Polyform
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:
- Two basic polygons may be joined only along a common edge, and must share the entirety of that edge.
- No two basic polygons may overlap.
- 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.
- 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.
| Sides | Basic polygon (monoform) | Monohedral tessellation |
Polyform | Applications | |
|---|---|---|---|---|---|
| 3 | 
|
equilateral triangle |  Deltille |
Polyiamonds: moniamond, diamond, triamond, tetriamond, pentiamond, hexiamond | Blokus Trigon |
| 4 | 
|
square |  Quadrille |
Polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, nonomino, decomino | Tetris, Fillomino, Tentai Show, Ripple Effect (puzzle), LITS, Nurikabe, Sudoku, Blokus |
| 6 | 
|
regular hexagon |  Hextille |
Polyhexes: monohex, dihex, trihex, tetrahex, pentahex, hexahex | Tantrix |
| Sides | Basic polygon (monoform) | Monohedral tessellation |
Polyform | Applications | |
|---|---|---|---|---|---|
| 1 | File: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 | File:Monodrafter.png | 30°-60°-90° triangle | File:1-uniform 3 dual.svg Kisrhombille |
Polydrafters: monodrafter, didrafter, tridrafter, tetradrafter, pentadrafter, hexadrafter | Eternity puzzle |
| File:Monoabolo.png | right isosceles (45°-45°-90°) triangle | File:1-uniform 2 dual.svg Kisquadrille |
Polyaboloes: monabolo, diabolo, triabolo, tetrabolo, pentabolo, hexabolo, heptabolo, octabolo, enneabolo, decabolo | Tangram | |
| 30°-30°-120° isosceles triangle | File:Tiling truncated 6 dual simple.svg Kisdeltille |
Polypons: tripon, tetrapon | |||
| golden triangle | Polyores | ||||
| 4 |
|
square (connected at edges or corners) | 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 | File:Stacked bond.png Stacked bond |
Polyrects: tetrarect, pentarect, hexarect, heptarect | Brickwork | ||
| trapezoid | Polytraps: tritrap | ||||
| File:Monominoid.svg | rhombus | File:Rhombic star tiling.svg Rhombille |
Polyrhombs | ||
| 60°-90°-90°-120° kite | File: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 | File:Regular polygon 5 annotated.svg | regular pentagon | - | Polypents: monopent, dipent, tripent, tetrapent, pentapent, hexapent, heptapent | |
| File:Pentagonal Cairo Snub Square Tile.svg | Cairo pentagon | File:Equilateral Cairo tiling.svg 4-fold pentille |
Polycairoes | ||
| flaptile[2] | File:1-uniform 8 dual.svg Iso(4-)pentille |
Polyflaptiles: diflaptile, triflaptile, tetraflaptile | |||
| 120°-120°-120°-120°-60° pentagon | File:Tiling snub 3-6 left dual simple.svg 6-fold pentille |
Polyflorets | |||
| 6 | Rombik[3] | Polyrombiks[4] | |||
| 8 | File:Regular polygon 8 annotated.svg | regular octagon (with squares) | Polyocts: dioct | ||
| - | quarter of circular arc | Polybends | |||
| File: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 | ||||
| Edges | Basic polytope (monoform) | Monohedral honeycomb |
Polyform | Applications | |
|---|---|---|---|---|---|
| 12 | File:Cube-h.svg | cube | File: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 | File:Hypercube.svg | tesseract | File:Tesseractic tetracomb.png Tesseractic honeycomb |
Polytesseracts[5] | |
See also
References
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- ^ Page Module:Citation/CS1/styles.css has no content."The Poly Pages". www.recmath.com. Retrieved 2025-11-25.
- ^ Page Module:Citation/CS1/styles.css has no content."Rombix - Illustrated booklet" (PDF). Archived from the original (PDF) on 2016-05-06.
- ^ Page Module:Citation/CS1/styles.css has no content."A Periodic Table of Polyform Puzzles" (PDF). Archived (PDF) from the original on 2020-09-27.
- ^ Page Module:Citation/CS1/styles.css has no content."PolyHypercubes". www.iread.it. Retrieved 2025-11-25.
External links
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- The Poly Pages at RecMath.org, illustrations and information on many kinds of polyforms.
- Poly Puzzles, for polyform related puzzles.
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