Tilings by Isosceles Right Triangles

IRTS by Geoffrey Morley

Geoffrey Morley calls a tiling;

A fourth catalogue is for 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.

Tilings of Rectangles can also be constructed. The attached pdf, by Geoffrey Morley, shows Simple Perfect Isosceles Right Triangled Rectangles (SPIRTRs) with aspect ratios of 1x2, 1x3, 1x4, 1x5 and 1x6.

A Pseudotriangle (Pseudoquadrangle) is a polygon with three (four) convex corners and an arbitrary number of reflex angles.
Precisely which orders of which catalogues are complete is not known.
Any letters which appear before order:side depict properties of the tiling: