University of California, Berkeley Bridges 1999 CS 39R Simple 2-Manifolds Carlo H. Séquin University of California, Berkeley Today’s topic: Starting from simple 2-manifolds -- all the way to the surface classification theorem.

Deforming a Rectangle 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. We will study thos five ways of connecting the edges of a rectangle in different ways. cylinder Möbius band torus Klein bottle cross surface

Five Important Two-Manifolds X = V-E+F = Euler Characteristic; G = genus Cylinder Möbius band X=0 X=0 G=0 G=1 X=0 X=0 X=1 G=1 G=2 G=1 This is a quick overview of what shapes will result. Now we discuss each one in some detail. X = Euler Characteristic: V - E + F; G = genus. Torus Klein bottle Cross surface

Cylinder Construction Bend and join one way.

Möbius Band Construction Bend, twist through 180 degrees, join. ==> Single-sided surface.

Cylinders as Sculptures A quick side-step into Art. Max Bill John Goodman

The Cylinder in Architecture And architecture. Chapel

Möbius Sculpture by Max Bill A famous stone sculpture (in Baltimore).

Möbius Sculptures by Keizo Ushio Split Möbius band -- not really a single-sided surface any more!

More Split Möbius Bands Another example. Split Moebius band by M.C. Escher And a maquette made by Solid Free-form Fabrication

Torus Construction Now we are going to connect two pairs of opposite edges on the rectangle. ==> 2-sided: an inside and an outside! Glue together both pairs of opposite edges on rectangle Surface has no borders Double-sided surface

“Bonds of Friendship” J. Robinson Again, a look at art. 1979

Torus Sculpture by Max Bill Something quite special by Max Bill.

Virtual Torus Sculpture Note: Surface is represented by a loose set of bands ==> yields transparency Red band forms a Torus-Knot. Red band forms a “Torus-Knot”. More later in the course when we will talk about Knot Theory… “Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd.

Klein Bottle -- “Classical” Aka “inverted-sock” Klein bottle. Connect one pair of edges straight and the other with a twist Single-sided surface -- (no boundaries)

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

Many More Klein Bottle Shapes ! Cliff Stoll lives here in Berkeley. (Check out YouTube video on “Numberphiles”) Klein bottles in glass by Cliff Stoll, ACME

Klein-Stein (Bavarian Bear Mug) Another variation of the Klein-bottle shape. 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.” You cannot embed a Klein bottle in 3D space. You will always have some intersections -- unless you add punctures (holes through the surface with sharp borders)

Klein Bottle Skeleton (FDM) In the display case on the 6th floor of Soda Hall.

Klein Bottle Skeleton (FDM) I had to carefully adjust the position and bending of the handle. Struts don’t intersect !

Another Type of Klein Bottle Here is a different way of making a Klein bottle. And it is really different from the classical Klein bottle, insofar as it cannot be smoothly deformed into that one. Cannot be smoothly deformed into the classical Klein Bottle Still single sided -- no boundaries

Figure-8 Klein Bottle Woven by Carlo Séquin, 16’’, 1997 This model is currently in the Geometry exhibit at the Exploratorium. Woven by Carlo Séquin, 16’’, 1997

Triply Twisted Figure-8 Klein Bottle It also works if the figure-8 cross section makes 3 half-turns; -- or any odd number of half-turns.

Triply Twisted Figure-8 Klein Bottle Another model with a gridded surface.

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. Put the filament crossings of the other branch (= outer blue edges) at the circle locations. 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 Inside the Fused-Deposition Modeling machine.

Single-thread Figure-8 Klein Bottle The build-phase has finished. As it comes out of the FDM machine

Single-thread Figure-8 Klein Bottle Still need to remove the grey support material.

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

Cross-cap Construction Gradual deformation of the rectangle…

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.

The Projective Plane Projective Plane is single-sided; has no edges. Follow the pair of rays going to the head and feet of the figure… -- 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 Depiction by a famous artist. (But not in isolation; attached to a torus -- morfe on this later) Helaman Ferguson ( Torus with Crosscap = Klein Bottle with Crosscap )

“Four Canoes” by Helaman Ferguson Here the cross-over happens in the torus loop itself. This makes it a Klein bottle.

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 Dyck’s surface (genus3). The cross cap on a sphere (cross-surface) models the projective plane (genus 1), but has some undesirable singularities. Can we avoid these singularities ? Can we get more symmetry ?

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

Construction of Steiner Surface Start with three orthonormal squares … … connect the edges (smoothly). --> forms 6 “Whitney Umbrellas” (pinch points with infinite curvature) Bend 3 corner flaps together and fuse them smoothly. Pinch-points on the 6 tetrahedral edges.

Steiner Surface Parametrization Steiner surface can best be built from a hexagonal domain. Let the surface pass through itself, so that the three marked yellow points coincide; but do NOT form a connection! One is not allowed to step from one surface branch to any of the other two! Glue opposite edges with a 180º twist.

Again: Alleviate Self-intersections Strut passes through hole

Skeleton of a Steiner Surface

Steiner Surface has more symmetry; but still has singularities (pinch points). Hilbert and other mathematicians have puzzled over this question for many years. Finally Hilbert asked a graduate student to prove that it cannot be done! 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 ! 3-fold symmetry based on hexagonal domain Result is now known as the Boy Surface!

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 On the web you can find some wonderful movies depicting this surface. Computer graphics by John Sullivan

Another “Map” of the “Boy Planet” From book by Jean Pierre Petit “Le Topologicon” (Belin & Herscher) A more whimsical rendering of Boy’s surface.

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

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

Revisit Boy Surface Sculptures One more rendering by H. Ferguson. Helaman Ferguson - Mathematics in Stone and Bronze

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