CSE325
Computers
and Sculpture
Prof. George Hart
Symmetry
•
•
•
•
Intuitive notion – mirrors, rotations, …
Mathematical concept — set of transformations
Possible 2D and 3D symmetries
Sculpture examples:
– M.C. Escher sculpture
– Carlo Sequin’s EscherBall program
• Constructions this week based on symmetry
Intuitive uses of “symmetry”
• left side = right side
– Human body or face
• n-fold rotation
– Flower petals
• Other ways?
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, …
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.
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.
Wallpaper Groups
o
2222
xx
**
*2222
22*
Wallpaper Groups
22x
x*
*442
2*22
442
4*2
Wallpaper Groups
333
*333
632
3*3
*632
Images by
Xah Lee
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
Roll up a Frieze into a Cylinder
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
Symmetries of cube = Symmetries of octahedron
In “dual position” symmetry axes line up
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:
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–
–
–
–
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
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
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.
Tetrahedron Rotations
Four 3-fold axes (vertex to opposite face center).
Three 2-fold axes.
Tetrahedral Mirrors
• Regular tetrahedron has 6 mirrors (1 per edge)
• “Squiggled” tetrahedron has 0 mirrors.
• “Pyrite symmetry” has tetrahedral rotations but 3
mirrors:
Symmetry in Sculpture
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People Sculpture (G. Hart)
Sculpture by M.C. Escher
Replicas of Escher by Carlo Sequin
Original designs by Carlo Sequin
People
Candy Box
M.C. Escher
Sphere with Fish
M.C. Escher, 1940
Carlo Sequin, after Escher
Polyhedron with Flowers
M.C. Escher, 1958
Carlo Sequin, after Escher
Sphere with Angels and Devils
M.C. Escher, 1942
Carlo Sequin, after Escher
M.C. Escher
Construction this Week
• Wormballs
– Pipe-cleaner constructions
– Based on one line in a 2D tessellation
The following slides are borrowed from
Carlo Sequin
Escher Sphere Construction Kit
Jane Yen
Carlo Séquin
UC Berkeley
I3D 2001
[1] M.C. Escher, His Life and Complete Graphic Work
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
Spherical Tilings

Spherical Symmetry is difficult
– Hard to understand
– Hard to visualize
– Hard to make the final object
[1]
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
Interface Design

How can we make the system intuitive and
easy to use?

What is the best way to communicate how
spherical symmetry works?
[1]
Spherical Symmetry

The Platonic Solids
tetrahedron
R3
R5
octahedron
R3
cube
R5
dodecahedron
R3
icosahedron
R2
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
R3
R3
R2
R2
R3
R3
R2
R3
Tile 1
Tile 2
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
[2]
Tile 1
Tile 2
Tile 2
Modifying the Tile

Insert and move boundary points
– system automatically updates the tile based on symmetry

Add interior detail points
Adding Bas-Relief
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Stereographically projected and triangulated

Radial offsets can be given to points
– individually or in groups
– separate mode from editing boundary points
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
Video
Fabrication Issues

Many kinds of manufacturing technology
– we use two types based on a layer-by-layer approach
Fused Deposition Modeling (FDM) Z-Corp 3D Color Printer
- parts made of plastic
- each part is a solid color
assembly
- starch powder glued together
- parts can have multiple colors
FDM Fabrication
moving
head
Inside the FDM machine
support
material
Z-Corp Fabrication
de-powdering
infiltration
Results
FDM
Results
FDM | Z-Corp
Results
FDM | Z-Corp
Results
Z-Corp
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)