mosaic, seattle, aug. 2000 turning mathematical models into sculptures carlo h. séquin university...
TRANSCRIPT
![Page 1: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/1.jpg)
CHSCHS
UCBUCB MOSAIC, Seattle, Aug. 2000MOSAIC, Seattle, Aug. 2000
Turning Mathematical Models
into Sculptures
Carlo H. Séquin
University of California, Berkeley
![Page 2: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/2.jpg)
CHSCHS
UCBUCB Boy Surface in OberwolfachBoy Surface in Oberwolfach
Sculpture constructed by Mercedes Benz
Photo from John Sullivan
![Page 3: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/3.jpg)
CHSCHS
UCBUCB Boy Surface by Helaman FergusonBoy Surface by Helaman Ferguson
Marble
From: “Mathematics in Stone and Bronze”by Claire Ferguson
![Page 4: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/4.jpg)
CHSCHS
UCBUCB Boy Surface by Benno ArtmannBoy Surface by Benno Artmann
From home page of Prof. Artmann,TU-Darmstadt
after a sketch byGeorge Francis.
![Page 5: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/5.jpg)
CHSCHS
UCBUCB Samples of Mathematical SculptureSamples of Mathematical Sculpture
Questions that may arise:
Are the previous sculptures really all depicting the same object ?
What is a “Boy surface” anyhow ?
![Page 6: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/6.jpg)
CHSCHS
UCBUCB The Gist of my TalkThe Gist of my Talk
Topology 101:
Study five elementary 2-manifolds
(which can all be formed from a rectangle)
Art-Math 201:
The appearance of these shapes as artwork
(when do math models become art ? )
![Page 7: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/7.jpg)
CHSCHS
UCBUCB What is Art ?What is Art ?
![Page 8: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/8.jpg)
CHSCHS
UCBUCB Five Important Two-ManifoldsFive Important Two-Manifolds
cylinder Möbius band
torus Klein bottle cross-cap
X=0 X=0
X=0 X=0 X=1G=1 G=2 G=1
![Page 9: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/9.jpg)
CHSCHS
UCBUCB Deforming a RectangleDeforming a Rectangle
All five manifolds can be constructed by starting with a simple rectangular domain and then deforming it and gluing together some of its edges in different ways.
cylinder Möbius band torus Klein bottle cross-cap
![Page 10: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/10.jpg)
CHSCHS
UCBUCB Cylinder ConstructionCylinder Construction
![Page 11: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/11.jpg)
CHSCHS
UCBUCB Möbius Band ConstructionMöbius Band Construction
![Page 12: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/12.jpg)
CHSCHS
UCBUCB Cylinders as SculpturesCylinders as Sculptures
Max Bill John Goodman
![Page 13: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/13.jpg)
CHSCHS
UCBUCB The Cylinder in ArchitectureThe Cylinder in Architecture
Chapel
![Page 14: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/14.jpg)
CHSCHS
UCBUCB Möbius Sculpture by Max BillMöbius Sculpture by Max Bill
![Page 15: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/15.jpg)
CHSCHS
UCBUCB Möbius Sculptures by Keizo UshioMöbius Sculptures by Keizo Ushio
![Page 16: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/16.jpg)
CHSCHS
UCBUCB More Split Möbius BandsMore Split Möbius Bands
Typical lateral splitby M.C. Escher
And a maquette made by Solid Free-form Fabrication
![Page 17: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/17.jpg)
CHSCHS
UCBUCB Torus ConstructionTorus Construction
Glue together both pairs of opposite edges on rectangle
Surface has no edges
Double-sided surface
![Page 18: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/18.jpg)
CHSCHS
UCBUCB Torus Sculpture by Max BillTorus Sculpture by Max Bill
![Page 19: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/19.jpg)
CHSCHS
UCBUCB ““Bonds of Friendship” J. RobinsonBonds of Friendship” J. Robinson
1979
![Page 20: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/20.jpg)
CHSCHS
UCBUCB Proposed Torus “Sculpture”Proposed Torus “Sculpture”
“Torus! Torus!” inflatable structure by Joseph Huberman
![Page 21: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/21.jpg)
CHSCHS
UCBUCB ““Rhythm of Life” by John RobinsonRhythm of Life” by John Robinson
“DNA spinning within the Universe” 1982
![Page 22: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/22.jpg)
CHSCHS
UCBUCB Virtual Torus SculptureVirtual Torus Sculpture
“Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd.
Note:
Surface is representedby a loose set of bands
==> yields transparency
![Page 23: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/23.jpg)
CHSCHS
UCBUCB Klein Bottle -- “Classical”Klein Bottle -- “Classical”
Connect one pair of edges straightand the other with a twist
Single-sided surface -- (no edges)
![Page 24: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/24.jpg)
CHSCHS
UCBUCB Klein Bottles -- virtual and realKlein Bottles -- virtual and real
Computer graphics by John Sullivan
Klein bottle in glassby Cliff Stoll, ACME
![Page 25: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/25.jpg)
CHSCHS
UCBUCB Many More Klein Bottle Shapes !Many More Klein Bottle Shapes !
Klein bottles in glass by Cliff Stoll, ACME
![Page 26: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/26.jpg)
CHSCHS
UCBUCB Klein MugsKlein Mugs
Klein bottle in glassby Cliff Stoll, ACME
Fill it with beer --> “Klein Stein”
![Page 27: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/27.jpg)
CHSCHS
UCBUCB Dealing with Self-intersectionsDealing with Self-intersections
Different surfaces branches should “ignore” one another !
One is not allowed to step from one branch of the surface to another.
==> Make perforated surfaces and interlace their grids.
==> Also gives nice transparency if one must use opaque materials.
==> “Skeleton of a Klein Bottle.”
![Page 28: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/28.jpg)
CHSCHS
UCBUCB Klein Bottle Skeleton (FDM)Klein Bottle Skeleton (FDM)
![Page 29: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/29.jpg)
CHSCHS
UCBUCB Klein Bottle Skeleton (FDM)Klein Bottle Skeleton (FDM)
Struts don’t intersect !
![Page 30: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/30.jpg)
CHSCHS
UCBUCB Fused Deposition ModelingFused Deposition Modeling
![Page 31: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/31.jpg)
CHSCHS
UCBUCB Looking into the FDM MachineLooking into the FDM Machine
![Page 32: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/32.jpg)
CHSCHS
UCBUCB Layered Fabrication of Klein BottleLayered Fabrication of Klein Bottle
Support material
![Page 33: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/33.jpg)
CHSCHS
UCBUCB Another Type of Klein BottleAnother Type of Klein Bottle
Cannot be smoothly deformed into the classical Klein Bottle
Still single sided -- no edges
![Page 34: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/34.jpg)
CHSCHS
UCBUCB
Woven byCarlo Séquin,16’’, 1997
Figure-8 Figure-8 Klein BottleKlein Bottle
![Page 35: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/35.jpg)
CHSCHS
UCBUCB Triply Twisted Fig.-8 Klein BottleTriply Twisted Fig.-8 Klein Bottle
![Page 36: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/36.jpg)
CHSCHS
UCBUCB Triply Twisted Fig.-8 Klein BottleTriply Twisted Fig.-8 Klein Bottle
![Page 37: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/37.jpg)
CHSCHS
UCBUCB Avoiding Self-intersectionsAvoiding Self-intersections
Avoid self-intersections at the crossover line of the swept fig.-8 cross section.
This structure is regular enough so that this can be done procedurally as part of the generation process.
Arrange pattern on the rectangle domain as shown on the left.
After the fig.-8 - fold, struts pass smoothly through one another.
Can be done with a single thread for red and green !
![Page 38: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/38.jpg)
CHSCHS
UCBUCB Single-thread Figure-8 Klein BottleSingle-thread Figure-8 Klein Bottle
Modelingwith SLIDE
![Page 39: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/39.jpg)
CHSCHS
UCBUCB Zooming into the FDM MachineZooming into the FDM Machine
![Page 40: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/40.jpg)
CHSCHS
UCBUCB Single-thread Figure-8 Klein BottleSingle-thread Figure-8 Klein Bottle
As it comes out of the FDM machine
![Page 41: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/41.jpg)
CHSCHS
UCBUCB Single-thread Figure-8 Klein BottleSingle-thread Figure-8 Klein Bottle
![Page 42: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/42.jpg)
CHSCHS
UCBUCB The Doubly Twisted Rectangle CaseThe Doubly Twisted Rectangle Case
This is the last remaining rectangle warping case.
We must glue both opposing edge pairs with a 180º twist.
Can we physically achieve this in 3D ?
![Page 43: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/43.jpg)
CHSCHS
UCBUCB Cross-cap ConstructionCross-cap Construction
![Page 44: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/44.jpg)
CHSCHS
UCBUCB Significance of Cross-capSignificance of Cross-cap
< 4-finger exercise >
What is this beast ?
A model of the Projective Plane An infinitely large flat plane.
Closed through infinity, i.e., lines come back from opposite direction.
But all those different lines do NOT meet at the same point in infinity; their “infinity points” form another infinitely long line.
![Page 45: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/45.jpg)
CHSCHS
UCBUCB The Projective PlaneThe Projective Plane
C
PROJECTIVE PLANE
-- Walk off to infinity -- and beyond … come back upside-down from opposite direction.
Projective Plane is single-sided; has no edges.
![Page 46: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/46.jpg)
CHSCHS
UCBUCB Cross-cap on a SphereCross-cap on a Sphere
Wood and gauze model of projective plane
![Page 47: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/47.jpg)
CHSCHS
UCBUCB ““Torus with Crosscap”Torus with Crosscap”
Helaman Ferguson
( Torus with Crosscap = Klein Bottle with Crosscap )
![Page 48: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/48.jpg)
CHSCHS
UCBUCB ““Four Canoes” by Helaman FergusonFour Canoes” by Helaman Ferguson
![Page 49: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/49.jpg)
CHSCHS
UCBUCB Other Models of the Projective PlaneOther Models of the Projective Plane
Both, Klein bottle and projective planeare single-sided, have no edges.(They differ in genus, i.e., connectivity)
The cross cap on a torusmodels a Klein bottle.
The cross cap on a spheremodels the projective plane,but has some undesirable singularities.
Can we avoid these singularities ?
Can we get more symmetry ?
![Page 50: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/50.jpg)
CHSCHS
UCBUCB Steiner Surface Steiner Surface (Tetrahedral Symmetry)(Tetrahedral Symmetry)
Plaster Model by T. Kohono
![Page 51: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/51.jpg)
CHSCHS
UCBUCB Construction of Steiner SurfaceConstruction of Steiner Surface
Start with three orthonormal squares …
… connect the edges (smoothly).
--> forms 6 “Whitney Umbrellas” (pinch points with infinite curvature)
![Page 52: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/52.jpg)
CHSCHS
UCBUCB Steiner Surface ParametrizationSteiner Surface Parametrization
Steiner surface can best be built from a hexagonal domain.
Glue opposite edges with a 180º twist.
![Page 53: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/53.jpg)
CHSCHS
UCBUCB Again: Alleviate Self-intersectionsAgain: Alleviate Self-intersections
Strut passesthrough hole
![Page 54: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/54.jpg)
CHSCHS
UCBUCB Skeleton of a Steiner SurfaceSkeleton of a Steiner Surface
![Page 55: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/55.jpg)
CHSCHS
UCBUCB Steiner SurfaceSteiner Surface
has more symmetry;
but still hassingularities(pinch points).
Can such singularities be avoided ? (Hilbert)
![Page 56: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/56.jpg)
CHSCHS
UCBUCB Can Singularities be Avoided ?Can Singularities be Avoided ?
Werner Boy, a student of Hilbert,was asked to prove that it cannot be done.
But found a solution in 1901 ! 3-fold symmetry
based on hexagonal domain
![Page 57: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/57.jpg)
CHSCHS
UCBUCB Model of Boy SurfaceModel of Boy Surface
Computer graphics by François Apéry
![Page 58: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/58.jpg)
CHSCHS
UCBUCB Model of Boy SurfaceModel of Boy Surface
Computer graphics by John Sullivan
![Page 59: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/59.jpg)
CHSCHS
UCBUCB Model of Boy SurfaceModel of Boy Surface
Computer graphics by John Sullivan
![Page 60: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/60.jpg)
CHSCHS
UCBUCB Quick Surprise TestQuick Surprise Test
Draw a Boy surface
(worth 100% of score points)...
![Page 61: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/61.jpg)
CHSCHS
UCBUCB Another “Map” of the “Boy Planet”Another “Map” of the “Boy Planet”
From book by Jean Pierre Petit “Le Topologicon” (Belin & Herscher)
![Page 62: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/62.jpg)
CHSCHS
UCBUCB Double Covering of Boy SurfaceDouble Covering of Boy Surface
Wire model byCharles Pugh
Decorated by C. H. Séquin:
Equator
3 Meridians, 120º apart
![Page 63: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/63.jpg)
CHSCHS
UCBUCB Revisit Boy Surface SculpturesRevisit Boy Surface Sculptures
Helaman Ferguson - Mathematics in Stone and Bronze
![Page 64: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/64.jpg)
CHSCHS
UCBUCB Boy Surface by Benno ArtmannBoy Surface by Benno Artmann
Windows carved into surface reveal what is going on inside. (Inspired by George Francis)
![Page 65: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/65.jpg)
CHSCHS
UCBUCB Boy Surface in OberwolfachBoy Surface in Oberwolfach
Note:parametrization indicated by metal bands; singling out “north pole”.
Sculpture constructed by Mercedes Benz
Photo by John Sullivan
![Page 66: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/66.jpg)
CHSCHS
UCBUCB Boy Surface SkeletonBoy Surface Skeleton
Shape defined by elastic properties of wooden slats.
![Page 67: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/67.jpg)
CHSCHS
UCBUCB Boy Surface Skeleton (again)Boy Surface Skeleton (again)
![Page 68: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/68.jpg)
CHSCHS
UCBUCB Goal: A “Regular” TessellationGoal: A “Regular” Tessellation
“Regular” Tessellation of the Sphere (Buckminster Fuller Domes.)
![Page 69: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/69.jpg)
CHSCHS
UCBUCB ““Ideal” Sphere ParametrizationIdeal” Sphere Parametrization
Buckminster Fuller Dome: almost all equal sized triangle tiles.
![Page 70: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/70.jpg)
CHSCHS
UCBUCB ““Ideal” Sphere ParametrizationIdeal” Sphere Parametrization
Epcot Center Sphere
![Page 71: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/71.jpg)
CHSCHS
UCBUCB Tessellation from Surface EvolverTessellation from Surface Evolver
Triangulation from start polyhedron.
Subdivision and merging to avoid large, small, and skinny triangles.
Mesh dualization.
Strut thickening.
FDM fabrication.
Quad facet !
Intersecting struts.
![Page 72: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/72.jpg)
CHSCHS
UCBUCB Paper Model with Regular TilesPaper Model with Regular Tiles
Only meshes with 5, 6, or 7 sides.
Struts pass through holes.
Only vertices where 3 meshes join.
--> Permits the use of a modular component...
![Page 73: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/73.jpg)
CHSCHS
UCBUCB The Tri-connectorThe Tri-connector
![Page 74: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/74.jpg)
CHSCHS
UCBUCB Tri-connector ConstructionsTri-connector Constructions
![Page 75: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/75.jpg)
CHSCHS
UCBUCB Tri-connector Ball Tri-connector Ball (20 Parts)(20 Parts)
![Page 76: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/76.jpg)
CHSCHS
UCBUCB ExpectationsExpectations
Tri-connector surface will be evenly bent,with no sharp kinks.
It will have intersections that demonstrate the independence of the two branches.
Result should be a pleasing model in itself.
But also provides a nice loose model of the Boy surface on which I can study various parametrizations, geodesic lines...
![Page 77: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/77.jpg)
CHSCHS
UCBUCB HopesHopes
This may lead to even better modelsof the Boy surface:
e.g., by using the geodesic linesto define ribbons that describe the surface
(this surface will keep me busy for a while yet !)
![Page 78: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/78.jpg)
CHSCHS
UCBUCB ConclusionsConclusions
There is no clear line that separatesmathematical models and art work.
Good models are pieces of art in themselves.
Much artwork inspired by such modelsis no longer a good model for understandingthese more complicated surfaces.
My goal is to make a few great modelsthat are appreciated as good geometric art,and that also serve as instructional models.
![Page 79: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/79.jpg)
CHSCHS
UCBUCB End of TalkEnd of Talk
![Page 80: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/80.jpg)
CHSCHS
UCBUCB === spares ====== spares ===
![Page 81: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/81.jpg)
CHSCHS
UCBUCB Rotating TorusRotating Torus
![Page 82: MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley](https://reader036.vdocuments.mx/reader036/viewer/2022062519/5697bfb61a28abf838c9e17b/html5/thumbnails/82.jpg)
CHSCHS
UCBUCB Looking into the FDM MachineLooking into the FDM Machine