Squared Squares
There are catalogues by order, there are also listings by discoverer on the downloads page.
There are 3 main kinds of squared squares which have been of interest and recorded.
- SPSS's; Simple Perfect Squared Squares
- SISS's; Simple Imperfect Squared Squares
- CPSS's; Compound Perfect Squared Squares
SPSS's (Simple Perfect Squared Squares)
SPSS (Simple Perfect Squared Square), Order 21: 112 x 112 (AJWD)
Simple Perfect Squared Squares (SPSS's) are defined as Simple by having no smaller squared squares or squared rectangles in the dissection. and Perfect if the squares are all different sizes. They are considerably rare compared to perfect simple squared rectangles of the same order. According to David Gale[1], "Bouwkamp says that there are about 5,000,000 perfect simple squared rectangles to every such [squared] square (for order greater than 20)!". The lowest order SPSS appears alone in order 21.
SISS's (Simple Imperfect Squared Squares)
SISS (Simple Imperfect Squared Square), Order 13: 23 x 23
Simple Imperfect Squared Squares (SISS's) are Imperfect as not all the squares are different sizes (at least two squares are the same size). These are more numerous than Simple Perfect Squared Squares. The first one appears in order 13. Having squares of the same size can result in SISS's having symmetrical arrangements. SISS's can be used to 'derive' SPSS's two orders down.
CPSS's (Compound Perfect Squared Squares)
CPSS (Compound Perfect Squared Square), Order 24: 175 x 175(THW)
Compound Perfect Squared Squares (CPSS's) are defined as Compound by having smaller squared squares or squared rectangles in the dissection. Despite being composed of lower order SPSR's (Simple Perfect Squared Rectangles), these squared squares are rarer at any given order (>= 24) than SPSS's.
REFERENCES
- 'Tracking the Automatic Ant, and other Mathematical Explorations', by David Gale, A collection of Mathematical Entertainment Columns from the Mathematical Intelligencer, Ch 12, p87 & Ch 9