Dave Richeson

I just made this design, based on the "72 Pencils" sculptures by George Hart: https://www.georgehart.com/sculpture/pencils.html. George's design consists of 72 pencils arranged in a triangular lattice. His bundles of 18 pencils are arranged in a hollow hexagon pattern and are glued together. In this version, I create a 3D-printable polyhedron (a truncated octahedron) to hold the standard-sized hexagonal pencils. Each bundle consists of 19 pencils in a filled hexagonal pattern. Thus, it requires 76 pencils.

Here's the 3D-printable file: https://www.thingiverse.com/thing:7313100

My first attempt was to make the faces with the hexagonal holes. The holes had to be offset in a certain way so that, when joined into a polyhedron, the pencils would line up correctly inside it. I kept making errors. Then, I decided to completely restart the project. I made the pencils and the polyhedron in OpenSCAD, then "subtracted" the pencils from the polyhedron to create the holes.

This is so cool! Given only the "exponential minus log" function, elm(x,y)=exp(x)-ln(y), and the constant 1, you can perform +, -, x, ÷, exp, ln, trig, powers, roots, etc. You can also obtain e and π! See https://arxiv.org/pdf/2603.21852 and https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformation.pdf

In Multivariable Calculus, we're going to talk about limits, continuity, and differentiability of functions of two variables. I just printed these three examples of functions that do not have a limit at the origin. Their equations are:

f₁(x, y) = ((x² – y²)/(x² + y²))²
f₂(x, y) = x²/(x² + y⁴)
f₃(x, y) = xy/(x² + y²)

Also, if we define f₁(0, 0) = 1 and f₃(0, 0) = 0, then these two functions also have the property that both partial derivatives exist at the origin, but the function is not differentiable there.

You can download the files here: https://www.thingiverse.com/thing:7314735

At Gathering 4 Gardner, Colm Mulcahy suggested I try the following 3D-printing project.

Recall the "napkin ring problem": Take a sphere of radius r and drill out a hole along a diameter so the remaining shape has height h. Then the volume, V = πh³/6, does not depend on r.

He thought that if we print these shapes as solids (i.e., with 100% fill), they should all weigh the same. See the following post for the results.

Still playing with these impossible cylinders. Here I have an ellipse with the "Batman curve" in the reflection.

Gathering 4 Gardner is this week. I'll be running an activity where we'll make one of these giant cardioids (and a giant nephroid). Today, I'm taking advantage of the sunny, above-freezing day to laser-cut the pieces outside.

I designed and 3D printed a couple of models of Riemann sums for multivariable functions—the monkey saddle and the sombrero function. (I also got to try out this cool filament.) The files are on Thingiverse. https://www.thingiverse.com/thing:7019158 https://www.thingiverse.com/thing:7019184

I was sad to learn about the passing of Sheldon Newhouse. Even though we were not close, I can honestly say he changed the course of my life. I was on the job market in anticipation of finishing my PhD in 1998. It was a brutal market.

I gave a talk at "Smalefest" during my last year of grad school. I had recently found out that my "sure thing" postdoc (in my mind) fell through, and I was not having much luck elsewhere in my job search. After my talk, Sheldon, whom I didn't know but who was my "academic uncle," approached me to chat about my work. He asked what I was doing the next year. I said I had nothing yet. He said to FedEx a copy of my application to him at Michigan State. I did, and I was hired for a 2-year position. That kept me in academia, it was where I met my wife, and it was surely helpful when I was on the job market again. He was so kind to me. My thoughts are with his wife, Patti Lamm, and the rest of his family.
https://www.dignitymemorial.com/obituaries/san-diego-ca/sheldon-newhouse-12777624

One more, which I call "I ❤️ the Cardioid." (A cardioid from one direction and a heart in the other.)