# Mosaic

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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
Prof. Artmann,
   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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Modeling
with SLIDE
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UCB   Zooming into the FDM Machine
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As it comes out of the FDM machine
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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

… 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.
   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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