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.


The SPIIRTS catalogues are available as pdfs from this page.

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


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.