Chaotic Attractors Fall 2020

 Here are chaotic attractors created by students in Mathematics Through 3D Printing Fall 2020. 

Boys Surface

 This Boys surface is a demonstration piece for how to avoid orientability problems when trying to create 3D prints of non-orientable surfaces. Instead of spending a lot of time fixing your normals on your STL file, just make it a mesh -- which also allows you to see the interesting internal geometries. 

AMS Notices Article

The article Modeling Dynamical Systems for 3D Printing by Stephen K. Lucas, Evelyn Sander, and Laura Taalman by the cover of the  December 2020 issue of the AMS Notices. If you've ever wanted to print your very own chaotic attractor, check this out for instructions. 

Iterated Function Systems Fall 2020

 Iterated function systems are formed by iterating multi-valued affine maps of the plane or three dimensional space. These are created by students from Mathematics Through 3D Printing Fall 2020. 

Iterated function systems in the plane exhibiting 5- and 6-fold symmetries

The initial condition has maximum height, and the subsequent iterates are shorter and shorter. 

Iterates of the iterated function system appear like buildings in a cityscape - in fact IFS are often used for simulation of natural landscapes

Starting at with the bottom disk, each subsequent height is a new iterate

 

Four-fold symmetry

Sierpinsky structure

Partial Derivatives But Not Differentiable

This object was created this year by a student from last year's Mathematics Through 3D Printing (2019). It is the graph of a function with partial derivatives in both directions but which is not differentiable. 

Mathematics Through 3D Printing

Mathematics Through 3D Printing Fall 2020 -  Forgotten keys resulted in an outdoor class time. Lucky it was a beautiful day with a pile of newly printed objects.  

Mathematica Objects

These objects were made by students in Mathematics Through 3D Printing, Fall 2020. The assignment was to make something neat in order to learn to use Mathematica.

Mandelbrot and Julia Sets

 Mandelbrot and Julia sets related to asymptotic behavior of iterated function in the complex plane. These 3D printed objects represent Mandelbrot and Julia sets not just of the standard quadratic function but also for other functions as well (some people refer to these generalized cases by other names such as multibrot sets). 

A stack of multiple Julia sets for z^2 + c along a fixed line of c values in the complex plane. 

Another view of the previous object above

Mandelbrot set for z^(3.5)+c

Mandelbrot set for (Sin(z^2)/Exp(z))-(c/2)

Left: Mandelbrot set for z^(15)+c - this set has 14 bulbs.
Right: Mandelbrot set for z^8+c, resulting in 7 bulbs. 

Filled Julia set for f(z) = z⁷ + c, where c= -0.8+-0.2i

Filled Julia set for z^2+z+(-1+.2i) - presumably the spikeyness is due to this being outside the Mandelbrot set, meaning that the Julia set is not connected.

Additional view of above object

Pentagon Tilings of the Plane

 These irregular pentagon tilings of the plane are created by students in Mathematics Through 3D Printing, Fall 2020. Each is a member of a distinct class of pentagon tilings. The history of pentagon tilings began in 1918 when the first 5 classes were discovered. The 15th class was found in 2015. In 2017 when it was shown that there are only 15 distinct classes, all of which are known. 

Ambiguous and Reflexively Fused Objects

 These mathematical optical illusions are created by students in Mathematics Through 3D Printing, Fall 2020. Using a concept of Kokichi Sugihara, the reflection in the mirror is not the same as the object itself. This is due to projection of a 3-dimensional curve designed to trick the eye.