@OpenAI's new 4o image generation is pretty amazing. It also fails on some utterly basic tasks. Maybe a good metaphor for how easily we can be misled by algorithms based largely on fitting empirical data (i.e., machine learning) rather than deductive reasoning. @UsePenrose

Great work by the @UsePenrose team building Bloom: a lightweight way to make interactive diagrams: penrose.cs.cmu.edu/blog/bloo… All coordinates are automatically figured out by the Penrose layout engine, and diagram specifications can be re-used for different content (like HTML/CSS).

A standard way to draw planar graphs is to draw nodes as dots and edges as line segments. But the Koebe–Andreev–Thurston theorem also shows that any planar graph can be visualized by drawing nodes as circular disks, which are tangent if they share an edge:

In a moment nobody was waiting for, I've released meshes from Mark Kilgard's classic OpenGL / GLUT demo "dinoshade.c": cs.cmu.edu/~kmcrane/Projects… The image below re-imagines this example as a vectorized SVG generated in @UsePenrose—read how it was done here! github.com/penrose/penrose/t…

Tired of making the same kind of diagrams over and over by hand (e.g., in PowerPoint)? The @UsePenrose team has been working away on Penrose 3.0, an automated notation-to-diagram tool, finally released today! Check it out here: penrose.cs.cmu.edu/blog/v3

The winding number of a polygon is, quite literally, the number of times a polygon "winds" around a given point x. It can be used for inside/outside tests by just summing the angles made by each (oriented) edge: zero if x is outside, ±2π if x is inside. [Made with @UsePenrose.]

Animation of how the winding number changes as the center point x moves inside/outside a shape. Notice how the signed angle of an edge suddenly jumps from +π to -π (or vice versa) as x crosses an edge, changing the total signed angle by ±2π. [Made with @UsePenrose]

Thanks to @sgestep, we squashed a 4+ year old bug in @UsePenrose that makes it *asymptotically* faster! 🪲 Pro tip: if you're using L-BFGS to avoid building a dense matrix… don't build the dense matrix!! Try the new, way faster version of Penrose here: penrose.cs.cmu.edu/try/

What's the best way to pack N circles into a square? Using less than 20 lines of code in @UsePenrose, we can reproduce the best-known solutions ever found: en.wikipedia.org/wiki/Circle… Try it out for yourself! penrose.cs.cmu.edu/try/?gist…

You can perfectly pack 16 unit squares into a larger 4x4 square. But what’s the smallest square that can contain 17 squares? Check out this thread for an interactive exploration, via @UsePenrose!

[1/n] There's been a lot of hubbub lately about the best known packing of 17 unit squares into a larger square, owing to this post: xitter.cf/KangarooPhysics/status… I realized this can be coded up in < 5 minutes in the browser via @UsePenrose, and gave it a try. Pretty darn close! 🧵

Little experiment with direct editing in @UsePenrose. The beauty of Penrose is that it automatically generates diagrams for you, from simple CSS-like code. But what if you want to make some small tweaks—while respecting constraints? We're working on it… penrose.cs.cmu.edu/siggraph2…

Correct typography is something you notice only when it *doesn't* work! Lesson of the day: to draw a bounding box around text (assuming an alphabetic baseline), you must subtract half the bounding box height, then add the descender height.

Another fun little @UsePenrose example: exterior algebra: github.com/penrose/penrose/t… I just implemented a basic "wedge" operation on vectors—the image shows some random example diagrams generated from the displayed Substance code. Can you take it further? :-)