A couple of previous posts (here and here) looked at Heesch classification of tessellation types. As I dug deeper into the geometry of tessellations, I found that there was another classification scheme that is popular. Branko Grünbaum and Geoffrey Shepard in their book Tilings and Patterns (Freeman, 1987) found 93 isohedral classes of tilings. Tiles in isohedral tilings must have the same shape and size, must fit together to fill the plane with no overlaps or gaps, and the tile must play the same role throughout the pattern, which means that it must fit with its neighbors in only one way. (So a pattern in which some of the tiles have five neighbors and some have six is not isohedral.) Twenty eight are the familiar Heesch types and another 20 fit Heesch categories but have additional restrictions on how some edges are formed. Some of these restricted cases result in tilings that fit into more than one Heesch type and about half result in tiles that have mirror symmetry. The other 45 have one or more lines that must be straight. For a complete list and a much more detailed explanation of these 93 types, see freespace.virgin.net/tom.mclean/index.html.
IH8 is a restricted TTTTTT/TCCTCC hexagon. Each pair of opposite sides is equal and parallel. The sides have a central point rotation and are translated to opposite side. It can be constructed with the template for the pentagon TCCTC type with the final C side eliminated. It is a challenge to find visually interesting patterns for this type as you can see in this example from TesselManiac!. (Note that the straight line needs to be eliminated, which means that the design must be copied in another drawing program. The same note holds for all of the types discussed in this and the next post.)
TesselManiac! not only gives the Heesch type for each of its templates but also the isohedral class (IH) number. In addition to the 28 Heesch types, TesselManiac! has templates for eight isohedral classes that have mirroring. (Twelve other restricted Heesch types do not have templates in TesselManiac! but most can, with only a little work, be formed using TesselManiac!, though to really see them properly the design must be taken out of TesselManiac! and redone in a different drawing program, eliminating unwanted lines. (I do this in Fontagrapher if I need to do it.) This and a future post illustrate these twelve restricted types and explain how Tesselmaniac can be made to produce them. (Kevin Lee seems not to have omitted these twelve from TesselManiac! because his focus was on Escher-like tessellations, not ornamental or abstract tessellations. )
IH8 is a restricted TTTTTT/TCCTCC hexagon. Each pair of opposite sides is equal and parallel. The sides have a central point rotation and are translated to opposite side. It can be constructed with the template for the pentagon TCCTC type with the final C side eliminated. It is a challenge to find visually interesting patterns for this type as you can see in this example from TesselManiac!. (Note that the straight line needs to be eliminated, which means that the design must be copied in another drawing program. The same note holds for all of the types discussed in this and the next post.)
IH11 is a restricted hexagon of that fits several Heesch types and is a special case of IH8 with more symmetry and thus more visual interest. Each of the six sides is identical with a central point rotation. It can be created in TesselManiac! in at least two ways. One way is with the CC6C6 template using only the C sides.
It can also be done with the C3C3C6C6 template.
The IH18 type is a more restricted variant of IH 10. It is a regular hexagon with all sides the same shape, but the sides also have central symmetry. It takes a little artistic talent to get it just right in TesselManiac! (but is easy in Fontographer because of its line manipulation tools). Below is an attempt to form it in TesselManiac!. (The straight lines forming the regular hexagons need to be eliminated.)
IH90 is the only restricted triangle in the IH classification system. It is a Heesch CCC type with all sides identical. It can be done in at least two ways in TesselManiac!. One way is with the CC3C3 template using only the C or central rotation sides.
It can also be done with C3C3C6C6 template using only the C6 sides. (Of course the straight lines need to be eliminated to get the final result.)
Most of what I term "restricted Heesch" classes change the symmetry of the tessellation. Tessellations that fit into most of these classes have additional symmetry, usually with with reflection. These tessellations belong to symmetry groups that have an "m" in their name such as cmm, pmg, and p31m.
Playing with this aspect of tessellations was an early step in what eventually has become a book unlike any of the others that I have published via CreateSpace. The title is Exploring Tessellations: A Journey through Heesch Types and Beyond. It is available from CreateSpace and Amazon. More information about it will be included in future posts.
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