Assemblable Barth Sextics

by silviana amethyst
Winter 2023-24

First I present this unedited photograph of an assemblable Barth Sextic, where colors are numbers of plugs / sockets. This is a random assembly of five each 0's, 1's, 2's, and 3's. When I say "random", I mean I just did whatever came to my mind as I put it together.

Red is 0 plugs (aside: I find the plugs easier to count, which disappoints me as a post-op trans woman)
Black is 1
Blue is 2
White is 3

I made this set of pieces during the Illustrating Mathematics semester program at ICERM in Fall 2019. That semester was incredible and forever changed my career. If you find me at a conference, it's likely I have this set with me. Ask me and you can play with it!

Symmetries

I've been working in late 2023 to make snap-together models illustating all of them. I completed the rotations, antipodal, tetrahedra, and cube, but did not complete the mirror (I ran out of time before JMM 2024, and it would take six days to print the 12 models to make the mirror symmetry).

🔗 Use the links below for real-life versions!!!

The Barth Sextic has 120 affine symmetries:

2-fold rotational about pairs of node joining pieces
3-fold rotational about tips of antipodal pieces
5-fold rotational about centers of antipodal pentagons
Mirror (uncomplete so far)

The Barth Sextic additionally has some geometry hiding in it:

About the pieces

The Barth Sextic has 20 finite pieces (and 20 infinite pieces which I'm omitting). Each piece can have 0, 1, 2, or 3 plugs/sockets (choose one to focus on). Alexandr Holroyd computed a nice table of numbers of distinct assemblies for me in Fall 2019. That table has not yet been published by me/us, though I am sure the numbers are already known somewhere. I'm not publishing that table here for now.

But I will say that if you use five 0's, five 1's, five 2's, and five 3's, you can assemble them in 140,240 ways.

For many of the photographs below, I randomly assembled a 5555 and then colored the pieces on paper, and then partitioned the 20 pieces from a suitable set of different colors to illustate a symmetry or neat property of the Sextic.

I have more on the Barth Sextic elsewhere on my site, and other people have described it well elsewhere on the internet. I'm focusing here on my work on making puzzle-like objects from it.

In case you like equations, here's a defining equation for this projective algebraic surface of degree 6:

\( 4(\phi^2 x^2-y^2) (\phi^2 y^2-z^2) (\phi^2 z^2-x^2) - (1+2\phi) (x^2+y^2+z^2-1)^2 w^2 = 0 \)

To compute the affine version I'm working with, I use an affine patch: the simplest, in a sense, is \(w = 1\).