Individual tilings are accessible from the menus on the left. All collections of tilings can also be downloaded. The SPIIRTS's catalogs are available as pdfs from this page.

- pdf of SPIIRTS order 2 (1 tiling) 4k
- pdf of SPIIRTS order 8 (2 tilings) 7k
- pdf of SPIIRTS order 9 (1 tiling) 5k
- pdf of SPIIRTS order 10 (10 tilings) 15k
- pdf of SPIIRTS order 11 (37 tilings) 41k
- pdf of SPIIRTS order 12 (141 tilings) 144k
- pdf of SPIIRTS order 13 (410 tilings) 418k
- pdf of SPIIRTS order 14 (945 tilings) 977k

We call a tiling *primitive* if it does not contain a triangle or 45-45-45-225 pseudotriangle subdivided into more than one or two triangles respectively. Otherwise we call the tiling *derivative*. Previously a Simple Perfect Right-Angled Isosceles Triangled Square (SPRAITS), as it was called in [3], was called *secondary* if it had property 's' (explained below). Otherwise it was called *primary*.

*Isomers* (tilings with the same set of tiles) are ordered by their XB code elements (giving
side length and orientation, not coordinates). The sets of isomers with the same order:side are ordered by the XB code elements of the first (or only) tiling in each set. For each order:side, tilings in the same set are allocated the first or next 2-letter id from a range associated with the catalog.

Property 'b' described below is currently indicated instead by an asterisk (erroneously omitted from some tilings). **The following remarks, including all the other properties, will apply only when the expanded and new catalogs have been implemented.**

Besides additions and corrections, the January 2005 catalogs of primary and secondary SPRAITS's have been expanded to include all known primitive (order < 20) and derivative (order < 19) ultraperfect squares respectively (*PPIRTS's* and *DUIRTS's*). There are new catalogs for Non ultraperfect Primitive IRTS's (*NPIRTS's*) and Simple Primitive Imperfect IRTS's (*SPIIRTS's*). Only DUIRTS's and NPIRTS's are likely to be complete, or nearly so.

The possible properties of a tiling which precede "order:side" are:

- b = is a subdivision of an instance of the same edge-to-edge deformable tiling T (by 4 triangles, 2 pseudotriangles and 2 pentagonal pseudoquadrangles) as is 20:129AD. In every b-tiling of order<19 T is degenerate, with only one corner element. There are thousands of b-tilings of order 20: only the few in which T is nondegenerate are cataloged.
- q = has isomer(s) ineligible for this catalog.
- r = includes a subrectangle. Only simple PPIRTS's (SPPIRTS's) are cataloged for order 20.
- s = includes a 45-45-45-225 pseudotriangle subdivided into more than two triangles.
- t = includes a subdivided triangle. Omitted from NPIRTS's.
- x = excludes 45-45-45-225 pseudotriangles. Such tilings may be more aesthetically pleasing.
- z = includes a two-element non-square parallelogram.

Only ultraperfect triangled squares and the two order 7 perfect squares (each of which includes an order 6 perfect triangle) are attributed to discoverers:

- J. Douglas and E.P. Starke (
**D&S**, United States) (7:10PA only) - Geoffrey H. Morley (
**GHM**, England) - Jasper D. Skinner, II (
**JDS**, United States) - Arthur H. Stone (
**AHS**, United States, 1916-2000) (7:7PA only) - William T. Tutte (
**WTT**, Canada, 1917-2002) (15:44AI, 17:136AJ and 19:56AJ only)

**References**

- R.L. Brooks, C.A.B. Smith, A.H. Stone, and W.T. Tutte, The dissection of rectangles into squares,
*Duke Math. J.***7**(1940) 312-340. - A.H. Stone, proposer; M. Goldberg, solver. Problem E476,
*Amer. Math. Monthly***48**(1941) 405 and**49**(1942) 198-199. - J.D. Skinner II, C.A.B. Smith, and W.T. Tutte, On the Dissection of Rectangles into Right-Angled Isosceles Triangles,
*Journal of Combinatorial Theory Series B***80**(2000), No.2, 277-319. [ISSN: 0095-8956] - John Wilson: Axiomatic circuit theory (abstract of two lectures on this topic with "an application to the dissection of rectangles into right-angled isosceles triangles") at www.math.fau.edu/cgtc/cgtc36/se36.html (March, 2005).

Geoff Morley, 2007.