geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite

polyhedron 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 ...

or a

tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling * Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People * Heinrich Sylvester ...

of the plane.Conway (2008), p.263

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...

called it a ''2-dimensional semi-check'', like a single row of a

checkerboard A checkerboard (American English) or chequerboard (British English; see spelling differences) is a board of checkered pattern on which checkers (also known as English draughts) is played. Most commonly, it consists of 64 squares (8×8) of altern ...

. If the sides are

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 a ...

s, it is a

uniform tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and Hyperbolic space, hyperbolic plane. Uniform tilings ar ...

. If colored with two sets of alternating squares it is still uniform. File:Infinite prism alternating.svg, Uniform variant with alternate colored square faces. File:Infinite_bipyramid.svg, Its

dual tiling In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

is an ''apeirogonal bipyramid''.

Related tilings and polyhedra

The apeirogonal tiling is the arithmetic limit of the family of prisms t or ''p''.4.4, as ''p'' tends to

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...

, thereby turning the prism into a Euclidean tiling. An alternation operation can create an

apeirogonal antiprism In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane. If the sides are equilateral triangles, i ...

composed of three triangles and one

apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to ...

at each vertex. : Similarly to the

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also f ...

and the

s, eight uniform tilings may be based from the regular

apeirogonal tiling In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include: * Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces * Order ...

. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the

Notes

References

* T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, Apeirogonal tilings Euclidean tilings Isogonal tilings Prismatoid polyhedra {{polyhedron-stub