Building a trivalent graph of harmonic relations among major and minor triads.

If you take two unit squares stacked one atop the other, and rotate one through an angle of $$\theta$$ about its center, the area in the intersection of the two squares is an octagon. I found it a pleasant exercise to express the area of the octagon in terms of $$\theta$$.

In times of rapid change and constant uncertainty, it's comforting to know that Dror Bar-Natan's web page will, until the heat death of the universe, look just as it did when I was a happy, bright-eyed graduate student.

A problem I wrote for a competition some time ago (but decided not to use):
Let $$a_n$$ denote the nearest integer to $$\log_{168}\left(927^n\right)$$, and let $$b_n=a_{n+1}-a_n$$. The first few terms in the sequence $$\{b_n\}$$ are $$\{b_n\} = 2,1,1,2,1,1,2,1,1,2,1,1,\ldots$$
which appears to be a sequence of period 3. Does the repeating 2,1,1 pattern continue indefinitely? Prove your answer.

My ultimate reaction to any given Putnam problem is either disappointment that I couldn't solve it or disappointment that I solved it (hence, it was too easy).

I wish there were more modern dance and pop music in 3/4 time. (On topic, because '3' and '4' are integers.)

I discovered a fun identity the other day. Probably every Russian schoolchild knows this, already but I didn't: For every positive integer m,
$$\sum_{k=1}^m \lfloor m/k\rfloor = \sum_{k=1}^m \sigma_0(k)$$,
where $$\sigma_0(k)$$ is the number of positive divisors of k.

I was working on something that led me to realize this must be true, but if I were given the identity cold and asked to prove it I'm not sure how well I'd have fared!

Nothing special, just a somewhat pleasing (I thought) tiling I drew.

jsiehler boosted

If I write fiction where Theseus and the minotaur fall in love
And then someone else takes my fic, swaps out the labyrinth for a coffeeshop setting
And then someone swaps the minotaur for another monster
And then someone swaps Theseus for another hero
Is it still the same ship?

This situation makes me want to yell "Hey! You kids! Clean up that mess you made!" except the kids involved are smarter and older than I am and in fact mostly dead.
en.wikipedia.org/wiki/Group_of

Abstract algebra is a very serious subject and one must, at all times, endeavor to uphold the timeless gravity of this hallowed matter.

"You'll be working with uranium," said Ben, as the other forest animals looked on, "and I must caution you to keep this matter utterly secret."
amazon.com/Red-Ben-Fox-Oak-Rid

Tiling with a Necker Cube-like perceptual flip.

Spoiler

Anyone find any that I've missed here?

I need to make a playable version of this for the browser, but it is a fun puzzle to see how many convex polygons you can form by assembling the seven puzzle pieces here, tangram-fashion. They're drawn on a grid of 30-60-90 triangles.

I have a habit of referring to "eigenstuff" and "eigencritters" in linear algebra, and a student illustrated this.

Today I ran across the site "WikiWand" which I would describe as "a pretty-printer for Wikipedia," which is to say, it serves up wikipedia articles in an eyeball-friendly way with some reader conveniences. I kind of like it, I think - even nicer-looking than Firefox's Reader view which I like a lot. Something feels sketchy about it, though, and I'm not sure why.

jsiehler boosted

I did some creative coding the other day. I'm quite happy with the results :) 1/2

Cantor's diagonal argument for non-denumerability boils down to "Every function from the natural numbers to the reals fails to be surjective."

Is there an equally lovely but underappreciated argument that does "Every function from the real numbers to the natural numbers fails to be injective?" If not, why is that direction more difficult?

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