In
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
, a polyform is a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
figure or solid compound constructed by joining together identical basic
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s. The basic polygon is often (but not necessarily) a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
plane-filling polygon, such as a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
or a
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
. 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
polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in pop ...
es.
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
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgrap ...
,
connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties tha ...
). 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
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
can be joined along congruent faces. Joining
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
s in this way produces the
polycube
upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
A puzzle involving arranging nine L tricubes into a 3×3 cube
A polycube is a solid figure formed by j ...
s, and joining
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
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
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
; in the case of polyominoes, this results in
polyominoid
In geometry, a polyominoid (or minoid for short) is a set of equal Square (geometry), squares in Three-dimensional space, 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the polyominoes, which are just the pla ...
s.
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-polyomino
A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a plane geometric figure formed by joining one or more equal squares edge-to-edge or corner-to-corner at 90°. It is a polyform with square cells. The polyominoes are a ...
s, also known as polyplets or polykings.
Types and applications
Polyforms are a rich source of problems,
puzzle
A puzzle is a game, Problem solving, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together (Disentanglement puzzle, or take them apart) in a logical way, in order to arrive at th ...
s and
game
A game is a structured form of play (activity), play, usually undertaken for enjoyment, entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator s ...
s. The basic
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
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
upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
A puzzle involving arranging nine L tricubes into a 3×3 cube
A polycube is a solid figure formed by j ...
*
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in pop ...
*
Polyominoid
In geometry, a polyominoid (or minoid for short) is a set of equal Square (geometry), squares in Three-dimensional space, 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the polyominoes, which are just the pla ...
References
External links
*
''The Poly Pages'' at RecMath.org illustrations and information on many kinds of polyforms.
{{Polyforms