# Interesting rectangles

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?

• Fold any A paper so that the upper side meets the left one. You'll see half of a square and a rectangle below. Now fold the lower rectangle along its diagonal from lower left to upper right -- you'll get a triangle with two equal sides ... Try this with a letter paper as well. – Michael Hoppe Oct 16 '16 at 9:17

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 mathematics here and here.

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.

• Perhaps not so much an interesting property of a $2\times3$ rectangle than an interesting proof that uses a $2\times3$ rectangle in its construction. Still, +1. (The last sentence would be much clearer if it said "This proves that..." rather than "Prove that...") – Rahul Nov 13 '16 at 23:45