A SPIIRTS is an imperfect isosceles right triangled square with no properly contained pseudotriangle/rectangle/triangle subdivided into 3/2/2 or more elements respectively. Two equal elements may constitute a non-square parallelogram but not a properly contained square.
The SPIIRTS catalogues are available as pdfs from this page.
A tiling is said to be crossed when there is a tile-corner traversed by two lines. There is no known crossed SPIIRTS of order < 15. The properties below may precede "order:side" in a tiling's title:
d = double-pentagon patterned. Every such tiling is a subdivision of an instance of the same deformable tiling by two 45-90-90-90-225 pentagons with a shared side, four triangles and two pseudotriangles. The number of pentagons which are degenerate, in the sense that one or more sides have shrunk to zero length, may be 0 (as in 13:15TA and 13:15TB), 1 (as in 11:9TC and 11:10TD) or 2 (as in 8:4TA).
e = elegant. No tile-corner is just a T-junction. Such tilings may be considered aesthetically pleasing.
f = as few as two elements in every subquadrilateral (or an SPIIRTS with no subquadrilateral).
i = isomers exist which are ineligible for this catalogue. They are not included in the isomer count which follows 'of' in the tiling id.
z = zigzag by shorter sides of two or more equal tiles, pairs of which form parallelograms.
The catalogue was built by Geoffrey H. Morley but no SPIIRTSs are attributed to discoverers.