MOSAIC, Seattle, Aug. 2000

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MOSAIC, Seattle, Aug. 2000

• Turning Mathematical Models
• into Sculptures
• Carlo H. Séquin
• University of California, Berkeley

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Boy Surface in Oberwolfach

• Sculpture constructed by Mercedes Benz
• Photo from John Sullivan

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Boy Surface by Helaman Ferguson

• Marble
• From Mathematics in Stone and Bronzeby
  Claire Ferguson

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Boy Surface by Benno Artmann

• From home page of Prof. Artmann,TU-Darmstadt
• after a sketch byGeorge Francis.

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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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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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What is Art ?
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Five Important Two-Manifolds
cylinder
Möbius band

• X0
  X0X0 X0
  X1G1 G2
  G1

torus
Klein bottle
cross-cap
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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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Cylinder Construction
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Möbius Band Construction
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Cylinders as Sculptures
John Goodman
Max Bill
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The Cylinder in Architecture
Chapel
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Möbius Sculpture by Max Bill
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Möbius Sculptures by Keizo Ushio
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More Split Möbius Bands
Typical lateral splitby M.C. Escher
And a maquette made by Solid Free-form Fabrication
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Torus Construction

• Glue together both pairs of opposite edges on
  rectangle
• Surface has no edges
• Double-sided surface

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Torus Sculpture by Max Bill
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Bonds of Friendship J. Robinson
1979
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Proposed Torus Sculpture
Torus! Torus! inflatable structure by Joseph
Huberman
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Rhythm of Life by John Robinson
DNA spinning within the Universe 1982
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Virtual Torus Sculpture
Note Surface is representedby a loose set of
bands gt yields transparency
Rhythm of Life by John Robinson, emulated by
Nick Mee at Virtual Image Publishing, Ltd.
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Klein Bottle -- Classical

• Connect one pair of edges straightand the other
  with a twist
• Single-sided surface -- (no edges)

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Klein Bottles -- virtual and real
Computer graphics by John Sullivan
Klein bottle in glassby Cliff Stoll, ACME
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Many More Klein Bottle Shapes !
Klein bottles in glass by Cliff Stoll, ACME
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Klein Mugs
Klein bottle in glassby Cliff Stoll, ACME Fill
it with beer --gt Klein Stein
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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.
• gt Make perforated surfaces and interlace their
  grids.
• gt Also gives nice transparency if one must use
  opaque materials.
• gt Skeleton of a Klein Bottle.

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Klein Bottle Skeleton (FDM)
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Klein Bottle Skeleton (FDM)
Struts dont intersect !
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Fused Deposition Modeling
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Looking into the FDM Machine
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Layered Fabrication of Klein Bottle

• Support material

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Another Type of Klein Bottle

• Cannot be smoothly deformed into the classical
  Klein Bottle
• Still single sided -- no edges

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Figure-8 Klein Bottle

• Woven byCarlo Séquin,16, 1997

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Triply Twisted Fig.-8 Klein Bottle
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Triply Twisted Fig.-8 Klein Bottle
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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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Single-thread Figure-8 Klein Bottle
Modelingwith SLIDE
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Zooming into the FDM Machine
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Single-thread Figure-8 Klein Bottle
As it comes out of the FDM machine
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Single-thread Figure-8 Klein Bottle
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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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Cross-cap Construction
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Significance of Cross-cap

• lt 4-finger exercise gtWhat 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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The Projective Plane
PROJECTIVE PLANE
C
-- Walk off to infinity -- and beyond come
back upside-down from opposite direction. Project
ive Plane is single-sided has no edges.
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Cross-cap on a Sphere

• Wood and gauze model of projective plane

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

• Both, Klein bottle and projective planeare
  single-sided, have no edges.(They differ in
  genus, i.e., connectivity)
• The cross cap on a torusmodels a Klein bottle.
• The cross cap on a spheremodels the projective
  plane,but has some undesirable singularities.
• Can we avoid these singularities ?
• Can we get more symmetry ?

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Steiner Surface (Tetrahedral Symmetry)

• Plaster Model by T. Kohono

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Construction of Steiner Surface

• Start with three orthonormal squares
  connect the edges (smoothly).--gt forms 6
  Whitney Umbrellas (pinch points with
  infinite curvature)

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Steiner Surface Parametrization

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

Glue opposite edges with a 180º twist.
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Again Alleviate Self-intersections
Strut passesthrough hole
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Skeleton of a Steiner Surface
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Steiner Surface

• has more symmetry
• but still hassingularities(pinch points).

Can such singularities be avoided ? (Hilbert)
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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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Model of Boy Surface

• Computer graphics by François Apéry

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Model of Boy Surface

• Computer graphics by John Sullivan

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Model of Boy Surface

• Computer graphics by John Sullivan

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Quick Surprise Test

• Draw a Boy surface(worth 100 of score
  points)...

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Another Map of the Boy Planet

• From book by Jean Pierre Petit Le
  Topologicon (Belin Herscher)

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Double Covering of Boy Surface

• Wire model byCharles Pugh
• Decorated by C. H. Séquin
• Equator
• 3 Meridians, 120º apart

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Revisit Boy Surface Sculptures

• Helaman Ferguson - Mathematics in Stone and Bronze

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Boy Surface by Benno Artmann

• Windows carved into surface reveal what is going
  on inside. (Inspired by George Francis)

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Boy Surface in Oberwolfach

• Noteparametrization indicated by metal bands
  singling out north pole.
• Sculpture constructed by Mercedes Benz
• Photo by John Sullivan

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Boy Surface Skeleton

• Shape defined by elastic properties of wooden
  slats.

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Boy Surface Skeleton (again)
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Goal A Regular Tessellation

• Regular Tessellation of the Sphere
  (Buckminster Fuller Domes.)

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Ideal Sphere Parametrization
Buckminster Fuller Dome almost all equal sized
triangle tiles.
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Ideal Sphere Parametrization

• Epcot Center Sphere

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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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Paper Model with Regular Tiles

• Only meshes with 5, 6, or 7 sides.
• Struts pass through holes.
• Only vertices where 3 meshes join.--gt Permits
  the use of a modular component...

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The Tri-connector
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Tri-connector Constructions
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Tri-connector Ball (20 Parts)
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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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Hopes

• This may lead to even better modelsof the Boy
  surface
• e.g., by using the geodesic linesto define
  ribbons that describe the surface
• (this surface will keep me busy for a while yet !)

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Conclusions

• There is no clear line that separatesmathematical
  models and art work.
• Good models are pieces of art in themselves.
• Much artwork inspired by such modelsis no longer
  a good model for understandingthese more
  complicated surfaces.
• My goal is to make a few great modelsthat are
  appreciated as good geometric art,and that also
  serve as instructional models.

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End of Talk
QUESTIONS ?
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spares
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Rotating Torus
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Looking into the FDM Machine