Polyform Tetrads

Then clap four slices of pilaster on 't,
That laced with bits of rustic makes a front.
—Alexander Pope, Epistle to Richard Boyle
  • Introduction
  • Polyominoes
  • Polyhops
  • Polyaboloes
  • Polyiamonds
  • Polypents
  • Polyhexes
  • Polyhepts
  • Polyocts
  • Polykites
  • Polycairos
  • Polydrafters
  • Arbitrary Shapes
  • Introduction

    A tetrad is a plane 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.

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

    Polyominoes

    The smallest polyomino tetrads are made from octominoes:

    The fifth tetrad was reported by Olexandr Ravsky in 2005.

    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:

    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:

    Restricted Motion

    These octominoes form tetrads without being reflected:

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

    Holeless

    The smallest holeless polyomino tetrad uses 11-ominoes:

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

    Polyhops

    The smallest polyhop or polybrick tetrad uses 7-hops:

    Holeless

    The smallest polyhops that form holeless tetrads use 10-hops:

    Polyaboloes

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

    Holeless

    Juris Čerņenoks found the smallest holeless polyabolo tetrads, using 16-aboloes:

    Symmetric Tiles

    The smallest tetrad for a symmetric polyabolo uses 22-aboloes:

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

    Polyiamonds

    The smallest polyiamond tetrads use 10-iamonds. According to Karl Scherer, Frank Rubin may have found the first solution. 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 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:

    Restricted Motion

    The smallest polyiamonds that form tetrads without being reflected are these 10-iamonds:

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

    Holeless

    The smallest holeless tetrad made from a polyiamond with mirror symmetry uses 22-iamonds. It was found independently by Robert Ammann, Greg Frederickson, and Jean L. Loyer.

    Polypents

    The two smallest polypent tetrads use a hexapent:

    Symmetric Tiles

    The smallest tetrads with symmetric polypents use 11-pents:

    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:

    Polyhexes

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

    Symmetric Tiles

    The smallest tetrad for a polyhex with birotary symmetry uses 6-hexes:

    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:

    Polyhepts

    The smallest polyhept tetrad uses 9-hepts:

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

    Polyocts

    The smallest polyoct tetrads use 6-octs:

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

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

    Polykites

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

    Symmetric Tiles

    … and the smallest tetrad with symmetric polykites, using 8-kites:

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

    Holeless

    Juris Čerņenoks found the smallest holeless polykite tetrad, using 9-kites:

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

    Polycairos

    The smallest polycairo tetrads use a 7-cairo:

    Holeless

    The smallest holeless polycairo tetrad uses a 9-cairo:

    Symmetric Tiles

    The smallest symmetric polycairo that forms a tetrad is this 16-cairo:

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

    Polydrafters

    The smallest polydrafter tetrad, allowing cells off the polyiamond grid, uses a tetradrafter:

    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:

    Arbitrary Shapes

    Livio Zucca found the polygon with the fewest edges that forms a tetrad:

    Last revised 2016-08-06.


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