Tag Archives: algebra

Back to Plato’s Line

Last February I posted about how my friend Tina, who writes the Diotima’s Ladder blog, asked for some help with a set of diagrams for her novel. The intent was to illustrate an aspect of Plato’s Divided Line — an analogy about knowledge from his worldwide hit, the Republic. Specifically, to demonstrate that the middle two (of four) segments always have equal lengths.

The diagrams I ended up with outlined a process that works, but I was never entirely happy with the last steps. They depended on using a compass to repeat a length as well as on two points lining up — concrete requirements that depend on drawing accuracy.

Last week I had a lightbulb moment and realized I didn’t need them. Lurking right in front of my eyes is a solid proof that’s simple, clear, and fully abstract.

Continue reading


Plato’s Divided Line

Recently my friend Tina, who writes the blog Diotima’s Ladder, asked me if I could help her with a diagram for her novel. (Apparently all the math posts I’ve written gave her ideas about my math and geometry skills!)

What she was looking for involved Plato’s Divided Line, an analogy from his runaway bestseller, the Republic (see her post Plato’s Divided Line and Cave Allegory for an explanation; I’m not going to go into it much here). The goal is a geometric diagram proving that the middle two segments (of four) must be equal in length.

This post explores and explains what I came up with.

Continue reading


It’s A Noether Day!

Last week we celebrated Albert Einstein’s birthday (he turned 140). Now we need another cake so we can celebrate the other March major mathematician’s birthday — Emmy Noether turns 137 today.

To my regret, despite that I frequently invoke her name (she co-starred with Albert in the Special Relativity series), her work in mathematics is pretty far above my head, and I’m simply not qualified to write about it. I can say that her work connects mathematical symmetry with physical conservation laws. She also made significant contributions to abstract algebra.

Just recently, I’ve begun to nibble at the edges of the latter in the form of group theory as a part of studying rotation.

Continue reading


Party Time!

beautiful dayI misspent my younger days in the warm climes of Southern California. In particular, I went to high school and college there.  I moved to the Midwest about seven years after college. For many, college was the end of anything resembling much in the way of time to call their own. I have many fond memories of idle times in perfect weather!

People who know me know I have a pretty intense work ethic. They also know I have a pretty intense party ethic. (Truth is: I’m just intense. Period. Work hard; play hard; relax hard.)  This past week—my first week into retirement—I’ve been relaxing hard.

And the weather has been just glorious this week. So far, retirement is aces!

Continue reading