Early impressions of @OpenAI's DALL-E 2. 🧵

All images below were produced by AI, with me feeding it the quoted prompt. I was most curious about how helpful such a tool might be in creative work.

"A sloth playing a guitar, photograph 35mm lens" It seems to try hard to keep an image consistent, often making creative choices to do so. I didn't ask for the cloud to be a wad of yarn, but given the prompt, it actually makes a ton of sense.

“Photo of a thunderstorm where lightning is made of yarn and raindrops are needles”

Fun puzzle! (This animation shows the first 5 iterations, the one lower in the thread shows the rest and spoils the answer.)

A hallway has lockers numbered 1,...,N, all initially closed.

Student 1 opens all of them.
Student 2 closes lockers 2, 4, 6, 8, etc.
... Student 3 toggles every third locker, meaning if it’s closed, they open it, if it’s open, they close it.

Student 4 toggles every fourth locker.

Etc., with the k'th student toggling every k’th locker.

After N students do this, which lockers are open, and which are closed?

I'd like to tell you about a game/puzzle to help celebrate today.

We'll call it "Death's Dice".

(1/9) Death finds you. You plead with him that it's too soon, and he agrees to a concession. Every year, he'll roll a set of dice, and if it turns up snake eyes (both 1's) he'll take your life, otherwise, you get one more year.

But it's not necessarily a normal pair of dice.

(2/9)

Niche/abstract topics rarely get highly illustrated explanations, much less in the form of a bedtime story.

A former prof of mine, Ravi Vakil, made this gem capturing how he thinks about exact sequences, cohomology, etc., which he kindly let me repost.

🧵

The announcement for SoME1 winners is now out!

Links to winners in the thread below (in no particular order)

That weird light at the bottom of a mug — ENVELOPES

James Schloss (@LeiosOS) and I decided to organize a casual discord server for what we're calling the first Summer of Math Exposition (SoME1).

It's a space for people interested in producing online math/physics/cs explanations to self-organized.

discord.gg/Q3cC2GXN2f There are so many people who have some great idea for a video or blog post or interactive experience that might help others learn a tricky topic, but who aren't sure where to start.

It's a very good question! What follows are many tweets attempting to answer in a possibly way-too-verbose manner. No pictures (sorry), but I'll trust in the readers' mind's eye.

https://twitter.com/saha_sidharth/status/1337519170011820032

Commentary from mathematicians is more than welcome at the bottom of the thread, which includes scattered thoughts on whether there's a purely geometric reason to expect any rotation to have complex eigenvalues, i.e. one that doesn't appeal to the fundamental theorem of algebra.

Okay, follow through with me on this one and I guarantee you're in for a fun surprise.

We'll start forming rows of numbers, kind of like Pascal's triangle, but with a different rule.

The first row is simple "1, 1" For row 2, copy the first row, but insert a 2 between any elements that add to 2.

In this case that just means sticking a 2 between the two 1's.

In light of the very sad news about Ron Graham's passing, I thought I'd share an interesting tidbit about his famous constant which I only learned recently (thread). This number has captured the imaginations of many people with its unfathomable size, myself included. I distinctly remember when I was a kid first learning about it spending hours trying to wrap my mind around it and using it as inspiration for writing ever-larger numbers.

Of all the videos I've made, one of my favorites topics to have covered was a proof of the inscribed rectangle problem by H. Vaughan using a Mobius strip.

But now there's a new result! @QuantaMagazine recently did a great article about recent work by Greene and Lobb using a beefed-up version of the same idea, letting a Mobius strip encode geometric properties of the curve to solve a more general result, check it out:

quantamagazine.org/new-geometric-…

The birthday paradox is very famous in probability. If you take 23 people, there's about a 50/50 chance that two of them share a birthday. With 50 people, it's a 97% chance.

We could make many other fun examples to illustrate the same counterintuitive phenomenon (thread). Choose a random card from a deck of 52 cards. Put it back, shuffle well, and choose another. Do this for only 9 draws, and more likely than not, you've pulled the same card twice.

Do it 16 times, and your chances are over 90%. Try it!