Polyform

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The 18 one-sided pentominoes: polyforms consisting of five squares. All 18 Pentominoes.svg
The 18 one-sided pentominoes: 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.

Contents

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. [1]

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.

Regular polyforms
SidesBasic polygon (monoform)Monohedral
tessellation
PolyformApplications
3 Monoiamond.png equilateral triangle Uniform triangular tiling 111111.svg
Deltille
Polyiamonds: moniamond, diamond, triamond, tetriamond, pentiamond, hexiamond Blokus Trigon
4 Monomino.png square Square tiling uniform coloring 1.svg
Quadrille
Polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, nonomino, decomino Tetris, Fillomino, Tentai Show, Ripple Effect (puzzle), LITS, Nurikabe, Sudoku, Blokus
6 Monohex.png regular hexagon Uniform tiling 63-t0.svg
Hextille
Polyhexes: monohex, dihex, trihex, tetrahex, pentahex, hexahex Tantrix
Other low-dimensional polyforms
SidesBasic polygon (monoform)Monohedral
tessellation
PolyformApplications
1 Monostick.png line segment (square)- Polysticks: monostick, distick, tristick, tetrastick, pentastick, hexastick Segment Displays
line segment (triangular) Polytrigs
line segment (hexagonal) Polytwigs: monotwig, ditwig, tritwig, tetratwig, pentatwig, hexatwig
3 Monodrafter.png 30°-60°-90° triangle 1-uniform 3 dual.svg
Kisrhombille
Polydrafters: monodrafter, didrafter, tridrafter, tetradrafter, pentadrafter, hexadrafter Eternity puzzle
Monoabolo.png right isosceles (45°-45°-90°) triangle 1-uniform 2 dual.svg
Kisquadrille
Polyaboloes: monabolo, diabolo, triabolo, tetrabolo, pentabolo, hexabolo, heptabolo, octabolo, enneabolo, decabolo Tangrams
30°-30°-120° isosceles triangle Tiling truncated 6 dual simple.svg
Kisdeltille
Polypons: tripon, tetrapon
golden triangle Polyores
4 Monomino.png square (connected at edges or corners) Square tiling uniform coloring 1.svg
Quadrille
Polykings: 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, hexaknight Knight in chess
rectangle Stacked bond.png
Stacked bond
Polyrects: tetrarect, pentarect, hexarect, heptarect Brickwork
trapezoid Polytraps: tritrap
Monominoid.svg rhombus Rhombic star tiling.svg
Rhombille
Polyrhombs
60°-90°-90°-120° kite Tiling small rhombi 3-6 dual simple.svg
Tetrille
Polykites: trikite, tetrakite, pentakite, hexakite, heptakite
half-squares Polyares: triare, tetrare, pentare, hexare
half-hexagons Polyhes: monohe, dihe, trihe, tetrahe
5 Regular polygon 5 annotated.svg regular pentagon - Polypents: monopent, dipent, tripent, tetrapent, pentapent, hexapent, heptapent
Pentagonal Cairo Snub Square Tile.svg Cairo pentagon Equilateral Cairo tiling.svg
4-fold pentille
Polycairoes
flaptile [2] 1-uniform 8 dual.svg
Iso(4-)pentille
Polyflaptiles: diflaptile, triflaptile, tetraflaptile
120°-120°-120°-120°-60° pentagon Tiling snub 3-6 left dual simple.svg
6-fold pentille
Polyflorets
6 Rombik [3] Polyrombiks [4]
8 Regular polygon 8 annotated.svg regular octagon (with squares) Polyocts: dioct
- quarter of circular arc Polybends
Circle-withsegments.svg circle (with concave circles as bridges) Polyrounds
quarter of circle, and quarter-circle sector removed from a square Polyarcs: monarc, diarc, triarc
High-dimensional polyforms
EdgesBasic polytope (monoform)Monohedral
honeycomb
PolyformApplications
12 Cube-h.svg cube Cubic honeycomb.png
Cubille
Polycubes: monocube, dicube, tricube, tetracube, pentacube, hexacube, heptacube, octacube Soma cube, Bedlam cube, Diabolical cube, Snake cube, Slothouber–Graatsma puzzle, Conway puzzle, Herzberger Quader
half-cubes Polybes: monobe, dibe, tribe, hexabe
32 Hypercube.svg tesseract Tesseractic tetracomb.png
Tesseractic honeycomb
Polytesseracts [5]

See also

References

  1. Weisstein, Eric W. "Polyplet". MathWorld .
  2. http://www.recmath.com/PolyPages/PolyPages/index.htm?Polyflaptiles.htm
  3. https://schoengeometry.com/b-fintil-media/little_red_book.pdf
  4. https://www.gamepuzzles.com/PeriodicTableofPolyformPuzzles.pdf
  5. https://www.iread.it/lz/polyhypercubes.html