# Simple Primitive Imperfect Isosceles Right Triangled Squares (SPIIRTSs)

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.

### Catalogues

The SPIIRTS catalogues are available as pdfs from this page.

- pdf of SPIIRTS order 2 (1 tiling) 2.9Kb
- pdf of SPIIRTSs order 8 (2 tilings) 3.8Kb
- pdf of SPIIRTS order 9 (1 tiling) 3.1Kb
- pdf of SPIIRTSs order 10 (10 tilings) 10.3Kb
- pdf of SPIIRTSs order 11 (37 tilings) 32.8Kb
- pdf of SPIIRTSs order 12 (145 tilings) 125Kb
- pdf of SPIIRTSs order 13 (423 tilings) 376.4Kb
- pdf of SPIIRTSs order 14 (966 tilings) 891.0Kb

### Properties

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.

### Credit for Discovery

The catalogue was built by Geoffrey H. Morley but no SPIIRTSs are attributed to discoverers.