways can five equal cubes be joined face to face?

The next year, in Volume 4, Number 5 (April 1940), the editor reported that R. J. French had proved that there are 23 such shapes, counting mirror images as the same shape.

In working with sets of pentacubes, mirror images are usually distinguished, since one image cannot be repositioned to make the other. The Fairy Chess Review reported in 1948 that F. Hansson and others had counted 29 pentacubes, counting mirror images as distinct. Some investigators have counted 30, overlooking the plane diagonal mirror symmetry of one of the nonflat pentacubes.

Here are the 29 pentacubes. The flat pentacubes appear first, then the nonflat pentacubes with mirror symmetry, then the six mirror pairs.

The 29 pentacubes have no generally accepted system of names. Below I describe some published ones. If you know of others, please let me know.

## Serena Sutton Besley Tollefson's Nomenclature

In 1962, Serena Sutton Besley Tollefson obtained a patent for a puzzle in which the thirty pentacubes were to be arranged to form various rectangular boxes. She got thirty by counting one piece twice. For her patent filing, she numbered the pieces as shown below:

## Kate Jones's Nomenclature

Kate Jones manufactures games and puzzles that use polyforms. She marketed her first product, Quintillions®, in 1979. It was a set of the 12 solid pentominoes—flat pentacubes. They could be used like pentominoes. They could also make 3D shapes.

Later Kate produced Super Quintillions®, an expansion set for Quintillions. It contains 18 non-flat pentacubes, one of each of the 17, and an extra copy of the pentacube labeled J3/L3. Since 29 is a prime number, there are advantages to having 30 pentacubes.

Kate's system names the solid pentominoes after Solomon's Golomb's names for the pentominoes. Most of the other names come from the names of solid tetrominoes that they contain. A number is added to tell which cell of the tetracube has a bump.

Torsten Sillke described Kate's system as human-friendly.

## C. J. Bouwkamp's Nomenclature

In 1980, Springer published a book called The Mathematical Gardner. It was a collection of articles produced to honor the recreational mathematics writer Martin Gardner.

One article, by mathematician Christoffel Bouwkamp, is called Packing Handed Pentacubes. It deals only with the 12 chiral pentacubes. Bouwkamp numbered them as shown:

## Stewart Coffin's Nomenclature

Stewart Coffin published his book The Puzzling World of Polyhedral Dissections in 1990. In chapter 3 he gives the 29 pentacubes labels of the form 5-N, where N is a number from 1 to 29.

## Ekkehard Künzell's Nomenclature

Around 1992 mathematics teacher Ekkehard Künzell published a game, Reservat, using pentacubes. He devised a system of two-digit numbers to identify the pentacubes. Künzell's system was adopted by Torsten Sillke, a pioneer of combinatorial geometry.

## Abaroth's Nomenclature

Abaroth maintains a vast website about polyforms and puzzles that use them. He published his numeration of the 29 pentacubes around 2015.

## Donald Knuth's Nomenclature

In 2022 Donald Knuth, the eminent computer scientist, published Volume 4B of his series The Art of Computer Programming. It gives much attention to polyforms as a subject for programming. (I can confirm that writing computer programs to solve polyform problems develops one's ingenuity, sometimes drastically.)

Knuth's system names the flat pentacubes after John Conway's names for the pentominoes. They are the letters from O to Z inclusive, which is convenient for programming. Knuth uses lower-case letters for the same reason, except that for chiral pairs of pentacubes he uses upper-lower-case pairs of letters:

## My Nomenclature

In 2006 I began to study polycubes. After a while I devised my own nomenclature for the pentacubes.

In those days I usually identified mirror images, so I had only 23 pentacubes to name. Naturally I used Roman letters. Later I distinguished chiral pairs by using a prime mark ():

## Reconciliation

This table shows the correspondence for the eight systems shown above:

 Besley Jones Bouwkamp Coffin Künzell Abaroth Knuth Sicherman 9 1 2 4 12 6 7 5 10 8 3 11 30 28 15/16 27 29 18 17 20 19 14 13 23 22 24 21 26 25 F I L N P T U V W X Y Z A T2 J3/L3 T1 Q L2 J2 L1 J1 L4 J4 S2 N2 S1 N1 V1 V2 5 6 2 1 4 3 9 10 11 12 7 8 5-6 5-12 5-9 5-11 5-1 5-3 5-7 5-2 5-5 5-8 5-10 5-4 5-28 5-25 5-15 5-24 5-29 5-18 5-14 5-17 5-13 5-19 5-16 5-23 5-21 5-22 5-20 5-26 5-27 70 10 11 40 60 80 90 13 30 50 12 20 37 82 81 51 61 71 72 21 22 41 42 33 34 31 32 36 35 6 1 2 4 5 7 8 9 10 11 3 12 26 27 25 28 29 21 22 15 16 13 14 19 20 17 18 23 24 r o q s p t u v w x y z j n l m k b B a A c C e E d D F f F I L N P T U V W X Y Z A B K M Q E E′ S S′ J J′ R R′ H H′ G G′

## Acknowledgments

I have relied heavily on Donald Knuth's historical research as presented in his Volume 4B, Combinatorial Algorithms, Part 2.

My copy of the book is a gift from Dr. Knuth.

Last revised 2024-01-19.

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