This page and the following two pages show the principle of extention, or segments, in the research of uniform hyperbolic honeycombs.
The figure on the left shows my rendition of the tiling that Ctrnáct Marek found, and to the right, my doodle using squares in place of triangles.
In hyperbolic geometry, if there is a repeatition of faces around a vertex or face, this period can be repeated any number of times. This produces an extended series of snub faces.
Here, Marek's triangles are replaced by squares. While constructing this example I realized this produced a new kind of snub.
While the principles of this extention may appear simple in hindsight, it has taken the effort of many people to draw these gems from the realm of hyperbolic space. Such explorations are made a lot easier with Tyler.
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