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
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
or a
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
. 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").
Generalizations
Polyforms can also be considered in higher dimensions. In 3-dimensional space, basic
polyhedra can be joined along congruent faces. Joining
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
s in this way produces the
polycube
image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricube ...
s, and joining
tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
s 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-polyominos, also known as polyplets or polykings.
Types and applications
Polyforms are a rich source of problems,
puzzles and
game
A game is a structured type of play usually undertaken for entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator sports or video games) or art ...
s. 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.
See also
*
Polycube
image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricube ...
*
Polyomino
*
Polyominoid
References
External links
*
''The Poly Pages'' at RecMath.org illustrations and information on many kinds of polyforms.
{{Polyforms