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Polyform

2D shape constructed by joining together identical basic polygons

Polyform
Summary

2D shape constructed by joining together identical basic polygons

The 18 one-sided [[pentomino]]es: polyforms consisting of five squares.

In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as detailed in the table below. For example, a square basic polygon results in the well-known polyominoes.

Construction rules

The rules for joining the polygons together may vary, and must therefore be stated for each distinct type of polyform. Generally, however, the following rules apply:

  1. Two basic polygons may be joined only along a common edge, and must share the entirety of that edge.
  2. No two basic polygons may overlap.
  3. A polyform must be connected (that is, all one piece; see connected graph, connected space). Configurations of disconnected basic polygons do not qualify as polyforms.
  4. The mirror image of an asymmetric polyform is not considered a distinct polyform (polyforms are "double sided").

These construction rules are not meant to be set in stone, but rather serve as general guidelines as to how polyforms may be constructed. Modifications of the first construction rule, for example, lead to different polyforms. Joining at a common vertex may lead to polykings, and being joined not by edge, but by the chess movement of the knight may lead to polyknights.

Generalizations

Polyforms can also be considered in higher dimensions. In 3-dimensional space, basic polyhedra can be joined along congruent faces. Joining cubes in this way produces the polycubes, and joining tetrahedrons in this way produces the polytetrahedrons. 2-dimensional polyforms can also be folded out of the plane along their edges, in similar fashion to a net; in the case of polyominoes, this results in polyominoids.

One can allow more than one basic polygon. The possibilities are so numerous that the exercise seems pointless, unless extra requirements are brought in. For example, the Penrose tiles define extra rules for joining edges, resulting in interesting polyforms with a kind of pentagonal symmetry.

When the base form is a polygon that tiles the plane, rule 1 may be broken. For instance, squares may be joined orthogonally at vertices, as well as at edges, to form hinged/pseudo-polyominoes, also known as polyplets or polykings.

Types and applications

Polyforms are a rich source of problems, puzzles and games. The basic combinatorial problem is counting the number of different polyforms, given the basic polygon and the construction rules, as a function of n, the number of basic polygons in the polyform.

SidesBasic polygon (monoform)MonohedraltessellationPolyformApplications3image:Monoiamond.png4image:Monomino.png6image:Monohex.png
equilateral triangle[[File:Uniform triangular tiling 111111.svg80px]]DeltillePolyiamonds: moniamond, diamond, triamond, tetriamond, pentiamond, hexiamondBlokus Trigon
square[[File:Square tiling uniform coloring 1.svg80px]]QuadrillePolyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, nonomino, decominoTetris, Fillomino, Tentai Show, Ripple Effect (puzzle), LITS, Nurikabe, Sudoku, Blokus
regular hexagon[[File:Uniform tiling 63-t0.svg80px]]HextillePolyhexes: monohex, dihex, trihex, tetrahex, pentahex, hexahexTantrix
SidesBasic polygon (monoform)MonohedraltessellationPolyformApplications1image:Monostick.png3image:Monodrafter.pngimage:Monoabolo.png4image:Monomino.png[[Image:Monominoid.svg60px]]5[[File:Regular_polygon_5_annotated.svg60px]][[Image:Pentagonal Cairo Snub Square Tile.svg60px]]68[[Image:Regular_polygon_8_annotated.svg60px]]-[[Image:Circle-withsegments.svg60px]]
line segment (square)-Polysticks: monostick, distick, tristick, tetrastick, pentastick, hexastickSegment Displays
line segment (triangular)Polytrigs
line segment (hexagonal)Polytwigs: monotwig, ditwig, tritwig, tetratwig, pentatwig, hexatwig
30°-60°-90° triangle[[File:1-uniform 3 dual.svg80px]]KisrhombillePolydrafters: monodrafter, didrafter, tridrafter, tetradrafter, pentadrafter, hexadrafterEternity puzzle
right isosceles (45°-45°-90°) triangle[[File:1-uniform 2 dual.svg80px]]KisquadrillePolyaboloes: monabolo, diabolo, triabolo, tetrabolo, pentabolo, hexabolo, heptabolo, octabolo, enneabolo, decaboloTangram
30°-30°-120° isosceles triangle[[File:Tiling_truncated_6_dual_simple.svg80px]]KisdeltillePolypons: tripon, tetrapon
golden trianglePolyores
square (connected at edges or corners)[[File:Square tiling uniform coloring 1.svg80px]]QuadrillePolykings: pentaking, hexaking, heptaking
square (connected at edges, shifted by half)Polyhops: dihop, trihop, tetrahop
square (connected at edges in 3D space)Polyominoids: monominoid
square (representing path of a chess knight)Polyknights: tetraknight, pentaknight, hexaknightKnight in chess
rectangle[[File:Stacked bond.png80px]]Stacked bondPolyrects: tetrarect, pentarect, hexarect, heptarectBrickwork
trapezoidPolytraps: tritrap
rhombus[[File:Rhombic star tiling.svg80px]]RhombillePolyrhombs
60°-90°-90°-120° kite[[File:Tiling_small_rhombi_3-6_dual_simple.svg80px]]TetrillePolykites: trikite, tetrakite, pentakite, hexakite, heptakite
half-squaresPolyares: triare, tetrare, pentare, hexare
half-hexagonsPolyhes: monohe, dihe, trihe, tetrahe
regular pentagon-Polypents: monopent, dipent, tripent, tetrapent, pentapent, hexapent, heptapent
Cairo pentagon[[Image:Equilateral_Cairo_tiling.svg80px]]4-fold pentillePolycairoes
flaptile[[File:1-uniform_8_dual.svg80px]]Iso(4-)pentillePolyflaptiles: diflaptile, triflaptile, tetraflaptile
120°-120°-120°-120°-60° pentagon[[Image:Tiling_snub_3-6_left_dual_simple.svg80px]]6-fold pentillePolyflorets
RombikPolyrombiks
regular octagon (with squares)Polyocts: dioct
quarter of circular arcPolybends
circle (with concave circles as bridges)Polyrounds
quarter of circle, and quarter-circle sector removed from a squarePolyarcs: monarc, diarc, triarc
EdgesBasic polytope (monoform)MonohedralhoneycombPolyformApplications12[[Image:Cube-h.svg60px]]32[[Image:Hypercube.svg60px]]
cube[[File:Cubic_honeycomb.png80px]]CubillePolycubes: monocube, dicube, tricube, tetracube, pentacube, hexacube, heptacube, octacubeSoma cube, Bedlam cube, Diabolical cube, Snake cube, Slothouber–Graatsma puzzle, Conway puzzle, Herzberger Quader
half-cubesPolybes: monobe, dibe, tribe, hexabe
tesseract[[File:Tesseractic_tetracomb.png80px]]Tesseractic honeycombPolytesseracts

References

References

  1. "Polyplet".
  2. "The Poly Pages".
  3. "Rombix - Illustrated booklet".
  4. "A Periodic Table of Polyform Puzzles".
  5. "PolyHypercubes".
Wikipedia Source

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