1 - Square / Ratio 2.
The rule is really easy : a=b, then one can try to add/subtract/multiply/divide by 2, 3.
2 - Golden ratio
The golden rule : 1.618 !
"At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing." from Wiki.
3 - Fibonacci number
Not far from the golden ratio, the sequence is as following:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89...
allowing the usual proportions 2/1, 3/2, 5/3, 8/5, 13/8...
See Wiki for more.
4 - Renard numbers.
Charles Renard (1847 - 1905) was a French military engineer. After the Franco-Prussian War of 1870/71 he started work on the design of air ships at the French army aeronautical department. He proposed a now widely used system of preferred numbers that was later named after him and became international standard ISO 3. It helped the French army to reduce the number of different balloon ropes kept on inventory from 425 to 17. From Wiki
Most important is the following sequence :
10 12 16 20 25 32 40 50 63 80 100
5 - The paper sheet ratio : ISO 216.
The international paper size standard, ISO 216, is based on the German DIN 476 standard for paper sizes. ISO paper sizes are all based on a single aspect ratio of square root of 2, or approximately 1/1.4142. From Wiki.
See more here :