The number of non-equivalent distinguishing coloring partitions of the path on n vertices (n>=1) with exactly k parts (k>=1). Regular triangle read by rows: the rows are indexed by n, the number of vertices of the path, and the columns are indexed by k, the number of parts.
I love this! It inspires me to start playing with these ideas again!
Check out @fdecomite.bsky.social’s incredible shelf based on Squaring the Square!
I did! Just got a 3D printer a few days ago!
I designed them in OpenSCAD and customized them for some 3mm x 1mm magnets that I bought for about 3¢ each (purchased on Amazon, for better or worse.)
I'll try to write a blog post about it soon, but reach out in the meantime if you want the files.
And a third stellation of a rhombic dodecahedron!
And his stellation!
Check out Robin's version of a rhombic dodecahedron!
Inspired by @robinhouston.mathstodon.xyz.ap.brid.gy's G4G16 gift of tetragonal disphenoidal blocks, I've made some magnetized versions, which are satisfyingly click-y!
Triangle read by rows: T(n,k) = number of step shifted (decimated) sequence structures of length n using exactly k different symbols.
That's right! This is a puzzle on the faces of the pentagonal icositetrahedron, which has three short edges and two long edges.
I wrote a blog post about my G4G16 gift. If you want a copy of your own, download and print the files!
The photo shows identical soft cell units assembled into a lattice structure. The shape seems to have the same qualities as a truncated octahedron, It won’t dense pack as efficient as the Kelvin or Weaire-Phelan structures because the deformation of the faces increases the surface area of the individual units and makes the ratio of the volume of the object to the edge length less efficient. But it looks pretty and I’m working on filling in the holes now .
Enlarging 3D printed models. I am imagining the structure of our physical universe just above the Planck scale, where quantum forces overcome the force of gravity. If you are exploring the fundamental structure of three dimensional form, this is the place to be!
Petition: Christopher Havens is an inmate in the Washington state prison system who has been collaborating on mathematics with people both inside and outside prison. He was recommended for clemency by the parole board. However, governor Bob Ferguson has not granted it yet. (link below)
Peter has a nice blog post about making his card. Check it out peterkagey.com/blog/2025/12...
I got some nice #ptpx cards in the mail today. Thanks @peterkagey.com @aotearoan!
Three podcasts that I loved that have since stopped releasing new episodes, but which I wish more people knew about.
– Hakai Magazine Audio Edition
– Ox Tales (by the Oxford Symposium on Food and Cookery)
– 70 Over 70 (by Pineapple Street Studios)
Please recommend defunct podcasts to me too!
I still do it, but they’re very low stakes—hardly worth more than a homework assignment.
A video of the pen plotter in action!
I wrote about the postcard I designed for the 2025 plotter postcard exchange #ptpx on my blog, based on a potato-based game that M.C. Escher used to play with his kids.
The sequence is defined as the following:
T(n,k) = length of longest carry sequence when adding k to n in binary representation, 1 ≤ k ≤ n (triangular array).
An illustration of a non-self-intersecting maximal loop on a torus.
I asked a Code Golf Stack Exchange question about this too, if you want to read more.
codegolf.stackexchange.com/q/181203/53884
Don Knuth just published OEIS sequence:
"A391478: T(m,n) is the number of Hamiltonian cycles on an m×n torus where each step goes north (N), east (E), or northeast (NE), with 2 ≤ n ≤ m."
It references three sequences, two of which are sequences I published in 2019!
I can’t wait to see what comes next!
I like this a lot! I wrote a blog post about a similar shelf that I prototyped using cardboard.
I don't think there was too much behind that question—I think I was just curious if someone could figure out a rule.
I first got to walk a robot when I played with a similar demo from Chase Meadors, which you can see here.
(It's based on this fun video!)
