Polyform Oddities

That is another of your odd notions, said the Prefect, who had the fashion of calling everything odd that was beyond his comprehension, and thus lived amid an absolute legion of oddities.

—Poe, The Purloined Letter

A polyform oddity is a geometric figure with binary symmetry or better, formed by joining an odd number of congruent polyforms. Oddities are also known as Sillke figures, after Torsten Sillke, who first studied them systematically.

This page shows minimal known oddities and similar constructions for various polyforms with various symmetries.

[Polyominoes and Polykings] [Polyiamonds and Polymings] [Polyhexes, Polyhings, and Polybrexes] [Polypents] [Polyhepts] [Polyocts] [Polyenns] [Polydecs] [Polyhendecs] [Polydodecs] [Polyaboloes/Polytans] [Polycairos] [Polykites] [Polygems] [Polykagomes] [Polybirds] [Polycubes]

Polyiamonds and Polymings

	Pentiamond, Heptiamond, and Enneiamond Oddities. Oddities for pentiamonds, heptiamonds, and enneiamonds. Oddities for polyiamonds of odd order can have only bilateral symmetry.
	Hexiamond Oddities. Hexiamond oddities with specific symmetries.
	Hexiamond Pair Oddities. Full-symmetry polyiamonds tiled with an odd number of two different hexiamonds.
	Pentiamond-Hexiamond Oddities. Full-symmetry polyiamonds tiled with an odd number of copies of a pentiamond and a hexiamond.
	Octiamond Oddities. Octiamond oddities with specific symmetries.
	Polyiamond Tri-Oddities. These are like oddities but with ternary symmetry, for orders 1–6.
	Hexiamond Pair Tri-Oddities. Tri-oddities formed with copies of two hexiamonds.
	Heptiamond Tri-Oddities. These are like oddities but with ternary symmetry, for order 7.
	Tri-Oddities for Heptiamond Pairs. Arrange copies of two heptiamonds to form a figure with ternary symmetry, with the tile count not a multiple of 3.
	Horizontally Symmetric Tri-Oddities for Heptiamond Pairs. Arrange copies of two heptiamonds to form a figure with ternary symmetry and horizontal mirror symmetry, with the tile count not a multiple of 3.
	Tetraming Oddities. Oddities for tetramings, or corner-connected tetriamonds.

Polyhexes, Polyhings, and Polybrexes

	Trihex Oddities. Trihex oddities with specific symmetries.
	Tetrahex Oddities. Tetrahex oddities with specific symmetries.
	Pentahex Oddities. Oddities for pentahexes, with specific symmetries.
	Pentahex Pair Oddities. Full-symmetry polyhexes tiled with an odd number of two different pentahexes.
	One-Sided Pentahex Oddities. Oddities for pentahexes that may not be reflected.
	The Magic Pentahex Pair Polyhex Oddity. The same odd polyhex is tiled with 153 different pairs of pentahexes.
	Tetrahex-Pentahex Oddities. Full-symmetry polyhexes tiled with an odd number of copies of a tetrahex and a pentahex.
	Trihex-Pentahex Oddities. Full-symmetry polyhexes tiled with an odd number of copies of a trihex and a pentahex.
	Hexahex Oddities. Hexahex oddities with specific symmetries.
	Heptahex Oddities. Oddities for polyhexes with 7 cells, with specific symmetries.
	Polyhex Tri-Oddities. Tri-oddities for polyhexes of order up to 5.
	Pentahex Pair Tri-Oddities. Tri-oddities for pairs of pentahexes.
	Pentahex Pair Tri-Oddities with Horizontal Symmetry. Horizontally symmetric tri-oddities for pairs of pentahexes.
	Pentahex Pair Tri-Oddities with Vertical Symmetry. Vertically symmetric tri-oddities for pairs of pentahexes.
	Hexahex Tri-Oddities. Tri-oddities for hexahexes.
	Polyhing Oddities. Oddities for polyhings.
	Tribrex Oddities. Oddities for tribrexes.
	Tetrabrex Oddities. Oddities for tetrabrexes.

Polypents

	Tetrapent Oddities. Oddities for tetrapents.
	Tetrapent Pair Oddities. Oddities for pairs of tetrapents.
	Tetrapent Triplet Oddities. Oddities for sets of three tetrapents.
	Pentapent Oddities. Oddities for pentapents.

	Polyabolo Oddities. Oddities for polyaboloes.
	Diabolo-Triabolo Pair Oddities. Full-symmetry oddities for a diabolo and a triabolo.
	Triabolo-Tetrabolo Pair Oddities. Full-symmetry oddities for a triabolo and a tetrabolo.
	Diabolo-Pentabolo Pair Oddities. Full-symmetry oddities for a diabolo and a pentabolo.

Polycairos

Polycairo Oddities. Oddities for polycairos.

Polykites

	Polykite Oddities. Oddities for polykites.
	Trikite-Tetrakite Pair Oddities. Joint oddities for a trikite and a tetrakite.

Polygems

	Polygem Oddities. Oddities for polygems.
	Trigem Pair Oddities. Oddities for pairs of trigems.

	Minimal Oddities for the L Tricube. Oddities for the L tricube with every even polycube symmetry.
	Minimal Oddities for the P Pentacube. Oddities for the P pentacube with every even polycube symmetry.
	Pentacube Oddities with Full Symmetry. Full-symmetric oddities for pentacubes.
	Pentacube Oddities with Orthogonal Mirror Symmetry. Orthogonal mirror-symmetric oddities for pentacubes.
	Pentacube Oddities with Diagonal Mirror Symmetry. Diagonal mirror-symmetric oddities for pentacubes.
	Pentacube Oddities with Orthogonal Rotary Symmetry. Orthogonal rotationally symmetric oddities for pentacubes.
	Pentacube Oddities with Plane Diagonal Rotary Symmetry. Plane diagonal rotationally symmetric oddities for pentacubes.
	Pentacube Oddities with Inverse Symmetry. Point-symmetric oddities for pentacubes.
	Pentacube Oddities with 4-Rotary Symmetry. Rotary 90° oddities for pentacubes.
	Pentacube Oddities with Dual Orthogonal Mirror Symmetry. Rectangular-symmetric oddities for pentacubes.
	Pentacube Oddities with Dual Diagonal Mirror Symmetry. Oblique rectangular-symmetric oddities for pentacubes.
	Pentacube Oddities with Square Symmetry. Square (dual orthogonal + 4-rotary) oddities for pentacubes.
	Pentacube Oddities with Square Box Symmetry. Dicubic (square-box) oddities for pentacubes.
	Pentacube Pair Full Oddities. Full-symmetry oddities formed by copies of two pentacubes.
	Tetracube-Pentacube Pair Full Oddities. Full-symmetry oddities formed by copies of a tetracube and a pentacube.

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Col. George Sicherman [ HOME | MAIL ]