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JIDI, Berkeley, May 22, 2019

Learn how to create tangible 3D models of mathematical concepts using CAD software and 3D printing. Explore the process of designing, printing, and assembling models for better understanding of mathematical concepts.

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JIDI, Berkeley, May 22, 2019

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  1. JIDI, Berkeley, May 22, 2019 Tangible Math Visualization Models Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. Basel, Switzerland M N G

  3. Logarithmic Spiral Jakob Bernoulli (1654‒1705)

  4. Leonhard Euler (1707‒1783) Imaginary Numbers

  5. Descriptive Geometry

  6. Geometry in every assignment . . . CCD TV Camera (1973) Soda Hall (1992) RISC 1 MicroChip (1982) 3D-Yin-Yang (2000)

  7. More Recent Designs and Models “Hilbert Cube” “Klein Bottle” “Evolving Trefoil” “Pax Mundi”

  8. 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.

  9. 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 Model • Use a low-end printer in our Maker-Space:Many different failure modes, manual clean-up! • Use advanced printers upstairs; pay for materials. • Send CAD model to https://www.shapeways.com/This costs some $$; one week turn-around.

  10. Brent Collins (1997) “Hyperbolic Hexagon II”

  11. Brent Collins’ Pax Mundi1997: Wood, 30”diam. 2006: Commission from H&R Block, Kansas City to make a 70”diameter version in bronze. My task: Define the master geometry. CAD tools play important role!

  12. How to Model Pax Mundi ... • Already addressed that issue in 1998: • Pax Mundicould not be done withSculpture Generator I • Needed a more general program ! • Used the Berkeley SLIDE environment. • First: Needed to find the basic paradigm   

  13. Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis- or base-ball;  “2-period Gabo curve.”

  14. 2-period “Gabo Curve” • Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used (symmetry!).

  15. 4-period “Gabo Curve” Same construction as for as for 2-period curve

  16. Pax Mundi Revisited • Can be seen as:“Amplitude modulated, 4-period Gabo curve”

  17. SLIDE-GUI for “Pax Mundi” Shapes Good combination of interactive 3D graphicsand parameterizable procedural constructs.

  18. Sweep Curve Generator: Gabo Curves as B-splines Cross Section Fine Tuner: Paramererized shapes Sweep / Twist Controller Modularity of Gabo Sweep Generator

  19. 2-period Gabo Sculpture Tennis ball – or baseball – seam used as sweep curve.

  20. Viae Globi Family (Roads on a Sphere) 2 3 4 5 periods

  21. Via Globi 5 (Virtual Wood) Wilmin Martono

  22. Target Geometry(2007) • Constraints: • Bronze, 70” diameter • Less than 1500 pounds • Less than $50’000 • Maintain beauty, strength • Minimize master geometry

  23. Use 4 copies. Emulation; Define Master Pattern • Master to make a mold from. Alignment tab

  24. Joe Valasek’s CNC Milling Machine • Styrofoam milling machine

  25. Machined Master Pattern #2

  26. (Cut) Master  Silicone Rubber Mold

  27. Mold Several (4) Wax Copies

  28. Spruing the Wax Parts for Casting

  29. Ceramic Slurry Shell Around Wax Part

  30. Taking the Shell out of the Kiln

  31. Shell Ready for Casting

  32. The Pour

  33. Casting with Liquid Bronze

  34. Freeing the Bronze Cast

  35. Assembling the Segments

  36. The “Growing” Ribbon

  37. Assembly Completed

  38. Front Door of the ... H&R Block Building

  39. Steve Reinmuth, Bronze Studio, Eugene OR • http://www.reinmuth.com/

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

  41. Extension: Free-form Curve on a Sphere Spherical Spline Path Editor (Jane Yen) Smooth interpolating curve through sparse data points

  42. Many Different ViaeGlobi Models

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

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

  45. Figure-8 KnotBronze, Dec. 2007Carlo Séquin 2nd Prize, AMS Exhibit 2009

  46. Paradigm Extension:Sweep Path is no longer confined to a sphere! Music of the Spheres (Brent Collins)

  47. Partitioning; Joint Design Alignment stubs 1/3 = unique geometry 18 pieces: fit in kiln!

  48. Assembly of Music of the Spheres

  49. Installation at MWSU, Feb. 2013 Steve Reinmuth Brent Collins

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

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