Bridges 2013 Girl’s Surface Sue Goodman, UNC-Chapel Hill Alex Mellnik, Cornell University Carlo H. Séquin U.C. Berkeley

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?

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

Key Features of a Boy Surface The triple point, the center of the skeleton Its “skeleton” or intersection neighborhood Boy surface and its intersection lines

The Complex Outer Disk

Boy’s Surface – 3-fold symmetric u From Alex Mellnik’s page:

A Topological Question: u Is Werner Boy’s way of constructing a 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 ? u 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 “Simple” Way ? (with one triple point and 3 intersection loops) u Are there others? -- How many? u Sue Goodman & co-workers asked this question in u There is exactly one other way! They named it: “Girl’s Surface” u 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 Neighborhood Twisted lobe! Boy Surface Girl Surface

How the Surfaces Get Completed Boy surface (for comparison) Girl Surface Red disk expands and gets warped; Outer gray disk gives up some parts.

Girl’s Surface – no symmetry u From Alex Mellnik’s page:

Transform Boy Surface into Girl Surface

The Crucial Transformation Step (b) Horizontal surface segment passes through a saddle r-Boy skeleton r-Girl skeleton

Compact Models of the Projective Plane l-Boy r-Boy Homeomorphism (mirroring) Homeomorphism (mirroring) l-Girl r-Girl Regular Homotopy twist one loop

Open Boy Cap Models Expanding the hole Final Boy-Cap Boy surface minus “North Pole” C2C2

A “Cubist” Model of an Open Boy Cap One of six identical components Completed Paper Model

C 2 -Symmetrical Open Girl Cap C2C2

The “Red” Disk in Girl’s Surface Paper model of warped red disk Intersection neighborhoods Boy- & Girl-

Cubist Model of the Inner “Red” Disk

Cubist 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 Cap Paper model Smoothed computer rendering

Epilogue: Apéry’s 2 nd Cubist Model Another model of the projective plane

Apery’s Net of the 2 nd Cubist Model ( somewhat “conceptual” ! )

My First Paper Model u Too small! – Some elements out of order!

Enhanced Apery Model u Add color, based on face orientation u Clarify and align intersection diagram

Enhanced Net u Intersection lines u Mountain folds u Valley folds

My 2 nd Attempt at Model Building The 3 folded-up components -- shown from two directions each.

Combining the Components u 2 parts merged

All 3 Parts Combined u Bottom face opened to show inside

Complete Colored Model u 6 colors for 6 different face directions u Views from diagonally opposite corners

The Net With Colored Visible Faces u Based on visibility, orientation

Build a Paper Model ! u The best way to understand Girl’s surface! u 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 - Connection “Heart of a Girl” Cubist Intersection Neighborhood

The Best Way to Understand Girl’s Surface! u Build a Paper Model ! u Description with templates available in a UC Berkeley Tech Report: EECS “Construction of a Cubist Girl Cap” by C. H. Séquin, EECS, UC Berkeley (July 2013) Q U E S T I O N S ?

S P A R E

Transformation Seen in Domain Space