Aspect ratio ordered squared rectangles reveal similar squared rectangles and suggest ways of combining them into squared squares. There are many coprime low integer aspect ratios which have no known corresponding simple perfect squared rectangle.
Recent Discoveries Brian Trial of Ferndale, Michigan, U.S.A. has discovered many 1:n aspect ratio simple perfect squared rectangles (SPSRs) ranging from 1:4 to 1:18.
Here is the attached pdf showing some of his discoveries (also arranged in portrait form for closer viewing).
Aspect ratios up to 1:2 can be found if simple perfect squared rectangle catalogues up to order 20's are searched for low integer aspect ratios. Some particular aspects which may be of interest include values near 1.618. This gives rectangle approximations to Fibonacci ratios which may be used in certain tiling constructions such as the Fibonacci golden rectangle and golden square.
Skinner has found a 1:3 simple perfect squared rectangle in order 26. Finding simple squared rectangles of the form 1:n, where n > 1 has been much harder than finding simple squared squares, but with recent results that has changed.
format; (width+height) (width-height) (widthxheight) (width/height [aspect ratio]) order width height e1 e2 e3 ... en (elements in bouwkamp code order);
for example
65 1 1056 1.03 9 33 32 9 10 14 8 1 7 4 18 15
Expect more low integer aspect ratio results from higher order SPSR to be included here in the near future.