“Lattice-Filling Curves, Prime Locations, Eulerian Circuits, and the Traveling Salesperson Problem”
Presentation at Bay Area Artists and Mathematicians (BAAM) meeting at Santa Clara University on 2/15/20
I’ve known about prime locations along Ulam’s Spiral in the xy-plane grid since I was young, but I’ve been wondering about a lattice-filling curve in three dimensions for a while now.
Starting with cubical shells, I extended the spiral idea (top) into a helix down the sides of a cube, and spiraling into the bottom (south pole). The first picture shows the primes in green along this path (orange). The next shell out can be covered by a similar path from bottom to top, and in this way, fill space.
Ulam’s Spiral extended along the sides and bottom of a cube
The same can be done with octahedra, arranging concentric shells |x| + |y| + |z| = n and winding from top to bottom, then bottom to top, or one triangular face at a time, as I show in the second picture:
Space can be filled with concentric octahedral shells. Each shell has lattice points that can be visited by a single route, and the path can then jump out to the next shell.
Now the octahedron’s edges can be traced once each by an Eulerian circuit (L), but the cube’s cannot (R), as seen from my pieces at Bridges and JMM. The octahedral path can then be modified to cover all lattice points near the edges, and it comes back to its starting point as well.
Eulerian String Figure Octahedron (L) and Sloppy Fourier Cube (R)
I passed out handouts so people could cut this shape out for the n=7 octahedral layer. It had the added surprising property that each face features the same zig-zaggy polygon in either black or white. This then folds up into the “Zigzag Octahedron” (last picture).
Net for tiled octahedron shown below, copies passed out to the audience!
“Zig-Zag Octahedron” - My entry for the Joint Mathematics Meetings 2021 (Virtual) Math Art Exhibition
I came out of 2017 with a new confidence, artistically. I submitted two artworks to the JMM 2018 in San Diego: Slinky Spheres and Prime Bead Spirals. This time only the first one was accepted, but one of my favorites.
“Slinky Spheres” by Dan Bach.
Twenty colored spheres are cradled inside a web of toroidal helices formed from the normal and binormal vectors along a central wiggle (not pictured). The wiggle also contains the centers of the spheres, which is only fair.
The conference was huge and vast, without being too personal. There were way too many interesting talks to attend, not to mention our hosting an artist friends party at the nearby AirBnB we stayed at.
But the Math Art Exhibition was where it was at, especially if you liked math or art or both. I got there early and staked out a prime location. The Prime Bead Spirals woulda looked good there.
After seeing the total eclipse in Oregon the summer before, the conference floor seemed even brighter.
“Prime Bead Spirals” by Dan Bach
Starting at the bottom, strings of 1000 beads are formed into a disk, cylinder, cone, and “parabowla.” Primes are green, composites are yellow. The last prime in each string is 997. Why do the green beads line up better on the cylinder? (There are 60 beads per rotation on the cylindrical helix.)
The Math Art community is getting more familiar to me now, and it’s nice to have my work appreciated and complimented by peers! I wonder if there’s a huge market for this mathematical art stuff, and how I can muscle in on the business.
After retiring from teaching in 2014, I migrated into interactive book production, using Apple’s iBooks Author. Making the graphics and movies for the book, I started concentrating on producing nice-looking images, pushing the limits of my computer saving or crashing. The Joint Mathematics Meetings in Atlanta was my first show as an artist. I actually had two pieces accepted but I didn’t scroll enough on the gallery site and only brought one canvas. I’ve included the other work here because I can.
“A Fine Mesh We’re In” by Dan Bach, 2016
A central curve (not shown) has tangent, normal, and binormal directions at each point, making a local {T, N, B} frame. Using a trigonometric combination of the N and B vectors, we describe a toroidal mesh of curves with hues of green, yellow, and orange. Spheres of varying colors and sizes are placed along an equatorial helix and some try to escape their bonds.
“Growing Parallel Normal Curves” by Dan Bach, 2016
The starting red curve has a center of curvature at each of its points. The colored parallel curves go toward that center in increasing amounts. Some groups of curves are rotated, some are reduced in number, but all originate from the same ball-shaped bundle at the top, appropriately used as the dansmath.com logo.
Last year (2015) I got my feet wet in the Interactive Multi-Touch Book world, controlled tightly by Apple iBooks. This year, I made a second trip to Nashville to promote my new iBook “3D Curves and Surfaces” at the iBooks Author Conference, put on by local entrepreneur Bradley Metrock.
And this time I was a featured speaker! My book and I were nominated for Best in Education and Best Book! It was great meeting keynote speaker Serenity Caldwell, and working with Hype and other authoring platforms in addition to iBooks Author, for which I earned a Certificate of Awesomeness, or the equivalent.
Above: The cover of my interactive book, part of the Math You Can Handle series.
Slides from my featured talk, creating 3D Math Art inside of an Apple iBook. They don’t want you to call it that.
Patty Leitner, award-winning college math teacher, has been making instructional math videos for years, and has allowed me to link to her awesome patty's calculus videos.
Let Patty take you on a video journey through the land of calculus, clearly explaining the topics in her friendly style.
The 47 videos are grouped into four playlists: Limits, Derivatives, Applications, and Integrals. Each part has 10 to 14 videos of about 35 to 60 minutes. If you look carefully you can locate some free PDF handouts to fill in while you watch the videos. It's a proven educational bonanza, all for free to you and/or your students.
Students: If you like these videos, let Patty know! On twitter at @pattysmathclass or email her at pattysmathclass@gmail.com
Teachers: If you use these videos as a class resource, tell Patty and she can put your name on the "approved" list! On twitter at @pattysmathclass or email her at pattysmathclass@gmail.com
Now I have figured out how to allow you, the viewer, to view my 3D viewable models in a Google Cardboard viewer!
Here's an example, but the general idea is that if you stick "/embed?cardboard=1" (without the quotes) after the model URL you will get a stereo viewable image for your Google Cardboard-compatible viewer.
Tap the image on your phone and put it in a Google Cardboard viewer.
I have one that I put together for under $8 (half of which was shipping). The Vel-cro was stronger than the Cardboard's cardboard, and it tore but still works, and the images look awesome.
I have now ordered a View-Master VR Viewer, usually $29.99, I got one for $21 (post-Xmas discount) and it should arrive in a couple of days. I'll let you know how that goes.
After 18 years on the web, the last 16 of them at dansmath.com, I have finally achieved a major goal: to create 3D math art in mathematica, and upload it to my site where anyone can view, appreciate it, spin it around, and say, "Now that's math!"
During this month I have been uploading several of my models to sketchfab.com, and just today I began embedding them here. Hope you like it/them!
One of my 3D models on Sketchfab, completely rotatable. Click image to go to my local models page.
Please feel free to follow me on Twitter, leave comments here or at my sketchfab page, and tell others about your experience! Happy mathing!
Now at last I have renewed my Squarespace site, and I am repopulating and updating my legacy pages from www.dansmath.com as well as adding major new content.
In the coming weeks and months I hope to offer math lessons, a math art and movie gallery, embedded 3D models, and eventually a storefront for digital art prints, rolled or framed canvases, and various 3D-printed objects.
