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 PatternsImagine 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

** *2222 22*

xx

Wallpaper Groups

22x x* *442

4*24422*22

Wallpaper Groups

333

*632632

3*3*333

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:– 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

• People Sculpture (G. Hart)

• Sculpture by M.C. Escher

• Replicas of Escher by Carlo Sequin

• Original designs by Carlo Sequin

People

Candy BoxM.C. Escher

Sphere with FishM.C. Escher, 1940

Carlo Sequin, after Escher

Polyhedron with FlowersM.C. Escher, 1958

Carlo Sequin, after Escher

Sphere with Angels and DevilsM.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

Jane YenCarlo Séquin

UC Berkeley

I3D 2001

[1] M.C. Escher, His Life and Complete Graphic Work

Escher Sphere Construction Kit

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 octahedron cube dodecahedron icosahedron

R3 R5 R5R3 R3 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

R2R3

R2

R3

R3

R3

R2

Tile 1 Tile 2

R3

R2

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 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 - starch powder glued together - each part is a solid color - parts can have multiple colors assembly

FDM Fabrication

supportmaterial

movinghead

Inside the FDM machine

Z-Corp Fabrication

infiltrationde-powdering

ResultsFDM

ResultsFDM | Z-Corp

ResultsFDM | Z-Corp

ResultsZ-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)