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

## Polyominoes

 Polyomino Oddities. Oddities for polyominoes of order up to 7. Pentomino Oddities. Pentomino oddities with specific symmetries. One-Sided Pentomino Oddities. One-sided pentomino oddities with specific symmetries. Pentomino Pair Full Oddities. Full-symmetry polyominoes tiled with an odd number of two different pentominoes. Pentomino Pair Dual Orthogonal Oddities. Polyominoes with dual orthogonal mirror symmetry tiled with an odd number of two different pentominoes. Pentomino Pair Dual Diagonal Oddities. Polyominoes with dual diagonal mirror symmetry tiled with an odd number of two different pentominoes. Pentomino Pair 4-Rotary Oddities. Polyominoes with 4-rotary symmetry tiled with an odd number of two different pentominoes. Tetromino-Pentomino Pair Oddities. Use copies of a tetromino and a pentomino to form a full-symmetry polyomino with an odd number of cells. Hexomino Oddities. Hexomino oddities with specific symmetries. Polyomino Semi-Oddities. Semi-oddities for polyominoes. A semi-oddity is a figure with quaternary symmetry and an even number of tiles that is not a multiple of 4. Heptomino Oddities. Heptomino oddities with specific symmetries.

## Polykings

 Triking Oddities. Oddities with specific symmetries for pseudopolyominoes of order 3. Pentaking Oddities. Oddities with specific symmetries for pseudopolyominoes of order 5.

## Polyiamonds

 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.

## Polymings

 Tetraming Oddities. Oddities for tetramings, or corner-connected tetriamonds.

## Polyhexes

 Polyhex Oddities. Oddities for polyhexes of order up to 5, with specific symmetries. Pentahex Pair Oddities. Full-symmetry polyhexes tiled with an odd number of two different pentahexes. 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. Polyhex Tri-Oddities. Tri-oddities for polyhexes of order up to 5. Pentahex Pair Tri-Oddities. Tri-oddities for pairs of pentahexes. Hexahex Tri-Oddities. Tri-oddities for hexahexes.

## 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.

## Polyhepts

 Tetrahept Oddities. Oddities for tetrahepts.
Pentahept Oddities. Oddities for pentahepts.

## Polyocts

 Polyoct Oddities. Oddities for polyocts of order 1 through 5.

## Polyenns

 Pentenn Oddities. Oddities for pentenns.

## Polydecs

 Polydec Oddities. Oddities for polydecs.

## Polyhendecs

 Tetrahendec Oddities. Oddities for tetrahendecs.

## Polydodecs

 Polydodec Oddities. Oddities for polydodecs with 1–4 cells.

## Polyaboloes/Polytans

 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.

## Polyhings

 Polyhing Oddities. Oddities for polyhings.

## Polykagomes

 Polykagome Oddities. Oddities for polykagomes.

## Polybirds

 Polybird Oddities. Oddities for polybirds.

## Polycubes

 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.

Back to Polyform Curiosities
Col. George Sicherman [ HOME | MAIL ]