Interesting rectangles

Question

Any rectangle representing a sheet of A series paper has an interesting property, i.e. when bisected, the two resulting rectangles are similar to the original one.

Generalizing the A series paper concept, any rectangle having area $A=k$, length $a=\sqrt[4]{n}\:k$ and width $b=\frac{k}{\sqrt[4]{n}}$ for $k\in \mathbb{R}^+,\:n\in\mathbb{Z}^+\setminus \left \{ 1 \right \}$ is interesting. Case $n=2$ corresponds to bisecting the rectangle and getting two rectangles similar to the original one, $n=3$ trisecting it and getting three rectangles similar to the original one, etc.

Another rectangle everybody knows is the golden rectangle as it is used in the simplest definition of the golden ratio $\varphi%$.

What other oblong rectangles have nice or interesting mathematical properties?

Answer 1

Japanese tatami mats have an aspect ratio of 2:1. This isn't a very interesting ratio, admittedly. But the many ways that tatami mats can be arranged in a room leads to some non-trivial combinatorics problems that have been considered by Donald Knuth, among others. Some patterns are said to be "auspicious" and some are "inauspicious".

Some real-life patterns.

Some of the "inauspicious" rules.

Some mathematics here and here.

Answer 2

I had a good one for a square, but the q asked for an oblong rectangle so I go with a 2x3 rectangle.

1) Divide it into unit squares.

2) Label the outside vertices of the grid of squares A, B, C, ..., J in rotational order, with ABCD aling one long edge of the large rectangle.

3) Draw triangle AEH which is right isosceles (the legs are corresponding diagonals of two perpendicularly oriented rectangles).

4) From the angles at E prove than the arctangents of one-half and one-third add up to $\pi/4$ radians or 45 degrees.

Answer 3

Besides the golden rectangle corresponding to the golden ratio, there are rectangles corresponding to other ratios called "metallic ratios" (with the golden ratio being one of them).

E.g., there is a rectangle corresponding to the "silver" ratio, "bronze" ratio, etc. They have an interesting property: in the case of the silver ratio, a silver rectangle is a rectangle such that if you remove two squares along its longer side, then the rectangle you end up with is also a silver rectangle. And so on with the other metallic ratio rectangles.