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UCBUCB MOSAIC, Seattle, Aug. 2000MOSAIC, Seattle, Aug. 2000

Turning Mathematical Models

into Sculptures

Carlo H. Séquin

University of California, Berkeley

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UCBUCB Boy Surface in OberwolfachBoy Surface in Oberwolfach

Sculpture constructed by Mercedes Benz

Photo from John Sullivan

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UCBUCB Boy Surface by Helaman FergusonBoy Surface by Helaman Ferguson

Marble

From: “Mathematics in Stone and Bronze”by Claire Ferguson

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UCBUCB Boy Surface by Benno ArtmannBoy Surface by Benno Artmann

From home page of Prof. Artmann,TU-Darmstadt

after a sketch byGeorge Francis.

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UCBUCB Samples of Mathematical SculptureSamples 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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UCBUCB The Gist of my TalkThe 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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UCBUCB What is Art ?What is Art ?

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UCBUCB Five Important Two-ManifoldsFive Important Two-Manifolds

cylinder Möbius band

torus Klein bottle cross-cap

X=0 X=0

X=0 X=0 X=1G=1 G=2 G=1

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UCBUCB Deforming a RectangleDeforming 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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UCBUCB Cylinder ConstructionCylinder Construction

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UCBUCB Möbius Band ConstructionMöbius Band Construction

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UCBUCB Cylinders as SculpturesCylinders as Sculptures

Max Bill John Goodman

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UCBUCB The Cylinder in ArchitectureThe Cylinder in Architecture

Chapel

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UCBUCB Möbius Sculpture by Max BillMöbius Sculpture by Max Bill

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UCBUCB Möbius Sculptures by Keizo UshioMöbius Sculptures by Keizo Ushio

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UCBUCB More Split Möbius BandsMore Split Möbius Bands

Typical lateral splitby M.C. Escher

And a maquette made by Solid Free-form Fabrication

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UCBUCB Torus ConstructionTorus Construction

Glue together both pairs of opposite edges on rectangle

Surface has no edges

Double-sided surface

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UCBUCB Torus Sculpture by Max BillTorus Sculpture by Max Bill

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UCBUCB ““Bonds of Friendship” J. RobinsonBonds of Friendship” J. Robinson

1979

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UCBUCB Proposed Torus “Sculpture”Proposed Torus “Sculpture”

“Torus! Torus!” inflatable structure by Joseph Huberman

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UCBUCB ““Rhythm of Life” by John RobinsonRhythm of Life” by John Robinson

“DNA spinning within the Universe” 1982

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UCBUCB Virtual Torus SculptureVirtual Torus Sculpture

“Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd.

Note:

Surface is representedby a loose set of bands

==> yields transparency

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UCBUCB Klein Bottle -- “Classical”Klein Bottle -- “Classical”

Connect one pair of edges straightand the other with a twist

Single-sided surface -- (no edges)

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UCBUCB Klein Bottles -- virtual and realKlein Bottles -- virtual and real

Computer graphics by John Sullivan

Klein bottle in glassby Cliff Stoll, ACME

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UCBUCB Many More Klein Bottle Shapes !Many More Klein Bottle Shapes !

Klein bottles in glass by Cliff Stoll, ACME

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UCBUCB Klein MugsKlein Mugs

Klein bottle in glassby Cliff Stoll, ACME

Fill it with beer --> “Klein Stein”

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UCBUCB Dealing with Self-intersectionsDealing 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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UCBUCB Klein Bottle Skeleton (FDM)Klein Bottle Skeleton (FDM)

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UCBUCB Klein Bottle Skeleton (FDM)Klein Bottle Skeleton (FDM)

Struts don’t intersect !

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UCBUCB Fused Deposition ModelingFused Deposition Modeling

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UCBUCB Looking into the FDM MachineLooking into the FDM Machine

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UCBUCB Layered Fabrication of Klein BottleLayered Fabrication of Klein Bottle

Support material

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UCBUCB Another Type of Klein BottleAnother Type of Klein Bottle

Cannot be smoothly deformed into the classical Klein Bottle

Still single sided -- no edges

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UCBUCB

Woven byCarlo Séquin,16’’, 1997

Figure-8 Figure-8 Klein BottleKlein Bottle

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UCBUCB Triply Twisted Fig.-8 Klein BottleTriply Twisted Fig.-8 Klein Bottle

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UCBUCB Triply Twisted Fig.-8 Klein BottleTriply Twisted Fig.-8 Klein Bottle

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UCBUCB Avoiding Self-intersectionsAvoiding 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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UCBUCB Single-thread Figure-8 Klein BottleSingle-thread Figure-8 Klein Bottle

Modelingwith SLIDE

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UCBUCB Zooming into the FDM MachineZooming into the FDM Machine

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UCBUCB Single-thread Figure-8 Klein BottleSingle-thread Figure-8 Klein Bottle

As it comes out of the FDM machine

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UCBUCB Single-thread Figure-8 Klein BottleSingle-thread Figure-8 Klein Bottle

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UCBUCB The Doubly Twisted Rectangle CaseThe 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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UCBUCB Cross-cap ConstructionCross-cap Construction

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UCBUCB Significance of Cross-capSignificance 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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UCBUCB The Projective PlaneThe Projective Plane

C

PROJECTIVE PLANE

-- 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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UCBUCB Cross-cap on a SphereCross-cap on a Sphere

Wood and gauze model of projective plane

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UCBUCB ““Torus with Crosscap”Torus with Crosscap”

Helaman Ferguson

( Torus with Crosscap = Klein Bottle with Crosscap )

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UCBUCB ““Four Canoes” by Helaman FergusonFour Canoes” by Helaman Ferguson

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UCBUCB Other Models of the Projective PlaneOther 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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UCBUCB Steiner Surface Steiner Surface (Tetrahedral Symmetry)(Tetrahedral Symmetry)

Plaster Model by T. Kohono

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UCBUCB Construction of Steiner SurfaceConstruction 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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UCBUCB Steiner Surface ParametrizationSteiner Surface Parametrization

Steiner surface can best be built from a hexagonal domain.

Glue opposite edges with a 180º twist.

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UCBUCB Again: Alleviate Self-intersectionsAgain: Alleviate Self-intersections

Strut passesthrough hole

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UCBUCB Skeleton of a Steiner SurfaceSkeleton of a Steiner Surface

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UCBUCB Steiner SurfaceSteiner Surface

has more symmetry;

but still hassingularities(pinch points).

Can such singularities be avoided ? (Hilbert)

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UCBUCB Can Singularities be Avoided ?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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UCBUCB Model of Boy SurfaceModel of Boy Surface

Computer graphics by François Apéry

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UCBUCB Model of Boy SurfaceModel of Boy Surface

Computer graphics by John Sullivan

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UCBUCB Model of Boy SurfaceModel of Boy Surface

Computer graphics by John Sullivan

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UCBUCB Quick Surprise TestQuick Surprise Test

Draw a Boy surface

(worth 100% of score points)...

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UCBUCB Another “Map” of the “Boy Planet”Another “Map” of the “Boy Planet”

From book by Jean Pierre Petit “Le Topologicon” (Belin & Herscher)

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UCBUCB Double Covering of Boy SurfaceDouble Covering of Boy Surface

Wire model byCharles Pugh

Decorated by C. H. Séquin:

Equator

3 Meridians, 120º apart

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UCBUCB Revisit Boy Surface SculpturesRevisit Boy Surface Sculptures

Helaman Ferguson - Mathematics in Stone and Bronze

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UCBUCB Boy Surface by Benno ArtmannBoy Surface by Benno Artmann

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

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UCBUCB Boy Surface in OberwolfachBoy 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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UCBUCB Boy Surface SkeletonBoy Surface Skeleton

Shape defined by elastic properties of wooden slats.

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UCBUCB Boy Surface Skeleton (again)Boy Surface Skeleton (again)

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UCBUCB Goal: A “Regular” TessellationGoal: A “Regular” Tessellation

“Regular” Tessellation of the Sphere (Buckminster Fuller Domes.)

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UCBUCB ““Ideal” Sphere ParametrizationIdeal” Sphere Parametrization

Buckminster Fuller Dome: almost all equal sized triangle tiles.

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UCBUCB ““Ideal” Sphere ParametrizationIdeal” Sphere Parametrization

Epcot Center Sphere

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UCBUCB Tessellation from Surface EvolverTessellation 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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UCBUCB Paper Model with Regular TilesPaper 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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UCBUCB The Tri-connectorThe Tri-connector

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UCBUCB Tri-connector ConstructionsTri-connector Constructions

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UCBUCB Tri-connector Ball Tri-connector Ball (20 Parts)(20 Parts)

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UCBUCB ExpectationsExpectations

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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UCBUCB HopesHopes

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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UCBUCB ConclusionsConclusions

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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UCBUCB End of TalkEnd of Talk

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UCBUCB === spares ====== spares ===

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UCBUCB Rotating TorusRotating Torus

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UCBUCB Looking into the FDM MachineLooking into the FDM Machine