Polyform Tetrads

Introduction

A tetrad is a figure made of four congruent shapes, joined so that each shares a boundary with each. Michael R. W. Buckley first used the name tetrad in an article in the Journal of Recreational Mathematics, volume 8.

Martin Gardner's book Penrose Tiles to Trapdoor Ciphers (Freeman, 1989; ISBN 0-7167-1987-8) shows some plane constructions by Scott Kim, including a holeless tetrad for a 12-omino, a holeless tetrad for a 26-iamond with mirror symmetry, and a holeless tetrad for a tetrahex. Karl Scherer shows many varieties of tetrads at Wolfram.

Polyominoes

The smallest polyomino tetrads are made from octominoes:

Holeless

The smallest holeless polyomino tetrad uses 11-ominoes:

Symmetric Tiles

The smallest tetrad for a polyomino with mirror symmetry uses 13-ominoes:

The smallest tetrad for a polyomino with birotary symmetry also uses 13-ominoes:

Here are all the tetrads for symmetrical 14-ominoes:

Here are all the tetrads for symmetrical 15-ominoes:

Polyaboloes

The smallest polyabolo tetrads, found by Juris Čerņenoks, use 12-aboloes:

Polyiamonds

The smallest polyiamond tetrads use 10-iamonds. The purple solutions were found by Juris Čerņenoks.

Symmetric Tiles

The smallest tetrad made from a polyiamond with mirror symmetry uses 12-iamonds:

The smallest holeless tetrad made from a polyiamond with mirror symmetry uses 22-iamonds:

I am not sure that it is unique.

The smallest tetrad made from a polyiamond with birotary symmetry uses 16-iamonds:

The smallest tetrads made from polyiamonds with horizontal mirror symmetry use 17-iamonds:

Polypents

The two smallest polypent tetrads use a hexapent:

Polyhexes

The smallest polyhex tetrads use tetrahexes. The first, by Scott Kim, is from Gardner's book.

Here is the smallest tetrad for a polyhex with birotary symmetry:

These are the smallest tetrads for polyhexes with vertical mirror symmetry:

Polyocts

The smallest polyoct tetrads use hexocts:

Polykites

Juris Čerņenoks found the smallest polykite tetrad, using 7-kites …

… and the smallest holeless polykite tetrad, using 9-kites:

Last revised 2013-01-13.

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Col. George Sicherman [ HOME | MAIL ]