The ratio is known.
Golden ratio - Wikipedia, the free encyclopedia
Was a big deal to Euclid, Plato, Pythagoras, da Vinci, etc, etc, etc. Called the Divine Proportion. The Parthenon, etc.
"Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics."
For example,
Amazon.com: golden ratio
16:9 is so close, yet so far.

It is closer to 16:10. In time between 4:3 and 16:9, wide screen monitors were 16:10.