u/FriddyHumbug

Love bats and now there are some here

Pulled this today and didn't notice until later. Left side of card is ok. But there seems to be something wrong with the cut geometry on the top side of the top-right corner and right side of the bottom-right corner. No edge damage.

Left figure "Trapezille" is derived from the Deltoidal Trihexagonal Tiling. All vertices where four faces meet of the DTT lie on the edges of a Triangular Tiling. The Trapezille is created by sliding these four-faced vertices left or right along this edge until the face morphs from a kite, to a trapezoid, which can be divided into three equilateral triangles. The resulting tiling has 3-fold chiral symmetry, but the outer shells of the trapezoids combine into equilateral triangles and regular hexagons that have no such chirality.

Middle figure "Rhombiditrigonal Tiling" is the dual of the Trapezoidal Ditrigonal Tiling. Less elegant, but it bears a similarity to the Rhombitrihexagonal tiling, which is why it carries its name, except that the regular hexagons have morphed into ditrigons, meeting short edge to long, the squares turned to stout trapezoids, and the triangles picking up a subtle twistedness and asymmetry from the operation. The three-faced vertices in the Trapezille are the only ones that are equiangular and whose edges are identical, so only the triangles in the Rhombiditrigonal Tiling are regular. It is also chiral, but unlike the Snub Trihexagonal Tiling the ditrigons are connected have one face between eachother. The figure is vertex-transitive, but because the Trapezille is not vertex-regular, the dual is not face-regular.

Right figure "Kistrapezille" is the result of applying the kis operator to the Trapezille which was an intermediate step to obtaining the dual, but I liked it enough to keep it around. It produced irregular faces in each polygon since the trapezoid is not compatible with the traditional operation, but produced quite a striking figure made of various scalene and isosceles triangles meeting in myriad ways. I shall have my bathroom tiled this way and the tile setter is going to hate me for it.