bridges 2013
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Bridges 2013. Girl ’ s Surface. 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!. - PowerPoint PPT PresentationTRANSCRIPT
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Bridges 2013Bridges 2013
Girl’s Surface
Sue Goodman,UNC-Chapel Hill
Alex Mellnik,Cornell University
Carlo H. SéquinU.C. Berkeley
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The Projective PlaneThe Projective Plane
-- Equator projects to infinity.-- Walk off to infinity -- and beyond … come back from opposite direction: mirrored, upside-down !
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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 . . .
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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 ?
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Cross-Surface ConstructionCross-Surface Construction
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Wood / Gauze Model of Projective PlaneWood / Gauze Model of Projective Plane
Cross-Surface = “Cross-Cap” + punctured sphere
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Cross-Cap ImperfectionsCross-Cap Imperfections
Has 2 singular points with infinite curvature.
Can this be avoided?
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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.
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Various Models of BoyVarious Models of Boy’’s Surfaces Surface
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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
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The Complex Outer DiskThe Complex Outer Disk
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BoyBoy’’s Surface – 3-fold symmetrics Surface – 3-fold symmetric
From Alex Mellnik’s page: http://surfaces.gotfork.net/
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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.
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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!
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New Intersection NeighborhoodNew Intersection Neighborhood
Twisted lobe!
Boy Surface Girl Surface
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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.
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GirlGirl’’s Surface – no symmetrys Surface – no symmetry
From Alex Mellnik’s page: http://surfaces.gotfork.net/
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Transform Boy Surface into Girl SurfaceTransform Boy Surface into Girl Surface
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The Crucial Transformation StepThe Crucial Transformation Step
(b) Horizontal surface segment passes through a saddle
r-Boy skeleton r-Girl skeleton
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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
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Open Boy Cap ModelsOpen Boy Cap ModelsExpanding the hole
Final Boy-Cap
Boy surface minus “North Pole”
C2
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A A ““CubistCubist”” Model of an Open Boy Cap Model of an Open Boy Cap
One of sixidentical
components
Completed Paper Model
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CC22-Symmetrical Open Girl Cap-Symmetrical Open Girl Cap
C2
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The “Red” Disk in Girl’s SurfaceThe “Red” Disk in Girl’s Surface
Paper model of warped red disk
Intersection neighborhoods
Boy- & Girl-
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Cubist Model of the Inner “Red” DiskCubist Model of the Inner “Red” Disk
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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
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The Whole Cubist Girl CapThe Whole Cubist Girl Cap
Paper model
Smoothed computer rendering
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Epilogue: Apéry’s 2Epilogue: Apéry’s 2ndnd Cubist Model Cubist Model
Another model of the projective plane
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Apery’s Net of the 2Apery’s Net of the 2ndnd Cubist Model Cubist Model
( somewhat “conceptual” ! )
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My First Paper ModelMy First Paper Model
Too small! – Some elements out of order!
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Enhanced Apery ModelEnhanced Apery Model
Add color, based on face orientation
Clarify and align intersection diagram
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Enhanced Net Enhanced Net Intersection lines
Mountain folds
Valley folds
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My 2My 2ndnd Attempt at Model Building Attempt at Model Building
The 3 folded-up components -- shown from two directions each.
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Combining the ComponentsCombining the Components
2 parts merged
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All 3 Parts CombinedAll 3 Parts Combined
Bottom face opened to show inside
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Complete Colored ModelComplete Colored Model
6 colors for 6 different face directions
Views from diagonally opposite corners
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The Net With Colored Visible FacesThe Net With Colored Visible Faces
Based on visibility, orientation
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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)
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Art - ConnectionArt - Connection
“Heart of a Girl” Cubist Intersection Neighborhood
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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
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S P A R ES P A R E
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Transformation Seen in Domain SpaceTransformation Seen in Domain Space