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

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

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 3D ?

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 (Tetrahedral Symmetry) Plaster model by T. Kohono 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 everywhere! u One triple point, u 3 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 makes a versatile building block! u 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:

M ӧ bius Band into Boy Cap u Credit: Bryant-Kusner

Geometrical Surface Elements “Cross-Cap” “Boy 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.

Boy Cap + Disk = Boy Surface u In summary: M ӧ bius Band + Disk = Projective Plane TWO M ӧ bius Bands = Klein Bottle u And: See: 

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

Classical Klein Bottle from 2 Boy-Caps BcL BcR BcL + BcR = KOJ “Inverted Sock” Klein bottle:

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 u There is more than one type of Klein bottle ! BcL BcR BcR + BcR = K8R Twisted Figure-8 Klein Bottle:

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

Make a Gridded Sculpture!

Increase the Grid Density

Actual Sculpture Model

S 6 Klein Bottle Rendered by C. Mouradian

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 increases Euler Characteristic by one. u We can fill these holes again with: l Disks: Decreases 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. Euler Char. unchanged

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

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 (60°rotation between neighbors) Employ tetrahedral symmetry! ( 0°rotation between neighbors)