Transcript Bridges 2013, Enschede Cross-Caps ‒ Boy Caps ‒ Boy Cups Carlo H.

Bridges 2013, Enschede
Cross-Caps ‒ Boy Caps ‒ Boy Cups
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
NOT THIS !
“Cross-Cap”
“Boy Cap”
found on the Internet
“Boy Cup”
Geometrical Surface Elements
“Cross-Cap”
“Boy Cap”
“Boy Cup”

Single-sided surface patches with one rim.

Topologically equivalent to a Möbius band.

“Plug-ins” that can make any surface single-sided.

“Building blocks” for making non-orientable surfaces.

Inspirational design shapes for consumer products, etc.
Making a Single-Sided Surface
Twisting a ribbon into a Möbius band
Simple Möbius Bands

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 Moebius Shapes
Left-twisting versions shown – can be smoothly transformed into one another

Topologically, these are all equivalent:
They all are single-sided,
They all have ONE rim,
They all are of genus ONE.

Each shape is chiral:
its mirror image differs from the original.
Topological Surface Classification
The distinguishing characteristics:

Is it two-sided, orientable – or single-sided, non-orientable?

Does it have rims? – How many separate closed curves?

What is its genus? – How many handles or tunnels?

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?
 Yes,
we can!

Close off the rim of any of those Möbius bands
with a suitably warped patch (a topological disk).

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!

It is single-sided:
Flood-fill paint flows to both faces of the plane.

It is non-orientable:
Shapes passing through infinity get mirrored.

A straight line does not cut it apart!
One can always get to the other side of that line
by traveling through infinity.

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

Let’s represent the infinite plane with
a very large square.

Points at infinity in opposite directions
are the same and should be merged.

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
Closing the Möbius Band into a Shell
The rim of the band seen from the top
Fill in side from the bottom until the upper edges join in a
double intersection line.
Cross-Cap Imperfections
 Has
2 singular points
with infinite curvature.

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 !

It has 3 self-intersection loops.

It has one triple point, where 3 surface branches cross.

It may be modeled with 3-fold symmetry.
Various Models of Boy’s Surface
Main Characteristics of Boy’s Surface
Key Features:
 Smooth
 One
3
everywhere!
triple point,
intersection loops
emerging from it.
Projective Plane With a Puncture
The projective plane minus a disk is:
 a Möbius band;

or a cross-cap;

or a Boy cap.
This

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:
A 20-sided Hole Yields a 5-tunnel Boy Cap
One of the 5 crossing
strips doing a 180°flip
Mӧbius Band into Boy Cap

Credit: Bryant-Kusner
 In
summary:
Boy Cap + Disk = Boy Surface
Mӧbius Band + Disk = Projective Plane
 And:
TWO Mӧbius Bands = Klein Bottle
See: 
2 Möbius Bands Make a Klein Bottle
KOJ
=
MR
+
ML
Classical Klein Bottle from 2 Boy-Caps
BcL
BcR
“Inverted Sock” Klein bottle:
BcL + BcR = KOJ
Klein Bottle with S6 Symmetry

Take two complementary Boy caps.

Rotate left and right halves 180°against each other
to obtain 3-fold glide symmetry, or S6 overall.
Klein Bottle from 2 Identical Boy-Caps
BcL

BcR
There is more than one type
of Klein bottle !
Twisted Figure-8 Klein Bottle:
BcR + BcR = K8R
Model the Shape with Subdivision
 Start
Level 0
with a polyhedral model . . .
Level 1
Level 2
Make a Gridded Sculpture!
Increase the Grid Density
Actual Sculpture Model
S6 Klein Bottle Rendered by C. Mouradian
http://netcyborg.free.fr/
Fusing Two Identical Boy Surfaces

Both shapes have D3 symmetry;

They differ by a 60°rotation between the 2 Boy caps.
Building Blocks To Make Any Surface
A
sphere to start with;
 A hole-punch to make punctures:
Each increases Euler Characteristic by one.

We can fill these holes again with:

Disks: Decreases Euler Characteristic by one.

Cross-Caps: Makes surface single-sided.
Euler Char.
unchanged

Boy-caps: Makes surface single-sided.

Handles (btw. 2 holes): Orientability unchanged.

Cross-Handles (btw. 2 holes): Makes surface single-sided.
Constructing a Surface with

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)
Construction a Genus-8 Surface
Concept:
8 Boy caps
grafted onto sphere
in octahedral positions.
Octa-Boy Sculpture
Two half-shells made on an RP machine
Octa-Boy Sculpture
The two half-shells combined
Octa-Boy Sculpture
Seen from a different angle
A Frank Ghery Building ??

These fascinating geometries may serve as starting points
for designs of consumer products, furniture, architecture . . .
“Non-Orientable” Furniture
Inspiration: Ribbon chairs
My own: Möbius chair
&
&
Möbius Bench (V. Acconci)
Klein Kouch
Möbius Band Inspirations
Stuff found on the web under “Möbius basket”
Wearables: Klein-Bottle Caps
From ACME Klein bottles: http://www.kleinbottle.com/klein_bottle_hats.htm
Woolen, Wearable Cross-Cap
knitted by Margareta Séquin
Dress Rehearsal of Woolen Cross-Cap
Can We Make a Wearable Boy Cap?
Knitting a Boy Cap
Is the Projective Plane Hat Crazy Enough?
http://www.youtube.com/watch?v=UcnJ-INfXXQ
A Virtual Boy Cap Straw Hat
Cups, Mugs, & Steins
Boy cup
Boy mug
Boy stein
Boy Mug Realized !
Made on a fused deposition modeling (FDM) machine
Boy Buoys
Even though the Boy surface is single sided,
due to its self-intersections, it has closed compartments
that provide it with buoyancy.
Boy Houses

Inspired by:
Another Virtual Design

http://mima.museum/mathematik-jreality.php
Cubist Frameworks for Buildings
Cubist Boy Cap on a Mirror
Cubist Boy Surface (MSRI)
Check out the Art Exhibit !
Glass Boy’s Cube 4
Blue Boy’s Cube
by Roger Vilder
Boy Surface Pavillion by Marc Johnson
http://marcjohnson.fr/pavilion/
Conclusions

The Boy surface is a fascinating shape.
It has intrigued me for several decades.

This elegant form can readily lead to
attractive constructivist sculptures.

But it can also inspire architects,
designers of consumer products,
and creative minds who will develop
nifty knitting or crocheting patterns.
SPARE
Compact Models of the Projective Plane
in Different Ambient Isotopy Classes
R-BOY
Homeomorphism
(mirroring)
Twist
One loop
Homeomorphism
(mirroring)
L-GIRL R-GIRL
Regular Homotopy
Regular Homotopy
L-BOY