| Tilings Of Regular Polygons |
Shopping Tiling |
Website Links For Regular |
Information AboutTilings Of Regular Polygons |
| CATEGORIES ABOUT TILING BY REGULAR POLYGONS | |
| euclidean plane geometry | |
| tiling | |
|
REGULAR TILINGS Following Grünbaum and Shephard (section 1.3), a tiling is said to be ''regular'' if the Symmetry Group of the tiling Acts Transitively on the ''flags'' of the tiling, where a flag is a triple consisting of a mutually incident Vertex , edge and tile of the tiling. This means that for every pair of flags there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an Edge-to-edge Tiling by Congruent regular polygons. There must be six Equilateral Triangle s, four Square s or three regular Hexagon s at a vertex, yielding the three ''regular tessellations''.
ARCHIMEDEAN, UNIFORM OR SEMIREGULAR TILINGS Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second. If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as ''Archimedean'', '' Uniform '' or ''semiregular'' tilings.
Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. COMBINATIONS OF REGULAR POLYGONS THAT CAN MEET AT A VERTEX The Internal Angle s of the polygons meeting at a vertex must add to 360 degrees. A regular -gon has internal angle degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a ''species'' of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one ''types'' of vertex. Only fifteen of these can occur in any tiling of regular polygons. For example, 3.7.42 cannot occur because it is the only type involving a regular Heptagon , meaning that triangles and 42-gons would need to alternate around the heptagon, which is impossible since it has an odd number of sides. With 3 polygons at a vertex:
With 4 polygons at a vertex:
With 5 polygons at a vertex:
With 6 polygons at a vertex:
OTHER EDGE-TO-EDGE TILINGS Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by regular polygons may be drawn. Here are four examples:
Such periodic tilings may be classified by the number of Orbits of vertices, edges and tiles. If there are orbits of vertices, a tiling is known as -uniform or -isogonal; if there are orbits of tiles, as -isohedral; if there are orbits of edges, as -isotoxal. The examples above are four of the twenty 2-uniform tilings. Chavey lists all those edge-to-edge tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal. TILINGS THAT ARE NOT EDGE-TO-EDGE Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may also be known as uniform if they are vertex-transitive; there are eight families of such uniform tilings, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles. BEYOND THE PLANE These tessellations are also related to regular and semiregular polyhedra and tessellations of the Hyperbolic Plane . Semiregular polyhedra are made from regular polygon faces, but their angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry have angles smaller than they do in the plane. In both these cases, that the arrangement of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive. Some regular tilings of the hyperbolic plane (Using Poincaré disc model projection) SEE ALSO
REFERENCES EXTERNAL LINKS
|
|
|