I wanted to write a computer program that visualizes those tessellations, but I didn't find a good strategy which colors should be used. A regular tessellation is a pattern made by repeating a regular polygon. For example, a triangle’s three angles total 180 degrees which is a divisor of. Regular tessellations have interior angles that are divisors of 360 degrees. There are three types of regular tessellations: triangles, squares and hexagons. So my question is: Does every regular tessellation of the hyperbolic plane admit a nontrivial symmetric coloring? Regular tessellations are tile patterns made up of only one single shape placed in some kind of pattern. A shorthand notation (‘Schläfli symbol’) for this is 3.3.4.3.4. An example of a semi-regular tessellation is that with triangletrianglesquaretrianglesquare in cyclic order, at each vertex. The first ones are called Regular Tessellations. The three regular tessellations are for convenience included here as special cases of semi-regular ones. A common real-life example of tessellation patterns would be floor tiles. Are mosaics an example of tessellation Tessellation refers to a pattern of 2D shapes that fit perfectly together, without gaps. Certain shapes that are not regular can also be tiled. For the other platonic solids there are also those colorings that assign the same colors only to opposite faces. There are different types of tessellations. Regular polygons are tiled if the interior angles can be added to form 360°. The only nontrivial symmetric colorings of the tetrahedron, is the one, that assigns a different color to each face. ($p:G\rightarrow $Sym$(C)$ is a group homomorphism).Įxamples for such colorings are the trivial coloring $c:F\rightarrow \$ or the coloring of the plane as an infinite chessboard. Download scientific diagram Examples of regular tessellations in the plane When classifying various types of tessellations, the concept of vertex configuration is very useful. I want to define a symmetric coloring of the tessellation as a surjective map from $c:F\rightarrow C$ to a finite set of colors $C$, such that for each group element $G$ there is a permutation $p_g$ of the colors, such that $c(gx)=p_g\circ c(x)$. either a platonic solid (a tessellation of the sphere), the tessellation of the euclidean plane by squares or by regular hexagons, or a regular tessellation of the hyperbolic plane. Consider a two-dimensional tessellation with regular -gons at each polygon vertex.
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