MOSAIC, Seattle, Aug Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley

Boy Surface in Oberwolfach ä Sculpture constructed by Mercedes Benz ä Photo from John Sullivan

Boy Surface by Helaman Ferguson ä Marble ä From: “Mathematics in Stone and Bronze” by Claire Ferguson

Boy Surface by Benno Artmann ä From home page of Prof. Artmann, TU-Darmstadt ä after a sketch by George Francis.

Samples of Mathematical Sculpture Questions that may arise: ä Are the previous sculptures really all depicting the same object ? ä What is a “Boy surface” anyhow ?

The Gist of my Talk Topology 101: ä Study five elementary 2-manifolds (which can all be formed from a rectangle) Art-Math 201: ä The appearance of these shapes as artwork (when do math models become art ? )

What is Art ? What is Art ?

Five Important Two-Manifolds cylinderMöbius band torusKlein bottlecross-cap X=0 X=0 X=0 X=0 X=1 G=1 G=2 G=1

Deforming a Rectangle ä 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

Cylinder Construction

Möbius Band Construction

Cylinders as Sculptures Max Bill John Goodman

The Cylinder in Architecture Chapel

Möbius Sculpture by Max Bill

Möbius Sculptures by Keizo Ushio

More Split Möbius Bands Typical lateral split by M.C. Escher And a maquette made by Solid Free-form Fabrication

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

Torus Sculpture by Max Bill

“Bonds of Friendship” J. Robinson 1979

Proposed Torus “Sculpture” “Torus! Torus!” inflatable structure by Joseph Huberman

“Rhythm of Life” by John Robinson “DNA spinning within the Universe” 1982

Virtual Torus Sculpture “Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd. Note: Surface is represented by a loose set of bands ==> yields transparency

Klein Bottle -- “Classical” ä Connect one pair of edges straight and the other with a twist ä Single-sided surface -- (no edges)

Klein Bottles -- virtual and real Computer graphics by John Sullivan Klein bottle in glass by Cliff Stoll, ACME

Many More Klein Bottle Shapes ! Klein bottles in glass by Cliff Stoll, ACME

Klein Mugs Klein bottle in glass by Cliff Stoll, ACME Fill it with beer --> “Klein Stein”

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.”

Klein Bottle Skeleton (FDM)

Struts don’t intersect !

Fused Deposition Modeling

Looking into the FDM Machine

Layered Fabrication of Klein Bottle Support material

Another Type of Klein Bottle ä Cannot be smoothly deformed into the classical Klein Bottle ä Still single sided -- no edges

ä Woven by Carlo Séquin, 16’’, 1997 Figure-8 Klein Bottle

Triply Twisted Fig.-8 Klein Bottle

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

Single-thread Figure-8 Klein Bottle Modeling with SLIDE

Zooming into the FDM Machine

Single-thread Figure-8 Klein Bottle As it comes out of the FDM machine

Single-thread Figure-8 Klein Bottle

The Doubly Twisted Rectangle Case ä This is the last remaining rectangle warping case. ä We must glue both opposing edge pairs with a 180º twist. Can we physically achieve this in 3D ?

Cross-cap Construction

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

The 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.

Cross-cap on a Sphere Wood and gauze model of projective plane

“Torus with Crosscap” Helaman Ferguson ( Torus with Crosscap = Klein Bottle with Crosscap )

“Four Canoes” by Helaman Ferguson

Other Models of the Projective Plane ä Both, Klein bottle and projective plane are single-sided, have no edges. (They differ in genus, i.e., connectivity) ä The cross cap on a torus models a Klein bottle. ä The cross cap on a sphere models the projective plane, but has some undesirable singularities. ä Can we avoid these singularities ? ä Can we get more symmetry ?

Steiner Surface (Tetrahedral Symmetry) ä Plaster Model by T. Kohono

Construction of Steiner Surface ä Start with three orthonormal squares … … connect the edges (smoothly). --> forms 6 “Whitney Umbrellas” (pinch points with infinite curvature)

Steiner Surface Parametrization ä Steiner surface can best be built from a hexagonal domain. Glue opposite edges with a 180º twist.

Again: Alleviate Self-intersections Strut passes through hole

Skeleton of a Steiner Surface

Steiner Surface ä has more symmetry; ä but still has singularities (pinch points). Can such singularities be avoided ? (Hilbert)

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 ! ä 3-fold symmetry ä based on hexagonal domain

Model of Boy Surface Computer graphics by François Apéry

Model of Boy Surface Computer graphics by John Sullivan

Model of Boy Surface Computer graphics by John Sullivan

Quick Surprise Test ä Draw a Boy surface (worth 100% of score points)...

Another “Map” of the “Boy Planet” ä From book by Jean Pierre Petit “Le Topologicon” (Belin & Herscher)

Double Covering of Boy Surface ä Wire model by Charles Pugh ä Decorated by C. H. Séquin: ä Equator ä 3 Meridians, 120º apart

Revisit Boy Surface Sculptures Helaman Ferguson - Mathematics in Stone and Bronze

Boy Surface by Benno Artmann ä Windows carved into surface reveal what is going on inside. (Inspired by George Francis)

Boy Surface in Oberwolfach ä Note: parametrization indicated by metal bands; singling out “north pole”. ä Sculpture constructed by Mercedes Benz ä Photo by John Sullivan

Boy Surface Skeleton Shape defined by elastic properties of wooden slats.

Boy Surface Skeleton (again)

Goal: A “Regular” Tessellation ä “Regular” Tessellation of the Sphere (Buckminster Fuller Domes.)

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

“Ideal” Sphere Parametrization Epcot Center Sphere

Tessellation from Surface Evolver ä Triangulation from start polyhedron. ä Subdivision and merging to avoid large, small, and skinny triangles. ä Mesh dualization. ä Strut thickening. ä FDM fabrication. ä Quad facet ! ä Intersecting struts.

Paper Model with Regular Tiles ä Only meshes with 5, 6, or 7 sides. ä Struts pass through holes. ä Only vertices where 3 meshes join. --> Permits the use of a modular component...

The Tri-connector

Tri-connector Constructions

Tri-connector Ball (20 Parts)

Expectations ä Tri-connector surface will be evenly bent, with no sharp kinks. ä It will have intersections that demonstrate the independence of the two branches. ä Result should be a pleasing model in itself. ä But also provides a nice loose model of the Boy surface on which I can study various parametrizations, geodesic lines...

Hopes ä This may lead to even better models of the Boy surface: ä e.g., by using the geodesic lines to define ribbons that describe the surface ä (this surface will keep me busy for a while yet !)

Conclusions ä There is no clear line that separates mathematical models and art work. ä Good models are pieces of art in themselves. ä Much artwork inspired by such models is no longer a good model for understanding these more complicated surfaces. ä My goal is to make a few great models that are appreciated as good geometric art, and that also serve as instructional models.

End of Talk

=== spares ===

Rotating Torus

Looking into the FDM Machine