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.
The PPIRTS catalogues are available as pdfs from this page.
- pdf of PPIRTSs order 11 (1 tiling) 3.2Kb
- pdf of PPIRTSs order 12 (3 tilings) 5.0Kb
- pdf of PPIRTSs order 13 (13 tiling) 14.1Kb
- pdf of PPIRTSs order 14 (38 tilings) 36.9Kb
- pdf of PPIRTSs order 15 (128 tilings) 128.9Kb
- pdf of PPIRTSs order 16 (486 tilings) 484.5Kb
- pdf of PPIRTSs order 17 (1187 tilings) 1.2Mb
- pdf of PPIRTSs order 18 (3027 tilings) 3.1Mb
- pdf of PPIRTSs order 19 (8141 tilings) 8.6Mb
- pdf of SPPIRTSs order 20 (7506 tilings) 8.1Mb
- 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.
The properties below may precede "order:side" in a tiling's title:
- c = crossed. There is a tile-corner traversed by two lines. The only known crossed PPIRTSs below order 20 are 19:35AB1of4 and 19:35AB4of4.
- 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. All below order 19 are degenerate in the sense that one or more sides of underlying tiles have shrunk to zero length. The non-degenerate d-tilings of order 19 are 19:221AA, 19:229AB and 19:241AA.
- e = elegant. No tile-corner is just a T-junction. Such tilings may be considered aesthetically pleasing. The only known elegant PPIRTSs below order 16 are 13:21AA, 14:26AJ, 14:35AA and 15:55AA. 20:125AF is the only known elegant PPIRTS with property 'f'.
- f = as few as two elements in every 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.
- r = rectangular inclusion. The only known PPIRTSs below order 16 with a rectangular inclusion are 13:18AA1-4of4 and 15:44AA1-4of4.
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)