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Tangible Math Visualization Models - A 3D Approach

Explore the creation and printing of 3D models to aid in understanding mathematical concepts. Learn about the challenges and possibilities of using 3D printing technology in math education.

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Tangible Math Visualization Models - A 3D Approach

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

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

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

  4. Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin

  5. Assembly of Music of the Spheres

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

  7. Ribbon Sculptures Altamont Stelvio Collins: Pax Mundi(1994)

  8. SLIDE-GUI for “ViaeGlobi” Shapes Good combination of interactive 3D graphicsand parameterizable procedural constructs.

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

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

  11. Controls applied to the 2-period Gabo curve: Azimuth / Twist Control Torsion Minimization:Azimuth: tangential / normal 900 of twistadded. Natural orientationwith Frenet frame

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

  13. Many Different Viae Globi Models Maloya Stelvio Altamont

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

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

  16. ChineseButton Knot(Knot 940)Bronze, Dec. 2007Carlo Séquincast & patina bySteve Reinmuth

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

  18. Sculptures Defined by Key Features RimsFunnelsTunnels Marked features:

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

  20. Computer-Aided Design Process Modeling “Interruption”  3D-print

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

  22. An Improved NOME Model Two more tunnels added.

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

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

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

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

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

  28. Many Different Materials . . . Ceramic Metal Plastic Plaster

  29. “Wolly” by Eva Hild Free-form surfaces offer a bigger modeling challenge!

  30. E X T R A S

  31. Topology of “Wholly” 2-sided, single border, genus 4

  32. A Flexible, Parameterized Model Polyhedral model, with high-level edit-controls

  33. First, Not Very Successful Attempt An FDM model of “Wholly” ?

  34. Second Attempt with Same Set-Up Wholly_A2: a more carefully tuned model

  35. Comparison Wholly_A1 Wholly_A2 Tunnels still not very round

  36. Trying a New Approach Combining 8 partial toroids into a flexible chain

  37. Parameterization (2D concept proof) Radii, heights, azimuth angles, tilt are adjustable. Automatic adjustment of tunnel/tunnel separation.

  38. Resulting 3D CAD Model Wholly_B1

  39. First Result from the New Approach Wholly_B1 coming off the 3D printer

  40. Model after Clean-up Wholly_B1

  41. Comparison Wholly_A2 Wholly_B1 Roundtunnels. Other problems!

  42. Orientability Moebius band Cooling tower Hild: “Interruption” Dyck loop 2-sided, 1-sided,orientable non-orientable

  43. Not a “Hild Sculpture” Tetrahedral configuration of 6 “Super Dyck disks” with 4 stubs

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