CS 39 R SingleSided Surfaces Carlo H Squin

CS 39 R Single-Sided Surfaces Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

Making a Single-Sided Surface Twisting a ribbon into a Möbius band

Simple Möbius Bands u A single-sided surface with a single edge: A closed ribbon with a 180°flip. A closed ribbon with a 540°flip.

More Möbius Bands Max Bill’s sculpture of a Möbius band. The “Sue-Dan-ese” M. B. , a “bottle” with circular rim.

A Möbius Band Transfromation Widen the bottom of the band by pulling upwards its two sides, get a Möbius basket, and then a Sudanese Möbius band.

Many Different Möbius Shapes Left-twisting versions shown – can be smoothly transformed into one another u Topologically, these are all equivalent: They all are single-sided, They all have ONE rim, They all are of genus ONE, and E. C. = 0. u Each shape is chiral: its mirror image differs from the original.

These are NOT Möbius Bands ! u What you may find on the Web under “Möbius band” (1):

These are NOT Möbius Bands ! u What you may find on the Web under “Möbius band” (2):

These are NOT Möbius Bands ! u What you may find on the Web under “Möbius band” (3):

TWO Möbius Bands ! u Two Möbius bands that eventually get fused together:

Topological Surface Classification The distinguishing characteristics: u Is it two-sided, orientable – or single-sided, non-orientable? u Does it have rims? – How many separate closed curves? u What is its genus? – How many handles or tunnels? u Is it smooth – or does it have singularities (e. g. creases)? Can we make a single-sided surface with NO rims?

Classical “Inverted-Sock” Klein Bottle

Can We Do Something Even Simpler? u Yes, we can! u Close off the rim of any of those Möbius bands with a suitably warped patch (a topological disk). u The result is known as the Projective Plane.

The Projective Plane -- Equator projects to infinity. -- Walk off to infinity -- and beyond … come back from opposite direction: mirrored, upside-down !

The Projective Plane is a Cool Thing! u It is single-sided: Flood-fill paint flows to both faces of the plane. u It is non-orientable: Shapes passing through infinity get mirrored. u. A straight line does not cut it apart! One can always get to the other side of that line by traveling through infinity. u It is infinitely large! (somewhat impractical) It would be nice to have a finite model with the same topological properties. . .

Trying to Make a Finite Model u Let’s represent the infinite plane with a very large square. u Points at infinity in opposite directions are the same and should be merged. u Thus we must glue both opposing edge pairs with a 180º twist. Can we physically achieve this in 3 D ?

Possible “Rectangle Universes” u 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

Cross-Surface Construction

Wood / Gauze Model of Projective Plane Cross-Surface = “Cross-Cap” + punctured sphere

Cross-Cap Imperfections u Has 2 singular points with infinite curvature. u Can this be avoided?

Steiner Surface Plaster model by T. Kohono (Tetrahedral Symmetry) A gridded model by Sequin Can singularities be avoided ?

Can Singularities be Avoided ? Werner Boy, a student of Hilbert, was asked to prove that it cannot be done. But he found a solution in 1901 ! u It has 3 self-intersection loops. u It has one triple point, where 3 surface branches cross. u It may be modeled with 3 -fold symmetry.

Various Models of Boy’s Surface

Main Characteristics of Boy’s Surface Key Features: u Smooth u One u 3 everywhere! triple point, intersection loops emerging from it.

Projective Plane With a Puncture The projective plane minus a disk is: u a Möbius band; u or a cross-cap; u or a Boy cap. u. This u makes a versatile building block! with an open rim by which it can be grafted onto other surfaces.

Another Way to Make a Boy Cap Similar to the way we made a cross cap from a 4 -stick hole: Frame the hole with 3 opposite stick-pairs and 6 connector loops:

Mӧbius Band into Boy Cap u Credit: Bryant-Kusner

Geometrical Surface Elements “Cross-Cap” “Boy Cup” u Single-sided surface patches with one rim. u Topologically equivalent to a Möbius band. u “Plug-ins” that can make any surface single-sided. u “Building blocks” for making non-orientable surfaces. u Inspirational design shapes for consumer products, etc.

u In summary: Boy Cap + Disk = Boy Surface Mӧbius Band + Disk = Projective Plane Genus = 1; E. C. = 1 u And: TWO Mӧbius Bands = Klein Bottle Genus = 2; E. C. = 0 See:

2 Möbius Bands Make a Klein Bottle KOJ = MR + ML

Classical Klein Bottle from 2 Boy-Caps Bc. L Bc. R “Inverted Sock” Klein bottle: Bc. L + Bc. R = KOJ

Klein Bottle with S 6 Symmetry u Take two complementary Boy caps. u Rotate left and right halves 180°against each other to obtain 3 -fold glide symmetry, or S 6 overall.

Klein Bottle from 2 Identical Boy-Caps Bc. L u Bc. R There is more than one type of Klein bottle ! Twisted Figure-8 Klein Bottle: Bc. R + Bc. R = K 8 R

Model the Shape with Subdivision u Start Level 0 with a polyhedral model. . . Level 1 Level 2

Make a Gridded Sculpture!

Increase the Grid Density

Actual Sculpture Model

S 6 Klein Bottle Rendered by C. Mouradian http: //netcyborg. free. fr/

Fusing Two Identical Boy Surfaces u Both shapes have D 3 symmetry; u They differ by a 60°rotation between the 2 Boy caps.

Building Blocks To Make Any Surface u. A sphere to start with; u. A hole-punch to make punctures: Each decreases Euler Characteristic by one. u We can fill these holes again with: l Disks: Increases Euler Characteristic by one. l Cross-Caps: Makes surface single-sided. l Boy-caps: Makes surface single-sided. l Handles (btw. 2 holes): Orientability unchanged. l Cross-Handles (btw. 2 holes): Makes surface single-sided. Genus changes; E. C. unchanged.

Constructing a Surface with u χ=2‒h Punch h holes into a sphere and close them up with: cross-caps or Boy caps or Closing two holes at the same time: handles or cross-handles

Single-sided Genus-3 Surfaces Renderings by C. H. Séquin Sculptures by H. Ferguson

Concept of a Genus-4 Surface 4 Boy caps grafted onto a sphere with tetra symmetry.

Genus-4 Surface Using 4 Boy-Caps Employ tetrahedral symmetry! ( 0°rotation between neighbors) (60°rotation between neighbors)