CS 39R Simple 2-Manifolds Carlo H. Séquin University of California, Berkeley
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. cylinder Möbius band torus Klein bottle cross surface
Five Important Two-Manifolds X = V-E+F = Euler Characteristic; G = genus X=0 X=0 G=0 G=1X=0 X=0 X=1G=1 G=2 G=1 Cylinder Möbius band Torus Klein bottle Cross surface
Cylinders as Sculptures John Goodman Max Bill
The Cylinder in Architecture MIT Chapel
More Split Möbius Bands Split Moebius band by M.C. Escher And a maquette made by Solid Free-form Fabrication
Torus Construction • Glue together both pairs of opposite edges on rectangle • Surface has no borders • Double-sided surface
Virtual Torus Sculpture Note: Surface is representedby a loose set of bands ==> yields transparency Red band forms a Torus-Knot. “Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd.
Klein Bottle -- “Classical” • Connect one pair of edges straightand the other with a 180 flip • Single-sided surface; no borders.
Klein Bottles -- virtual and real Computer graphics by John Sullivan Klein bottle in glassby Cliff Stoll, ACME, Berkeley
Many More Klein Bottle Shapes ! Klein bottles in glass by Cliff Stoll, ACME
Klein-Stein (Bavarian Bear Mug) Klein bottle in glassby 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 !
Another Type of Klein Bottle • Cannot be smoothly deformed into the classical Klein Bottle • Still single sided, no borders
Figure-8 Klein Bottle • Woven byCarlo Séquin,16’’, 1997
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 Modelingwith SLIDE
Single-thread Figure-8 Klein Bottle As it comes out of the FDM machine
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 ?
Significance of Cross-cap • < 4-finger exercise >What is this weird surface ? • 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 C -- 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 )
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 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)
Steiner Surface Parametrization • Steiner surface can best be built from a hexagonal domain. Glue opposite edges with a 180º twist.
Again: Alleviate Self-intersections Strut passesthrough hole
Steiner Surface • has more symmetry; • but still hassingularities(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 ! • 3-fold symmetry • based on hexagonal domain
Characteristics of Boy’s Surface Key Features: • Smooth everywhere! • One triple point, • 3 intersection loops emerging from it.
Mӧbius Band into Boy Cap Credit: Bryant-Kusner
More Models of the Boy Surface By many artists