![]() This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. You can work out the interior measure of the angles for each of these polygons: Shape Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other – a slide (or a glide!) is involved. When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.We can’t show the entire plane, but imagine that these are pieces taken from planes that have been tiled. ![]() A regular tessellation means a tessellation made up of congruent regular polygons. The word “tessellate” is derived from the Ionic version of the Greek word “tesseres,” which in English means “four.” The first tilings were made from square tiles.Ī regular polygon has 3 or 4 or 5 or more sides and angles, all equal. Rev.A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.Īnother word for a tessellation is a tiling.Ī dictionary * will tell you that the word “tessellate” means to form or arrange small squares in a checkered or mosaic pattern. Stephenson, J.: Ising model with antiferromagnetic next-nearest-neighbor coupling: spin correlations and disorder points. Radványi, A.G.: On the rectangular grid representation of general CNN networks. Räbinä, J., Kettunen, L., Mönkölä, S., Rossi, T.: Generalized wave propagation problems and discrete exterior calculus. In: Barneva, R., Brimkov, V.E., Šlapal, J. Nagy, B.: Weighted distances on a triangular grid. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 20, 63–78 (2004) Nagy, B.: Generalized triangular grids in digital geometry. In: International Symposium on Image and Signal Processing and Analysis conjunction with 23rd International Conference on Information Technology Interfaces 2001, pp. Nagy, B.: Finding shortest path with neighbourhood sequences in triangular grids. Nagy, B.: Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids. Nagy, B., Abuhmaidan, K.: A continuous coordinate system for the plane by triangular symmetry. Luczak, E., Rosenfeld, A.: Distance on a hexagonal grid. Kovalevsky, V.A.: Geometry of locally finite spaces: Computer agreeable topology and algorithms for computer imagery. ![]() In: Kropatsch, W.G., Artner, N.M., Janusch, I. Kovács, G., Nagy, B., Vizvári, B.: Weighted distances on the trihexagonal grid. In: Normand, N., Guédon, J., Autrusseau, F. Kovács, G., Nagy, B., Vizvári, B.: On weighted distances on the Khalimsky grid. Kovács, G., Nagy, B., Turgay, N.D.: Distance on the Cairo pattern. Kirby, M., Umble, R.: Edge tessellations and stamp folding puzzles. Her, I.: Geometric transformations on the hexagonal grid. Her, I.: A symmetrical coordinate frame on the hexagonal grid for computer graphics and vision. Grünbaum, B., Shephard, G.C.: Tilings by regular polygons. Keywordsīorgefors, G.: A semiregular image grid. Moreover, we are also presenting formulae to compute the digital, i.e., path-based distance based on the length of a/the shortest path(s) through neighbor tiles for these specific grids. For some of those grids, including the tetrille tiling D(6, 4, 3, 4) (also called deltoidal trihexagonal tiling and it is the dual of the rhombihexadeltille, T(6, 4, 3, 4) tiling), the rhombille tiling, D(6, 3, 6, 3) (that is the dual of the hexadeltille T(6, 3, 6, 3), also known as trihexagonal tiling) and the kisquadrille tiling D(8, 8, 4) (it is also called tetrakis square tiling and it is the dual of the truncated quadrille tiling T(8, 8, 4) which is also known as Khalimsky grid) we give detailed descriptions. The properties of the coordinate systems used to address the tiles are playing crucial roles. We show a general method to obtain coordinate system to address the tiles of these tessellations. In this paper, we are interested to their dual tessellations. There are eight semi-regular tessellations, they are based on more than one type of tiles. The square grid is self-dual, and the two others, the hexagonal and triangular grids are duals of each other. There are three regular ones, each of them using a sole regular tile. There are various tessellations of the plane.
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