CSE325 Computers and Sculpture

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CSE325 Computersand Sculpture

• Prof. George Hart

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Symmetry

• Intuitive notion mirrors, rotations,
• Mathematical concept set of transformations
• Possible 2D and 3D symmetries
• Sculpture examples
• M.C. Escher sculpture
• Carlo Sequins EscherBall program
• Constructions this week based on symmetry

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Intuitive uses of symmetry

• left side right side
• Human body or face
• n-fold rotation
• Flower petals
• Other ways?

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Mathematical Definition

• Define geometric transformations
• reflection, rotation, translation (slide),
• glide reflection (slide and reflect), identity,
  
• A symmetry is a transformation
• The symmetries of an object are the set of
  transformations which leave object looking
  unchanged
• Think of symmetries as axes, mirror lines,

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Frieze Patterns
Imagine as infinitely long. Each frieze has
translations. A smallest translation generates
all translations by repetition and
inverse. Some have vertical mirror lines. Some
have a horizontal mirror. Some have 2-fold
rotations. Analysis shows there are exactly seven
possibilities for the symmetry.
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Wallpaper Groups

• Include 2 directions of translation
• Might have 2-fold, 3-fold, 6-fold rotations,
  mirrors, and glide-reflections
• 17 possibilities
• Several standard notations. The following slides
  show the orbifold notation of John Conway.

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Wallpaper Groups
o
2222
xx

2222
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Wallpaper Groups
442
x
22x
442
222
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Wallpaper Groups
333
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Images by Xah Lee
632
632
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3D Symmetry

• Three translation directions give the 230
  crystallographic space groups of infinite
  lattices.
• If no translations, center is fixed, giving the
  14 types of polyhedral groups
• 7 families correspond to a rolled-up frieze
• Symmetry of pyramids and prisms
• Each of the seven can be 2-fold, 3-fold, 4-fold,
• 7 correspond to regular polyhedra

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Roll up a Frieze into a Cylinder
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Seven Polyhedra Groups

• Octahedral, with 0 or 9 mirrors
• Icosahedral, with 0 or 15 mirrors
• Tetrahedral, with 0, 3, or 6 mirrors
• Cube and octahedron have same symmetry
• Dodecahedron and icosahedron have same symmetry

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Symmetries of cube Symmetries of octahedron
In dual position symmetry axes line up
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Cube Rotational Symmetry

• Axes of rotation
• Three 4-fold through opposite face centers
• four 3-fold through opposite vertices
• six 2-fold through opposite edge midpoints
• Count the Symmetry transformations
• 1, 2, or 3 times 90 degrees on each 4-fold axis
• 1 or 2 times 120 degrees on each 3-fold axis
• 180 degrees on each 2-fold axis
• Identity transformation
• 9 8 6 1 24

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Cube Rotations may or may not Come with Mirrors
If any mirrors, then 9 mirror planes. If put
squiggles on each face, then 0 mirrors
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Icosahedral Dodecahedral Symmetry
Six 5-fold axes. Ten 3-fold axes. Fifteen
2-fold axes There are 15 mirror planes. Or
squiggle each face for 0 mirrors.
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Tetrahedron Rotations
Four 3-fold axes (vertex to opposite face
center). Three 2-fold axes.
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Tetrahedral Mirrors

• Regular tetrahedron has 6 mirrors (1 per edge)
• Squiggled tetrahedron has 0 mirrors.
• Pyrite symmetry has tetrahedral rotations but 3
  mirrors

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Symmetry in Sculpture

• People Sculpture (G. Hart)
• Sculpture by M.C. Escher
• Replicas of Escher by Carlo Sequin
• Original designs by Carlo Sequin

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People
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Candy BoxM.C. Escher
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Sphere with FishM.C. Escher, 1940
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Carlo Sequin, after Escher
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Polyhedron with FlowersM.C. Escher, 1958
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Carlo Sequin, after Escher
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Sphere with Angels and DevilsM.C. Escher, 1942
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Carlo Sequin, after Escher
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M.C. Escher
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Construction this Week

• Wormballs
• Pipe-cleaner constructions
• Based on one line in a 2D tessellation

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• The following slides are borrowed from
• Carlo Sequin

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Escher Sphere Construction Kit
Jane YenCarlo SéquinUC BerkeleyI3D 2001
1 M.C. Escher, His Life and Complete Graphic
Work
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Introduction

• M.C. Escher
• graphic artist print maker
• myriad of famous planar tilings
• why so few 3D designs?

2 M.C. Escher Visions of Symmetry
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Spherical Tilings

• Spherical Symmetry is difficult
• Hard to understand
• Hard to visualize
• Hard to make the final object

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Our Goal

• Develop a system to easily design and manufacture
  Escher spheres - spherical balls composed of
  tiles
• provide visual feedback
• guarantee that the tiles join properly
• allow for bas-relief
• output for manufacturing of physical models

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Interface Design

• How can we make the system intuitive and easy to
  use?
• What is the best way to communicate how spherical
  symmetry works?

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Spherical Symmetry

• The Platonic Solids

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How the Program Works

• Choose a symmetry based on a Platonic solid
• Choose an initial tiling pattern to edit
• starting place
• Example
• Tetrahedron

R3
R2
Tile 1
Tile 2
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Initial Tiling Pattern
easier to understand consequences of moving
points guarantees proper tiling requires user
to select the right initial tile - can only
make monohedral tiles
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Tile 2
Tile 1
Tile 2
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Modifying the Tile

• Insert and move boundary points
• system automatically updates the tile based on
  symmetry
• Add interior detail points

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Adding Bas-Relief

• Stereographically projected and triangulated
• Radial offsets can be given to points
• individually or in groups
• separate mode from editing boundary points

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Creating a Solid

• The surface is extruded radially
• inward or outward extrusion, spherical or
  detailed base
• Output in a format for free-form fabrication
• individual tiles or entire ball

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Video
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Fabrication Issues

• Many kinds of manufacturing technology
• we use two types based on a layer-by-layer
  approach

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FDM Fabrication
moving head
Inside the FDM machine
support material
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Z-Corp Fabrication
infiltration
de-powdering
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Results
FDM
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Results
FDM Z-Corp
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Results
FDM Z-Corp
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Results
Z-Corp
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Conclusions

• Intuitive Conceptual Model
• symmetry groups have little meaning to user
• need to give the user an easy to understand
  starting place
• Editing in Context
• need to see all the tiles together
• need to edit the tile on the sphere
• editing in the plane is not good enough
  (distortions)
• Part Fabrication
• need limitations so that designs can be
  manufactured
• radial manipulation
• Future Work
• predefined color symmetry
• injection molded parts (puzzles)
• tessellating over arbitrary shapes (any genus)