Example of 4. a^2 + b^2 = c^2Scope is important. Start with: which of each of 1..5 has been terribly influential, a kind of big bang.

Example of 1 [$]e[$].

As an aside, I have nice little book called

*The Story of Mathematics*, which covers math from a cultural perspective. Thought the ages, some societies were focused on trade and accounting, others wanted to build fantastic structures and developed their math accordingly. Others looked at the skies and wanted to figure out what is happening out there. More math needed! Derivatives and quant finance came later, (but not as late as some people think!)

The books is not rigorous or laden with derivations, of course, but it is insightful and has many beautiful pictures of artifacts and mathematical art.

And to fill in another space in your list:

3. formula:

It will be fun to build the taxonomy and examples out in spare time this winter.