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					CHS
UCB   MOSAIC, Seattle, Aug. 2000


      Turning Mathematical Models
              into Sculptures

                   Carlo H. Séquin
           University of California, Berkeley
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UCB   Boy Surface in Oberwolfach




         Sculpture constructed by Mercedes Benz
         Photo from John Sullivan
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UCB   Boy Surface by Helaman Ferguson




                         Marble



                         From: “Mathematics
                          in Stone and Bronze”
                          by Claire Ferguson
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UCB   Boy Surface by Benno Artmann




                        From
                         home page of
                         Prof. Artmann,
                         TU-Darmstadt
                        after a sketch by
                         George Francis.
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UCB   Samples of Mathematical Sculpture


      Questions that may arise:

         Are the previous sculptures
          really all depicting the same object ?


         What is a “Boy surface” anyhow ?
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UCB   The Gist of my Talk

      Topology 101:
         Study five elementary 2-manifolds

          (which can all be formed from a rectangle)


      Art-Math 201:
         The appearance of these shapes as artwork

          (when do math models become art ? )
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UCB   What is Art ?
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UCB    Five Important Two-Manifolds



       cylinder                                Möbius band
       X=0                                      X=0




      X=0           X=0                  X=1
      G=1           G=2                  G=1
            torus         Klein bottle         cross-cap
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UCB   Deforming a Rectangle

         All five manifolds can be constructed by
          starting with a simple rectangular domain
          and then deforming it and gluing together
          some of its edges in different ways.




          cylinder   Möbius band   torus   Klein bottle   cross-cap
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UCB   Cylinder Construction
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UCB   Möbius Band Construction
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UCB
       Cylinders as Sculptures




      Max Bill    John Goodman
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UCB   The Cylinder in Architecture




                        Chapel
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UCB   Möbius Sculpture by Max Bill
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UCB   Möbius Sculptures by Keizo Ushio
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UCB   More Split Möbius Bands




       Typical lateral split    And a maquette made by
         by M.C. Escher        Solid Free-form Fabrication
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UCB   Torus Construction




                      Glue together both pairs of
                       opposite edges on rectangle
                      Surface has no edges
                      Double-sided surface
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UCB   Torus Sculpture by Max Bill
      “Bonds of Friendship” J. Robinson
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UCB




                                     1979
       Proposed Torus “Sculpture”
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UCB




      “Torus! Torus!” inflatable structure by Joseph Huberman
      “Rhythm of Life” by John Robinson
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UCB




      “DNA spinning within the Universe” 1982
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UCB    Virtual Torus Sculpture

                                     Note:

                                     Surface is represented
                                     by a loose set of bands

                                     ==> yields transparency




      “Rhythm of Life” by John Robinson, emulated by
      Nick Mee at Virtual Image Publishing, Ltd.
      Klein Bottle -- “Classical”
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UCB




                  Connect one pair of edges straight
                   and the other with a twist
                  Single-sided surface -- (no edges)
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UCB   Klein Bottles -- virtual and real




      Computer graphics   Klein bottle in glass
      by John Sullivan    by Cliff Stoll, ACME
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UCB   Many More Klein Bottle Shapes !




       Klein bottles in glass by Cliff Stoll, ACME
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UCB   Klein Mugs




                   Klein bottle in glass
                   by Cliff Stoll, ACME

                   Fill it with beer
                   --> “Klein Stein”
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UCB   Dealing with Self-intersections

      Different surfaces branches
        should “ignore” one another !
      One is not allowed to step from one
       branch of the surface to another.
      ==> Make perforated surfaces
        and interlace their grids.
      ==> Also gives nice transparency
        if one must use opaque materials.

      ==> “Skeleton of a Klein Bottle.”
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UCB   Klein Bottle Skeleton (FDM)
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UCB   Klein Bottle Skeleton (FDM)




      Struts don’t
      intersect !
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UCB   Fused Deposition Modeling
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UCB   Looking into the FDM Machine
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UCB   Layered Fabrication of Klein Bottle




                        Support material
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UCB       Another Type of Klein Bottle




         Cannot be smoothly deformed
          into the classical Klein Bottle
         Still single sided -- no edges
CHS   Figure-8
UCB
      Klein Bottle




         Woven by
          Carlo Séquin,
          16’’, 1997
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UCB   Triply Twisted Fig.-8 Klein Bottle
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UCB   Triply Twisted Fig.-8 Klein Bottle
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UCB   Avoiding Self-intersections

                           Avoid self-intersections at
                            the crossover line of the
                            swept fig.-8 cross section.
                           This structure is regular
                            enough so that this can be
                            done procedurally as part
                            of the generation process.
                           Arrange pattern on the
                            rectangle domain as
                            shown on the left.
                           After the fig.-8 - fold,
                            struts pass smoothly
                            through one another.
                           Can be done with a single
                            thread for red and green !
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UCB   Single-thread Figure-8 Klein Bottle




                              Modeling
                              with SLIDE
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UCB   Zooming into the FDM Machine
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UCB   Single-thread Figure-8 Klein Bottle




         As it comes out of the FDM machine
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UCB   Single-thread Figure-8 Klein Bottle
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UCB   The Doubly Twisted Rectangle Case


                   This is the last remaining
                    rectangle warping case.


                   We must glue both opposing
                    edge pairs with a 180º twist.



      Can we physically achieve this in 3D ?
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UCB   Cross-cap Construction
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UCB   Significance of Cross-cap

         < 4-finger exercise >

          What is this beast ?
         A model of the Projective Plane
             An infinitely large flat plane.
             Closed through infinity, i.e.,
              lines come back from opposite direction.
             But all those different lines do NOT meet
              at the same point in infinity;
              their “infinity points” form another infinitely long line.
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UCB    The Projective Plane


                        PROJECTIVE PLANE




                               C



      -- Walk off to infinity -- and beyond …
         come back upside-down from opposite direction.

      Projective Plane is single-sided; has no edges.
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UCB   Cross-cap on a Sphere




      Wood and gauze model of projective plane
       “Torus with Crosscap”
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UCB




                                    Helaman Ferguson
      ( Torus with Crosscap = Klein Bottle with Crosscap )
      “Four Canoes” by Helaman Ferguson
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UCB
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UCB   Other Models of the Projective Plane

         Both, Klein bottle and projective plane
          are single-sided, have no edges.
          (They differ in genus, i.e., connectivity)
         The cross cap on a torus
          models a Klein bottle.
         The cross cap on a sphere
          models the projective plane,
          but has some undesirable singularities.


         Can we avoid these singularities ?
         Can we get more symmetry ?
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UCB   Steiner Surface   (Tetrahedral Symmetry)




                                   Plaster Model
                                    by T. Kohono
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UCB   Construction of Steiner Surface

         Start with three orthonormal squares …




          … connect the edges (smoothly).

          --> forms 6 “Whitney Umbrellas”
             (pinch points with infinite curvature)
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UCB   Steiner Surface Parametrization

         Steiner surface can
          best be built from a
          hexagonal domain.




       Glue opposite edges with a 180º twist.
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UCB   Again: Alleviate Self-intersections




                                 Strut passes
                                 through hole
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UCB   Skeleton of a Steiner Surface
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UCB   Steiner Surface




                                   has more
                                    symmetry;
                                   but still has
                                    singularities
                                    (pinch points).

      Can such singularities be avoided ? (Hilbert)
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UCB   Can Singularities be Avoided ?

      Werner Boy, a student of Hilbert,
       was asked to prove that it cannot be done.



      But found a solution in 1901 !
         3-fold symmetry
         based on hexagonal domain
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UCB   Model of Boy Surface




      Computer graphics by François Apéry
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UCB   Model of Boy Surface




      Computer graphics by John Sullivan
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UCB   Model of Boy Surface




      Computer graphics by John Sullivan
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UCB   Quick Surprise Test

       Draw    a Boy surface
       (worth 100% of score points)...
      Another “Map” of the “Boy Planet”
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UCB



                           From book by
                            Jean Pierre Petit
                            “Le Topologicon”
                            (Belin & Herscher)
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UCB   Double Covering of Boy Surface

                             Wire model by
                              Charles Pugh



                             Decorated by
                              C. H. Séquin:
                             Equator
                             3 Meridians,
                              120º apart
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UCB    Revisit Boy Surface Sculptures




      Helaman Ferguson - Mathematics in Stone and Bronze
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UCB   Boy Surface by Benno Artmann




         Windows carved into surface reveal what is
          going on inside. (Inspired by George Francis)
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UCB   Boy Surface in Oberwolfach

                             Note:
                              parametrization
                              indicated by
                              metal bands;
                              singling out
                              “north pole”.


                             Sculpture
                              constructed by
                              Mercedes Benz

                             Photo by
                              John Sullivan
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UCB   Boy Surface Skeleton

      Shape defined by elastic properties of wooden slats.
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UCB   Boy Surface Skeleton (again)
      Goal: A “Regular” Tessellation
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UCB



         “Regular” Tessellation of the Sphere
           (Buckminster Fuller Domes.)
       “Ideal” Sphere Parametrization
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UCB




      Buckminster Fuller Dome: almost all equal sized triangle tiles.
      “Ideal” Sphere Parametrization
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UCB




      Epcot Center Sphere
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UCB   Tessellation from Surface Evolver

                             Triangulation from
                              start polyhedron.
                             Subdivision and
                              merging to avoid
                              large, small, and
                              skinny triangles.
                             Mesh dualization.
                             Strut thickening.
                             FDM fabrication.
                             Quad facet !
                             Intersecting struts.
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UCB   Paper Model with Regular Tiles

                                Only meshes
                                 with 5, 6, or 7
                                 sides.
                                Struts pass
                                 through holes.

                                Only vertices
                                 where 3
                                 meshes join.

                                 --> Permits
                                 the use of
                                 a modular
                                 component...
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UCB   The Tri-connector
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UCB   Tri-connector Constructions
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UCB   Tri-connector Ball   (20 Parts)
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UCB   Expectations

         Tri-connector surface will be evenly bent,
          with no sharp kinks.
         It will have intersections that demonstrate
          the independence of the two branches.


         Result should be a pleasing model in itself.
         But also provides a nice loose model of
          the Boy surface on which I can study
          various parametrizations, geodesic lines...
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UCB   Hopes

         This may lead to even better models
          of the Boy surface:
         e.g., by using the geodesic lines
          to define ribbons that describe the surface


         (this surface will keep me busy for a while yet !)
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UCB       Conclusions

         There is no clear line that separates
          mathematical models and art work.
         Good models are pieces of art in themselves.
         Much artwork inspired by such models
          is no longer a good model for understanding
          these more complicated surfaces.


         My goal is to make a few great models
          that are appreciated as good geometric art,
          and that also serve as instructional models.
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UCB   End of Talk
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UCB   === spares ===
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UCB   Rotating Torus
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UCB   Looking into the FDM Machine

				
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posted:11/18/2012
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