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Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. [6] Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner ...
One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
A regular tiling has one type of regular face. A semiregular or uniform tiling has one type of vertex, but two or more types of faces. A k-uniform tiling has k types of vertices, and two or more types of regular faces. A non-edge-to-edge tiling can have different-sized regular faces.
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a word play on ein Stein, German for "one stone". [2]
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon , 108°, is not a divisor of 360°, the angle measure of a whole turn .
In geometry, the rhombille tiling, [1] also known as tumbling blocks, [2] reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets ...
A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, [3] [4] ruling out any of the Platonic solids other than the cube. Five space-filling convex polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra: