To create a decoration of a semi-regular tessellation we again use reflection at additional lines. Each corner point is shared by n polygons. Using a kaleidoscope with a right-angled triangle characterized by the numbers (k 2 n) we get a decoration of a regular tessellation with regular polygons that have k corners. We can use the method discussed above to get semi-regular tessellations of the hyperbolic plane. Images of the yellow region and parts of it make a decoration of the semi-regular tessellation. The triangle of red lines shows the position of the mirrors of a kaleidoscope. Regular tessellation with hexagons (blue lines) and semi-regular tessellation with hexagons and triangles (brown lines). In this sense we have made a decoration of the semi-regular tiling. Thus we get an image with additional mirror symmetries that emphasize the semi-regular tessellation. We cover the other rose-coloured part of the triangle doing a mirror image at the border of the triangle. We note that one of these border lines cuts the red triangle in two parts and we only use the pixels of the input image that lie inside the larger part, which is coloured yellow. To get a decoration of the semi-regular tiling we need additional mirror symmetries at the borders between the triangles and the hexagons. This emphasizes the regular tessellation of hexagons because there are mirror symmetries at the borders of the hexagons. Mirror reflections cover the entire plane with copies of this image part. It is shown in red lines in the image and uses the part of an input image that fits into this triangle. A decoration of the regular tessellation arises from a triangular kaleidoscope with angles of 30°, 60° and 90°. Each triangle is made up of three corners cut off from the hexagons. Drawing lines between the mid points of adjacent sides we get a semi-regular tessellation of triangles and hexagons. Let us take as an example the regular tessellation of hexagons shown in the image below. On the other hand, with computer generated mirror symmetries this becomes possible. With real physical mirrors we probably can’t do it. But what about semi-regular tilings? Could we decorate them too using mirrors? This would give us new designs. Using (a) and (b), find all possible pairs $(m,n)$įor a regular tessellation of the plane.In the last posts I have shown kaleidoscopes that make repeating images in Euclidean, spherical and hyperbolic spaces. Show that for any such tesselation, we must have $m \geq 3$ and, using part (a), that $n \leq 6$. In this problem you will discover some very strong restrictions on possible tesselations of the plane, stemming from the fact that that each interior angle of an $n$ sided regular polygon measures $\frac\right) = 360. Of a regular tessellation which can be continued indefinitely in all directions: The checkerboard pattern below is an example If any two polygons in the tessellation either do not meet, share a vertex only, If all polygons in the tessellation are congruent regular polygons and For example, part of a tessellation with rectangles is A tessellation of the plane is an arrangement of polygons which cover the plane without gaps or overlapping.
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