Primitive Perfect Isosceles Right Triangled Squares (PPIRTSs)

Primitive Perfects are not just perfect but ultraperfect because they do not include triangled triangles. The main method of finding one by hand is to begin by drawing a plausible-looking dissection of a square into a pseudoquadrangle and at least 8 triangles, all of unspecified sizes. If the relative dimensions of the triangles can be determined by simple algebra (using 2 or 3 unknowns) and the triangles are unequal then, having done that, attempt to triangulate the pseudoquadrangle. If the dimensions are indeterminate then any triangulation of the pseudoquadrangle will make them determinate.

Catalogues

The PPIRTS catalogues are available as pdfs from this page.

  1. pdf of PPIRTSs order 11 (1 tiling) 3.2Kb
  2. pdf of PPIRTSs order 12 (3 tilings) 5.0Kb
  3. pdf of PPIRTSs order 13 (13 tiling) 14.1Kb
  4. pdf of PPIRTSs order 14 (38 tilings) 36.9Kb
  5. pdf of PPIRTSs order 15 (128 tilings) 128.9Kb
  6. pdf of PPIRTSs order 16 (486 tilings) 484.5Kb
  7. pdf of PPIRTSs order 17 (1187 tilings) 1.2Mb
  8. pdf of PPIRTSs order 18 (3027 tilings) 3.1Mb
  9. pdf of PPIRTSs order 19 (8141 tilings) 8.6Mb
  10. pdf of SPPIRTSs order 20 (7506 tilings) 8.1Mb
  11. pdf of SPPIRTSs order 20 (6598 tilings) 7.1Mb

For order 20, only Simple PPIRTSs (SPPIRTSs) are catalogued. Of the thousands of these with property 'd' (see below) only the 13 in which no side of any underlying tile has shrunk to zero length are catalogued.

Properties

The properties below may precede "order:side" in a tiling's title:

Credit for Discovery

Just three people are credited with the discovery of Primitive Perfects:

Geoffrey H. Morley (GHM, England)

Jasper D. Skinner, II (JDS, United States)

William T. Tutte (WTT, Canada, 1917-2002) (15:44AI, 17:136AJ and 19:56AJ only)