That is another of your odd notions,said the Prefect, who had the fashion of calling everything

oddthat 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]

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. | |

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. |

Triking Oddities. Oddities with specific symmetries for pseudopolyominoes of order 3. | |

Pentaking Oddities. Oddities with specific symmetries for pseudopolyominoes of order 5. |

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. |

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

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. | |

Hexahex Tri-Oddities. Tri-oddities for hexahexes. |

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. |

Tetrahept Oddities. Oddities for tetrahepts. |

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

Pentenn Oddities. Oddities for pentenns. |

Polydec Oddities. Oddities for polydecs. |

Tetrahendec Oddities. Oddities for tetrahendecs. |

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

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. |

Polycairo Oddities. Oddities for polycairos. |

Polykite Oddities. Oddities for polykites. | |

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

Polygem Oddities. Oddities for polygems. | |

Trigem Pair Oddities. Oddities for pairs of trigems. |

Polyhing Oddities. Oddities for polyhings. |

Polykagome Oddities. Oddities for polykagomes. |

Polybird Oddities. Oddities for polybirds. |

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 ]