Why Euclid is Beautiful

 The beauty in Euclid's work does not come from the math that he discovered. In fact, he discovered very little of the math. Instead, the beauty comes from the connections that he made between all known math at the time. One could call this the "meta-math." He was able to sort through postulates that were seemingly inconsequential to one another and see the structure beneath.

Euclid has been so important for millennia because his structure showed mathematicians how to rigorously prove their ideas. The result of one's theory is only as good as the assumptions one makes at the beginning. "Garbage in, Garbage out." But with Euclid's structure, math became safeguarded against garbage entering the system. This garbage-less system is the source of beauty in his works.

Another source of beauty comes from knowing how much can be done with so little. By making only five assumptions (and a few "common knowledge postulates") all of human mathematical knowledge was laid out on the table. This beauty is like the beauty of a tree's branches spreading out from its trunk.

The final source of beauty came far after his death, when the fifth postulate was removed. For after its removal, an entire new field of knowledge sprung into existence. Or perhaps it was already there waiting to be discovered. This miraculous event is in my mind equivalent to marveling at a tree, digging through the soil underneath, and popping out into another realm with trees of its own.

Beauty in a system is simplicity. Euclid became the first person to show how complex ideas are just made from simple statements, and how those statements have the power to change entire fields of math.