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 and Polynars
  • Polyhops
  • Polyaboloes
  • Polyiamonds
  • Polypents
  • Polyhexes
  • Polyhepts
  • Polyocts
  • Polykites
  • Polycairos
  • Polydrafters
  • Polydoms
  • Arbitrary Shapes
  • Introduction

    A tetrad is a plane figure made of four congruent shapes, joined so that each shares a boundary with each.

    The first to study tetrads was Walter Trump, in 1970 or 1971. He recorded his findings in notebooks, but did not publish them. In 2020 he presented them on his website Further Questions About Tetrads. It contains a great deal of general and specific information about tetrads.

    Michael R. W. Buckley first used the name tetrad in a 1975 article in the Journal of Recreational Mathematics, volume 8. In 1979 Trump sent many of his results to Martin Gardner, including the holeless 11-omino tetrad shown under Polyominoes and Polynars.

    Martin Gardner's book Penrose Tiles to Trapdoor Ciphers (Freeman, 1989; ISBN 0-7167-1987-8) also 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 and Polynars

    See Polyomino and Polynar Tetrads.

    Polyhops

    The smallest polyhop or polybrick tetrad uses 7-hops:

    Holeless

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

    Polyaboloes

    For similar or scaled tetrads, in which the four pieces are similar but need not be congruent, see Scaled Polytan Tetrads.

    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:

    Dr. Karl Scherer found this holeless tetrad for a symmetric polyabolo. It uses a 278-abolo with only 27 edges:

    Polyiamonds

    See Polyiamond Tetrads.

    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

    See Polyhex Tetrads.

    Polyhepts

    The smallest polyhept tetrad uses 9-hepts:

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

    The smallest tetrads for polyhepts with mirror symmetry around an edge use 16-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

    See Polydrafter Tetrads.

    Polydoms

    For similar or scaled tetrads, in which the four pieces are similar but need not be congruent, see Scaled Polydom Tetrads.

    The smallest polydom tetrad uses a hexadom:

    There are 28 heptadoms that can form tetrads:

    Arbitrary Shapes

    Around 1979 Walter Trump found the the fewest edges for a polygon that can form a tetrad:

    Later Livio Zucca found another pentagon solution:

    Last revised 2020-03-18.


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