Then clap four slices of pilaster on 't,

That laced with bits of rustic makes a front.

That laced with bits of rustic makes a front.

—Alexander Pope,

Epistle to Richard Boyle

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.

Here I consider only tetrads that are themselves polyforms of the same type as their tiles.

The fifth tetrad was reported by Olexandr Ravsky in 2005.

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

The smallest tetrads for polyominoes with birotary symmetry about an edge use 14-ominoes:

The smallest tetrads for polyominoes with mirror symmetry about an edge use 18-ominoes:

The smallest tetrads for polyominoes with birotary symmetry about a vertex also use 18-ominoes:

Juris Čerņenoks found the smallest tetrads for polyominoes with diagonal symmetry, which use 19-ominoes:

The smallest polyominoes that form tetrads without 90° rotation are 13-ominoes:

The smallest known holeless tetrad for a symmetric polyomino was found independently by Frank Rubin and Karl Scherer. It uses 34-ominoes:

The smallest tetrad for a polyabolo with mirror symmetry uses a 26-abolo or 13-omino:

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:

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

The smallest tetrads made from polyiamonds with ternary symmetry about a vertex use 27-iamonds:

They are also the smallest polyiamonds that form tetrads without being rotated 60° or 180°.

The smallest tetrad for a polypent with bilateral symmetry about an edge uses a 12-pent:

The smallest tetrad for a polypent with birotary symmetry uses a 12-pent:

The smallest tetrads for polyhexes with vertical mirror symmetry use 7-hexes:

The smallest tetrads for polyhexes with birotary symmetry around a cell use 9-hexes:

The smallest tetrad for polyhexes with ternary symmetry uses 9-hexes:

The smallest tetrad for polyhexes with ternary symmetry about a cell uses 13-hexes:

The smallest holeless tetrad for symmetric polyhexes uses 9-hexes:

The smallest tetrad for a symmetric polyhept uses 15-hepts:

The smallest tetrads for a symmetric polyoct use 10-octs:

The smallest tetrads for polyocts with diagonal symmetry use 13-octs:

The smallest tetrads with symmetric polykites of odd order use 13-kites:

The smallest holeless tetrad with symmetric polykites uses 16-kites:

The smallest tetrad for polycairos with bilateral symmetry uses 18-cairos.

The smallest polydrafter with all cells on the polyiamond grid that forms a tetrad with some cells off the grid is this hexadrafter:

The smallest polydrafters on the polyiamond grid that form tetrads on the grid are these octadrafters:

*Last revised 2016-08-06.*

Back to Polyform Curiosities.

Col. George Sicherman [ HOME | MAIL ]