These 4 tilings are in two types. The first type consists of the slab tilings, the apeirogonal prism and antiprism. The second type consists of blends formed by stacking these two slabs alternatingly. Slabs are a regular occurence in higher dimensional tilings, and things like the cyclosimplex antiprism will be used in stacks for time immemorial. These tilings all just barely see use in nonprismatic honeycombs. Azip's verf is a right isosceles triangle, azap's verf is a trapezoid, etrat's vrerf is a mirror symmetric pentagon formed by augmenting a trapezoid with a triangle, and retrat's verf is a mirror symmetric sombreroid formed by excavating a trapezoid with a triangle.
These two are the ones that are finite in one dimension. Both are convex, though they are often excluded from lists of convex uniform tesselations because of apeirogon discrimination.
35: Azip - Apeirogonal prism. Its faces are squares and two apeirogons. (Finally, we have an actual number of faces!) Symbols are x∞o x and x∞x x. Being wythoffian and not dodecagonal, this sees use in honeycombs regularly. Stacking this forever on itself makes squat.
36: Azap - Apeirogonal antiprism. Its faces are triangles and two apeirogons. Symbols are s∞o s and s∞s s. This is alternated azip. This is nonwythoffian, but its status as a simple flat tiling which is related to trat (which sees often use) leads it to be used often in snubby honeycombs anyway, including in the famous apsirsh. Stacking this forever on itself makes trat.
These two are formed by alternating stacking azips and azaps. The first is convex and has everything prograde, the second is nonconvex and has the azips and azaps alternate orientation such that everything gets very overlappy.
37: Etrat - Elongated triangular tiling. Its faces are triangles and squares. This is alternating azips and azaps, the naming comes from how you can consider it a trat with rows of squares inserted. It could also be considered a gyroelongated squat, being squat with rows of triangles inserted. This is our last convex tiling, and the only convex tiling to not be wythoffian or a wythoffian snub. It is only barely used in nonprismatic honeycombs due to nontrivial blends of its prisms.
38: Retrat - Retroelongated triangular tiling. Its faces are triangles and squares. This is also alternating azips and azaps, just retrograde now. Also can be considered a retrogyroelongated squat. Used in nonprismatic honeycombs in the same way etrat is.
Below shows how the laminates pair up as conjugates.
Conjuates:
The following have no conjugate: azap
The following are self-conjugate: azip
The following are conjugate pairs: etrat-retrat
Category 6 --- Back to Tilings Home --- Category 8
Site by ThePokemonkey123
contact me on discord you don't get my email
Images from Stella