# Dan Bach

3D math artist / interactive book author

dansmathshop

Oakland, California, USA

I'm Dan Bach (@dansmath on twitter), a former college mathematics teacher from California. I used Mathematica for 25 years in the classroom, teacher workshops, and conference talks. Students convinced me long ago that my nice-looking calculus graphs could qualify on their own as art. Math has been perceived by many as separate from art, but as creators and viewers of math art, we enjoy using both halves of our brains! After a 36-year teaching career, I am now a 3D math artist and interactive book author, trying to bring the joy of mathematics to an unsuspecting audience.

Saddle Circles

40 x 50 cm

Digital Print

2020

I love playing with space curves. Any smooth parametric curve has a unit tangent vector at each point, going in the same direction as the curve. The rate of change of this vector, with respect to arclength, gives the curvature at that point, and we can construct an 'osculating circle' tangent to the curve and having the same radius of curvature locally. The four circles shown here are the smallest ones along this saddle curve. Try to picture other osculating circles along this curve! (click the link for an interactive 3D experience!)

Zigzag Octahedron

40 x 50 cm

Digital Print

2020

In my attempt to draw a path starting at the origin and going through every lattice point in space, I decided on nested octahedral layers, each defined by |x| + |y| + |z| = n, with n = 0, 1, 2, 3, ... An Eulerian circuit traces all edges of each octahedron once, and these can be filled out zig-zag style, to pass through all the points on the layer, as in the traveling salesman problem. The seventh layer is shown here; it turns out all the paths from one vertex to the next have the same shape, and this creates a 'zigsaw puzzle' with one piece for each face of the octahedron. The stray piece wants to be reattached to that hole in the side. Will it fit? (follow the link for a 3D version!)