“Fair dice” might make you think of perfect cubes with equal frequencies (say, 1/6 on all sides) 🎲 But “fair” really just means you get the frequencies you expect (say, 1/4, 1/4 & 1/2) We can now design fair dice with any frequencies—and any shape! 🐉 hbaktash.github.io/projects/put...

Nicely produced clip by Matt Wein and Marylee Williams about our recent dice design project at @scsatcmu.bsky.social and @adobe.com youtube.com/shorts/jD0ag... 🎲 🎥 🐉 🪙

I got tired of mashing together tools to write long threads with 𝐫𝐢𝐜𝐡 𝑓𝑜𝑟𝑚𝑎𝑡𝑡𝑖𝑛𝑔 and ℳα†ℏ—so I wrote La𝑇𝑤𝑒𝑒𝑡! It converts Markdown and LaTeX to Unicode that can be used in “tweets”, and automatically splits long threads. Try it out! keenancrane.github.io/LaTweet/

“Everyone knows” what an autoencoder is… but there's an important complementary picture missing from most introductory material. In short: we emphasize how autoencoders are implemented—but not always what they represent (and some of the implications of that representation).🧵

I can't* fathom why the top picture, and not the bottom picture, is the standard diagram for an autoencoder. The whole idea of an autoencoder is that you complete a round trip and seek cycle consistency—why lay out the network linearly?

I don't have a strong opinion about whether video models “understand the world.” But I do think the first bar should be checking whether you can recover consistent geometry from video—not whether it makes accurate predictions of physics.

#geogram + #graphite + #blender Python scripting = robust CSG directly available in Blender ! Everything is a numpy array.

Awesome work by two awesome former students. 🦖🏁

Working on a Walk on Spheres tutorial for #SIGGRAPH2025, and love the ads I'm getting served. 😂 Stay tuned for more…👣

Quick “teaser” for a fun #SIGGRAPH2025 project, led by Hossein Baktash, on optimizing a shape to have the desired rolling statistics. Basically we can turn arbitrary objects into fair dice, or make dice which capture the statistics of other objects—like several coin flips.

Making good on this promise—in the fastest turnaround time ever—my collaborator Etienne Corman has just posted MATLAB code for #RectangularSurfaceParameterization here: github.com/etcorman/Rec... (C++ version is still in the works…)

Meshes with 90° angles are super useful, providing asymptotically faster convergence for finite element simulation, and optimal shape approximation (when aligned with curvature). Amazingly, no past quad meshing method could guarantee 90° angles under refinement—until now. #RSP

Very happy that Jiří Minarčík will join our research group at @cmu.edu, the Geometry Collective, as a Fulbright visiting scholar! 🥳 Jiří is a world expert in space curves, and one of the core contributors to Penrose (penrose.cs.cmu.edu). Check out his beautiful work here: minarcik.com

Folks in the #SIGGRAPH community: You may or may not be aware of the controversy around the next #SIGGRAPHAsia location, summarized here www.cs.toronto.edu/~jacobson/we... If you're concerned consider signing this letter docs.google.com/document/d/1... via this form docs.google.com/forms/d/e/1F...

Very happy to receive a SIGGRAPH 2025 Best Paper Award (Honorable Mention) for our RSP algorithm, which dices surfaces into near-perfect rectangles. Such meshes are useful for everything from retopology, to microfluidic simulation, to textile design, to architectural geometry.

Shout out today to parents, grandparents, aunts, uncles, or anyone else who helps raise a kid. It didn’t really click until I was a dad: without parents, human civilization ends. Only with good parents does the future look bright. So, give a nod to anyone putting in the work.

Mesh intersections is hard ! Three years of agonizing pain summarized in my article that just got accepted to ACM Transactions on Graphics dl.acm.org/doi/10.1145/... The implementation and the expansion-based arithmetic kernel are available in geogram: github.com/BrunoLevy/ge...

Hey @nervousjesse.bsky.social @nervousjessica.bsky.social, how come you don't sell a waste basket (or recycling bin) in your shop? Would be a welcome alternative to conventional wicker or wire baskets. (Maybe modeled after lysosomes or peroxisomes?)

Generative art circa 2004. Inspired by Piet Mondrian and rendered in acrylic. (Currently on display at my grandmother’s house)

More in-depth article, on Ars Technica, about our recent dice design/rigid body project led by Hossein Baktash at CMU: arstechnica.com/science/2025... You can find the full paper and 3D printable STL files on Hossein's webpage here: hbaktash.github.io

Terrific new work in designing fair dice with arbitrary shapes, by Hossein Baktash at CMU: • www.newscientist.com/article/2482... • hbaktash.github.io/files/rollin...

Anyone interested in 3D printing some funky dice—like a "fair dragon" which can land in three different ways, or a single die equivalent to to 2D6? We've now put printable STL files online: www.cs.cmu.edu/~kmcrane/Pro... More info here: hbaktash.github.io

Great writeup by @stokel.bsky.social about our #SIGGRAPH paper on dice design (and rigid body analysis), led by my awesome PhD student Hossein Baktash: hbaktash.github.io

Fun new paper at #SIGGRAPH2025: What if instead of two 6-sided dice, you could roll a single "funky-shaped" die that gives the same statistics (e.g, 7 is twice as likely as 4 or 10). Or make fair dice in any shape—e.g., dragons rather than cubes? That's exactly what we do! 1/n

Huge congratulations to my former PhD student Rohan Sawhney for winning the @acmsiggraph.bsky.social Outstanding Doctoral Dissertation Award! 🥳 Rohan's thesis develops a whole new class of Monte Carlo methods for solving physical equations, without needing to break the world up into a mesh.

If you're a prospective PhD student interested in geometry, topology, and applications in geometric/visual computing, check out the incredible Mark Gillespie, who is joining the faculty at graphics.cs.utah.edu — and looking for students! markjgillespie.com

Jensen's inequality gives the difference between the average value of a convex function φ, and its value at the center, where both “average” and “center” are defined in terms of some distribution p_X. When the function φ is flat, or the distribution is narrow, they agree.

After trying several broken repos/scripts, I made a Python script for rendering #ShaderToy shaders to antialiased video (or rather, image sequences): github.com/keenancrane/... (This one will surely be broken too someday, but for now it works nicely 😛)

I looked at this image many times as a kid, saw it at the Met, eventually met mathematician Tom Banchoff, who told stories of his encounters with Dalí & their discussions about the 4th dimension. Only today did I realize it’s Christ on the hypercube. m.youtube.com/watch?v=IiXc...

I'll let you in on one of my deep diagramming secrets: When I see a “square”, I replace it with a square. When I see a “cube”, I replace it with a cube. It's easy to forget where these words come from. The only mistake we made as a species is not calling numbers “segments.”

Here's a nice "proof without words": The sum of the squares of several positive values can never be bigger than the square of their sum. This picture helps make sense of how ℓ₁ and ℓ₂ norms regularize and sparsify solutions (resp.). [1/n]

We have entered an era where it pays to have decent handwriting. [This was truly the only prompt; I did not edit the output.] #o4

We often use discretization to approximate continuous laws of physics, but it also goes the other way: You can use continuous equations to approximate the behavior of discrete systems! Here we'll see how electrical circuits can be modeled using the Laplace equation Δφ=0. [1/n]

The Symposium on Geometry Processing is an amazing venue for geometry research: meshes, point clouds, neural fields, 3D ML, etc. Reviews are quick and high-quality. The deadline is in ~10 days. Consider submitting your work, I'm planning to submit! sgp2025.my.canva.site/submit-page-...

Stupid #Keynote trick: if you want the math font to match the text font, write expressions like \𝚝𝚎𝚡𝚝{𝚞}_\𝚝𝚎𝚡𝚝{𝚒} rather than just 𝚞_𝚒 when inserting equations. [Wish this were just the default behavior…]

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. penrose.cs.cmu.edu

As the spring semester is starting, this is the bi-yearly reminder of the collections of courses on geometry processing available here: mirelab6.wixsite.com/dgpslides. Ping me if you teach geometry processing and want your course added.

One of the absolute best students I've ever had (star @Caltech grad and @CarnegieMellon PhD in Computer Science) just had faculty interviews canceled because universities are on a hiring freeze, due to uncertainty about funding. Not a great move for technological supremacy💀💻🇺🇸

Poorly worded #SIGGRAPH author instructions. If you have negative scores (individual or average), 𝗴𝗼 𝗮𝗵𝗲𝗮𝗱 𝗮𝗻𝗱 𝘄𝗿𝗶𝘁𝗲 𝗮 𝘀𝘁𝗿𝗼𝗻𝗴 𝗿𝗲𝗯𝘂𝘁𝘁𝗮𝗹! Negative scores are not a mark of death. Plenty misunderstandings get cleared up in the rebuttal; plenty initially low-score papers are ultimately accepted.

This might look like an anisotropic texture filtering benchmark, but actually it’s just Detroit.

I don't know who needs to hear this but: "SDF" has, for some, become a catch-all term that actually means "a function that tells you if you're inside/outside a shape, and a number that roughly indicates how far." A true signed *distance* function really gives you *distance.*

After many years of giving talks, I no longer get nervous. Instead, I'm now nervous when my students give talks! Fortunately, they do an amazing job. Here's Mark Gillespie giving an extended talk on a new *harmonic* surface representation: www.youtube.com/watch?v=9h13...

This is what people mean when they talk about "readable code," right? (This is in fact a valid—and useful—line of #Mathematica code…)

It's been debated, considered, and ultimately rejected several times over the years. But as a dissenting voice, I will continue to say: The SIGGRAPH deadline should really come *before* the holidays. (Yes, I'm aware of all the logistical hurdles and counter-arguments…)

My superstar student Mark Gillespie is on the academic job market 🤩: markjgillespie.com Beyond lots of beautiful, deep, award-winning research in geometry processing, he's just a terrific person who we already miss having around. Twist his arm if you want him to apply!

Entropy is one of those formulas that many of us learn, swallow whole, and even use regularly without really understanding. (E.g., where does that “log” come from? Are there other possible formulas?) Yet there's an intuitive & almost inevitable way to arrive at this expression.

For a course I teach on Monte Carlo Methods & Applications, I've asked students to do some basic options pricing (Problem 4): gi1242.codeberg.page/cmu-math-cs-... I'd be curious to know, from computational finance folks, how much this (admittedly toy) exercise captures techniques used in practice.

I feel like any slide that introduces this algorithm needs to address the elephant in the room. 🌶️🍲

Yes. Tools like ChatGPT “clicked” for me when I started thinking of them as “database queries for the internet with a super nice natural language interface,” rather than as something that will creatively solve novel problems (like research questions 😅). It’s (mostly) just “better Googling.”

BTW as literally everybody alive is reading this, I am accepting PhD students through this program: mlsystems.uk/supervisors/ If you want to do geometry + ML with me come over we'll have a blast,