AAAS Pomona June 2018 Tangible Math Visualization Models

AAAS Pomona, June 2018 Tangible Math Visualization Models Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

Motivation u Some geometrical / topological mathematical concepts are easier to understand, if you can run your fingers along a physical model. u Helaman Ferguson: “Sculptures are meant to be touched!”

Creating a 3 D (Math-) Model u Phase ONE: Create a CAD Model This is NOT trivial and it is time-consuming. l Use tools such as: Blender, Maya, Sketchpad … l Augment them with your own software, written in Java, Python, C++ … u Phase TWO: Obtaining a 3 D-Print Model Requires some luck, or money, or both. l Use a low-end printer in some Maker-Space: Many different failure modes, manual clean-up! l Send CAD model to https: //www. shapeways. com/ This costs some $$; one week turn-around.

Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin

Assembly of Music of the Spheres

Installation at MWSU, Feb. 2013 “Music of the Spheres” Steve Reinmuth Brent Collins

Ribbon Sculptures Altamont Collins: Pax Mundi (1994) Stelvio

SLIDE-GUI for “Viae Globi” Shapes Good combination of interactive 3 D graphics and parameterizable procedural constructs.

Modularity of Gabo Sweep Generator u Sweep l Gabo Curves as B-splines: u Cross l Section Fine Tuner: Parameterized shapes: u Sweep l Curve Generator: / Twist Controller: How is cross section applied?

Azimuth / Twist Control u Applying azimuth = 0, twist = 0; the cross section: azimuth = 90, twist = 0; azimuth = 90, twist =180;

Azimuth / Twist Control u Controls applied to the 2 -period Gabo curve: Torsion Minimization: Azimuth: tangential / normal 900 of twist added. Natural orientation with Frenet frame

Extension: Free-form Curve on a Sphere Spherical Spline Path Editor (Jane Yen) Nice smooth interpolating curves through sparse data points

Many Different Viae Globi Models Maloya Stelvio Altamont

Extending the Paradigm: Aurora-M u Simple path on sphere, u but more play with the swept cross section. u This is a Möbius band. u It is morphed from a concave shape at the bottom to a flat ribbon at the top of the flower. “Sweep-Morph”

Paradigm Extension: Sweep Path is no longer confined to a sphere! Chinese Button Knot ( K-940 )

Chinese Button Knot (Knot 940) Bronze, Dec. 2007 Carlo Séquin cast & patina by Steve Reinmuth

“Hollow” by Eva Hild, Varberg, 2006 u This is a design task, where I felt it necessary to develop my own rudimentary CAD software…

Sculptures Defined by Key Features u Marked features: Rims Funnels Tunnels

“NOME” Non-Orientable Manifold Editor u Place key features: Rims, Funnels, Tunnels; u Connect u Smooth u Use their borders with surface patches; the assembly with CC-subdivision; offset-surfaces to thicken the 2 -manifold; u Create u Send finely tessellated B-rep (. STL-file); to 3 D-Printer.

Computer-Aided Design Process u Modeling “Interruption” 3 D-print

Room for Improvement u Here the tunnel is not as nicely rounded as in Hild’s original.

An Improved NOME Model u Two more tunnels added.

How to Get a 3 D Print? u Save CAD model to an “. STL”-file. l a wasteful representation of your geometry (all cut up into little triangles). l But it is the “lingua franca” of all 3 D printers! u Take l it to a low-end printer in a Maker-Space. Hope that the printer will do its job. . .

Possible Results. . . u You want to get this: u You may get this: or

Imperfect Surfaces u There may be ridges where the support structure touched the desired shape. u Or the printer may just spew out a little more plastic than needed, creating ugly bumps.

Finally – A Good Print Try different parameter settings: u Try different build orientations, u Try different bases (“raft” vs. just “brims”), u Try different support structures (“lines” / “grids”), u Try a reduced print speed, u Other options … u You may get this:

Pass the Headaches to Someone Else u Use an on-line print service, e. g. shapeways: https: //www. shapeways. com/ u Send a design to their web site; u Select a material and a size for the 3 D-print; u Get a quote within minutes; u Pay by credit card; u Get your part within a week.