SEMIREGULAR TESSELLATIONS Definition and notation We can extend the idea of a regular tessellation to the case where more than one regular polygon is used. In this case we have also to add a condition about the polygons circling each vertex. Example of semiregular tessellations are shown at right. A semiregular tessellation of the plane is a tessellation involving more than one type of regular polygon, in which the same arrangement of polygons occurs about every vertex. Notice that as before, we require at least three polygons at each vertex. Also, since the equilateral triangle has the smallest vertex angle amongst the regular polygons, there can be at most six polygons meeting at a vertex. We need a strategic approach. Here is an idea.
A simple way of exploring this is to write a simple computer program Here is a Pascal program that deals with Case 1 above. If three polygons {
Here is the program:
The number 50 in this program is rather arbitrary, but we shall see that the exact (largish) value is not important.
3.7.42, 3.8.24, 3.9.18, 3.10.15, 3.12.12, We now show that only the eight green sets correspond to semiregular tessellations. Thus far we have only looked at the polygon behaviours about a single vertex. But this behaviour has to be repeated at every vertex.
We observe that the question mark has to be replaced by You can now apply an exactly similar argument to eliminate 3.3.4.12, 3.4.3.12, 3.3.6.6, and 3.4.4.6. We are now left with eight possibilities – the green combinations. None of this guarantees that these will form semiregular tessellations: there may be some other constraint we haven’t investigated yet! But in fact we obtain precisely eight as shown below.
Some practical work |
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Duality
There is some interest in investigating the duals of the semiregular tessellations. However, whereas the duals of the regular tessellations are again regular, the duals of the semiregular tessellations are not themselves semiregular. For example, 4.4.8 gives rise to a tesselation of right-angled isosceles triangles. Real life occurrences It is harder to find occurrences of the semiregular tessellations in real life. The 4.4.8 tessellation is the most common, occurring in pavings, tilings and Japanese woodcraft. The first website below gives some further examples from Islanic art and old ornamentation. |
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References Semiregular tessellations Totally tessellated : http://library.thinkquest.org/16661/simple.of.regular.polygons/semiregular.1.html MathWorld : http://mathworld.wolfram.com/SemiregularTessellation.html |
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