![]() ![]() The most common type of tessellation is that formed by rectangular and particularly square mosaics. In the case that a single type of mosaic formed by a regular polygon is used, then a regular tessellation, but if two or more types of regular polygons are used then it is a semi-regular tessellation.įinally, when the polygons that form the tessellation are not regular, then it is a irregular tessellation. In this way, there are no spaces left uncovered and the tiles or mosaics do not overlap. Tiles or tiles are flat pieces, generally polygons with congruent or isometric copies, which are placed following a regular pattern. They are everywhere: in streets and buildings of all kinds. The tessellated are surfaces covered by one or more figures called tiles. Example 12: tessellation in video games. ![]() ![]() Example 6: rhombi-tri-hexagonal tessellation.Example 5: Blunt hexagonal tessellation. ![]() Note that in some cases it was easier to use angles of $60/120/180$ in the drawing, but be aware that any angle could be used.Video: 12.1 Tessellations of Regular and Irregular Polygons Content The one in the question corresponds to type IG88, and if I have not overlooked any, there are 15 other periodic equilateral isogonal tilings, as shown in this drawing: A few however have a degree of freedom remaining that allow some of the faces to be distorted into non-regular shapes. those that are k-isogonal for k>1), and it seems to me that the situation is much the same as with monohedral tilings in that it is hard to find them all without a lot of computer assistance.Īll 93 aforementioned isogonal tiling types are topologically equivalent to semi-regular tilings, so if you impose straight edges of the same length, most of them revert to those semi-regular tilings. I don't think much is known about other monogonal tilings (i.e. I don't know how many of these can be equilateral. In Grunbaum and Shephard's brilliant book Tilings and Patterns, they classify all isogonal tilings, finding 93 types, two of which need extra markings on the faces or edges to distinguish their symmetries from the others. So the tilings you are interested in are monogonal and equilateral (and presumably periodic). This is analogous to isohedral, and similarly the term corresponding to 2-isohedral is 2-isogonal, where the symmetries of the tiling as a whole splits the vertices into two equivalence classes, etcetera. Note that the edges do not need to be all the same length, but their arrangement around each vertex must be the same.Ī tiling is called isogonal if the tiling as a whole has symmetries that map one vertex onto any other vertex. This is analogous to the term monohedral for tilings that use a single type of tile shape. each vertex with its incident edges) are all the same are called monogonal. Tilings where the vertex arrangements (i.e.
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