# Simple Primitive Imperfect Isosceles Right Triangled Squares (SPIIRTS's)

### Catalogs

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.

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.

### Credit for Discovery

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

1. 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.
2. A.H. Stone, proposer; M. Goldberg, solver. Problem E476, Amer. Math. Monthly  48 (1941) 405 and 49 (1942) 198-199.
3. 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]
4. 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.