Bridges 2013Bridges 2013

Girl’s Surface

Sue Goodman,UNC-Chapel Hill

Alex Mellnik,Cornell University

Carlo H. SéquinU.C. Berkeley

The Projective PlaneThe 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!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 ModelTrying 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 ConstructionCross-Surface Construction

Wood / Gauze Model of Projective PlaneWood / Gauze Model of Projective Plane

Cross-Surface = “Cross-Cap” + punctured sphere

Cross-Cap ImperfectionsCross-Cap Imperfections

Has 2 singular points with infinite curvature.

Can this be avoided?

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 BoyVarious Models of Boy’’s Surfaces Surface

Key Features of a Boy SurfaceKey Features of a Boy Surface

The triple point,the center of the skeleton

Its “skeleton” orintersection

neighborhood

Boy surface and its

intersection lines

The Complex Outer DiskThe Complex Outer Disk

BoyBoy’’s Surface – 3-fold symmetrics Surface – 3-fold symmetric

From Alex Mellnik’s page: http://surfaces.gotfork.net/

A Topological Question:A Topological Question:

Is Werner Boy’s way of constructinga smooth model of the projective plane the simplest way of doing this? Or are there other ways of doing it that are equally simple -- or even simpler ?

Topologist have proven (Banchoff 1974)that there is no simpler way of doing this;one always needs at least one triple point and 3 intersection loops connected to it.

Is This the ONLY Is This the ONLY ““SimpleSimple”” Way ? Way ?(with one triple point and 3 intersection loops)(with one triple point and 3 intersection loops)

Are there others? -- How many?

Sue Goodman & co-workers asked this question in 2009.

There is exactly one other way!They named it: “Girl’s Surface”

It has the same number of intersection loops, but the surface wraps differently around them.Look at the intersection neighborhood: One lobe is now twisted!

New Intersection NeighborhoodNew Intersection Neighborhood

Twisted lobe!

Boy Surface Girl Surface

How the Surfaces Get CompletedHow the Surfaces Get Completed

Boy surface(for comparison)

Girl Surface

Red disk expands and gets warped;Outer gray disk gives up some parts.

GirlGirl’’s Surface – no symmetrys Surface – no symmetry

From Alex Mellnik’s page: http://surfaces.gotfork.net/

Transform Boy Surface into Girl SurfaceTransform Boy Surface into Girl Surface

The Crucial Transformation StepThe Crucial Transformation Step

(b) Horizontal surface segment passes through a saddle

r-Boy skeleton r-Girl skeleton

Compact Models of the Projective PlaneCompact Models of the Projective Plane

l-Boy r-Boy

Homeomorphism(mirroring)

Homeomorphism(mirroring)

l-Girl r-Girl

Regular

Hom

otopytwistone loop

Open Boy Cap ModelsOpen Boy Cap ModelsExpanding the hole

Final Boy-Cap

Boy surface minus “North Pole”

C2

A A ““CubistCubist”” Model of an Open Boy Cap Model of an Open Boy Cap

One of sixidentical

components

Completed Paper Model

CC22-Symmetrical Open Girl Cap-Symmetrical Open Girl Cap

C2

The “Red” Disk in Girl’s SurfaceThe “Red” Disk in Girl’s Surface

Paper model of warped red disk

Intersection neighborhoods

Boy- & Girl-

Cubist Model of the Inner “Red” DiskCubist Model of the Inner “Red” Disk

Cubist Model of the Outer AnnulusCubist Model of the Outer Annulus

The upper half of this is almost the same as in the Cubist Boy-Cap model

Girl intersection neighborhood

The Whole Cubist Girl CapThe Whole Cubist Girl Cap

Paper model

Smoothed computer rendering

Epilogue: Apéry’s 2Epilogue: Apéry’s 2ndnd Cubist Model Cubist Model

Another model of the projective plane

Apery’s Net of the 2Apery’s Net of the 2ndnd Cubist Model Cubist Model

( somewhat “conceptual” ! )

My First Paper ModelMy First Paper Model

Too small! – Some elements out of order!

Enhanced Apery ModelEnhanced Apery Model

Add color, based on face orientation

Clarify and align intersection diagram

Enhanced Net Enhanced Net Intersection lines

Mountain folds

Valley folds

My 2My 2ndnd Attempt at Model Building Attempt at Model Building

The 3 folded-up components -- shown from two directions each.

Combining the ComponentsCombining the Components

2 parts merged

All 3 Parts CombinedAll 3 Parts Combined

Bottom face opened to show inside

Complete Colored ModelComplete Colored Model

6 colors for 6 different face directions

Views from diagonally opposite corners

The Net With Colored Visible FacesThe Net With Colored Visible Faces

Based on visibility, orientation

Build a Paper Model !Build a Paper Model !

The best way to understand Girl’s surface!

Description with my templates available in a UC Berkeley Tech Report:“Construction of a Cubist Girl Cap”by C. H. Séquin, EECS, UC Berkeley(July 2013)

Art - ConnectionArt - Connection

“Heart of a Girl” Cubist Intersection Neighborhood

The Best Way to Understand Girl’s Surface!The Best Way to Understand Girl’s Surface!

Build a Paper Model !

Description with templates available in a UC Berkeley Tech Report: EECS-2013-130“Construction of a Cubist Girl Cap”by C. H. Séquin, EECS, UC Berkeley(July 2013)

Q U E S T I O N S ?

http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-130.pdf

S P A R ES P A R E

Transformation Seen in Domain SpaceTransformation Seen in Domain Space