Often attributed to Kepler is a statement of the nature of the following. I don’t seem to have a copy of the Mysterium Cosmographicum handy, so I’ll parrot what I found at goldennumber.net: “Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” –Johannes Kepler I’ve always been a fan of Kepler since I was in Holland and studied 16th century science. As Sagan said, he was the last of the mathematical astrologer or the first astronomer.

As a person who took Trigonometry as an adult (at 29!), I was amazed by how much more fascinating the relationship between π and radii became. I even spent some time building squares of golden sections with a compass, just like the ancients used to. In this post below, Vi Hart makes a compelling argument that we should move on from π and move to τ. It’s a fascinating concept, but what I like most about it is that it still preserves some of the aesthetic mysteries that fundamental trigonometry provides.

Unicode bumper sticker I assembled the following answer by finding several disparate pieces to the puzzle, but I didn’t find an all-in-one explanation. So here’s how I did it. As someone who has studied several European languages and a notation system or two, I really like having access to the special symbols unique to a language. While OSX makes it easy to enter the standard Latin character set “special” glyphs like é thanks to ( ⌥+e, e ), other characters are not so easy to find without a “character picker” like logic and set theory “?” or the Dutch “?

I try to be realistic about AI and its impact on our world. My previous post explored how well Claude performed as a movie companion and philosophical conversationalist.

But let’s balance that with an example where multiple AIs - Gemini from Google, OpenAI, and even my beloved Claude - failed spectacularly. The prompt was simple:

Show me the book that introduced the graphical Cartesian coordinate system to the world.

None of these whole-dinosaurs-of power-consuming, copyright-trouncing-systems could deliver. Instead, they wrapped their “NO RESULTS FOUND” or “I don’t know” in purple prose apologies.

I was intrigued by a recent post by Alex Tiniuc about tool creation in software development. In “Unexpected Benefits of Building Your Own Tools”, Tiniuc describes how he’s been making a video game (main task) but that he took a detour to work on a simple tool for editing the inventory of “assets” (game-developer-speak for “stuff in the world”) possible in the game (side task). But once he returned to the main task, he found he’d boosted his productivity in such a way that he wasn’t in the weeds anymore; he was working at a higher level and, freed of drudgery, seeing ways to make the game (main task) better.¹

Tiniuc observes that when we develop small tools that save us cognitive burden, we create, in aggregate, space for inspiration and intuition to blossom. I found a fascinating echo in the case of Tiniuc and that of William Oughtred a 16^th century Anglican clergyman and mathematician.