Solid Freeform Fabrication of Aesthetic Objects

George W. Hart

Stony Brook University

Dept. Computer Science

1. Sculpture

2. Mathematical Models:

• Fractals

• Goldberg Polyhedra

• 4D Polytope Projections

3. Historical Leonardo Models

4. Orderly Tangles

5. Puzzles

6. Student Projects

1. Sculpture

72 Pencils

Chronosynclastic Infundibulum

Just Two Cavities

Disk Combobulation

The Plastic Tableware of Damocles

Rainbow Bits

Fire and Ice

I’d like to make one thing perfectly clear

Battered Moonlight

Yin and Yang

Gazmogenesis

Rorschachohedron

Fat and Skinny

Whoville

Whoville -- detail

Roads Untaken

The Susurrus of the Sea

Loopy

Millennium Bookball

Twisted Rivers, Knotted Sea

Twisted Rivers, Knotted Sea

Twisted Rivers -- reconstruction

Twisted Rivers – RP design

Tangled Reindeer

This End Up

Knot Structured

Nessie

Quintessence of Hedgehog

Bouquet

Frabjous

Salamanders

SFF Model (2.5 inch diam.)

Complete!

“People”

Snarl — wood model, 14 inch

Snarl — SFF model, 1 inch

Snarl — SFF model, 2 inch

Star Corona

Compass Points

Compass Points

“Cagework”

Aardvarks

Spaghetti Code

Rendering

Rendering, along 5-fold axis

Solid Freeform Fabrication Model

Spaghetti Code

Deep Structure

Gonads of the Rich and Famous

Gonads of the Rich and Famous-- design --

Gonads of the Rich and Famous-- design --

Untitled

Untitled

Untitled

Untitled

Untitled

2. Mathematical Models

Fractal — Sierpinski Tetrahedron

Connected Structure

Snub Truncated Icosahedron

Zonohedron-61

4D Polytope Projection Models

FDM Machine with 120-Cell Model

120-Cell, before dissolving support

120-Cell

• 120 cells, 720 faces, 1200 edges, 600 vertices.

• Cell-first parallel projection,(shows less than half of the edges.)

Five-Legged-Bee Hive

120-Cell – perspective projection

(smallest ?) 120-Cell

Wax model, made on Sanders machine

120-Cell – perspective projection

Selective laser sintering

Truncated 120-Cell

Truncated 120-Cell - Stereolithography

Zometool Truncated 120-Cell

MathCamp 2000

Ambo 600-Cell

Bridges Conference, 2001

Ambo 120-Cell

Orthogonal projection

Stereolithography

Can do with Zome

48 Truncated Cubes

Poor FDM model

Goldberg Polyhedra(Fullerenes)

“Buckyballs”• C60, form of carbon

• 1985 discovery

• 1996 Nobel Prize in chemistry

• By:– Richard Smalley (Rice Univ.)– Robert Curl (Rice Univ.)– Sir Harold Kroto

Geodesic Dome from Icosahedron

Dual Tessellations

Triangular Subdivisions

Area = x2 + xt + t2

Goldberg Polyhedra

(2,0)

(2,1)

(3,0)

Goldberg(2,2) Goldberg(4,1)

Goldberg (3,3) Goldberg(6,0)

Goldberg (7,3)

Goldberg (7,3)

Ernst Haeckel

Ernst Haeckel

Artificial Radiolarian

Goldberg (2,1)

Goldberg (3,1)

Goldberg (3,1)

Metal Goldberg (8, 3)

Nested Spheres

Nested Balls

Untitled (Abraham Sharp)

Untitled (Abraham Sharp)

3. Historical Models Leonardo / Pacioli

Sphere Model from Euclid

Leonardo Model

Leonardo Model

4. Orderly Tangles

Regular Polylinks

• Symmetric linkages of regular polygons

• Alan Holden built models– Cardboard or dowels

• Holden wrote:– Shapes, Spaces and

Symmetry,1971– “Regular Polylinks”,

1980– Orderly Tangles, 1983

• Table of lengths 4 Triangles

Regular Polylinks

4 Triangles 6 Squares

Left and right hand forms

Regular Polylinks

12 Pentagons

Regular Polylinks

6 Pentagons - size scaled

Derivation from Regular Polyhedron

Rotate faces Slide in or out

Wooden Puzzles

• Taiwan– Teacher Lin– Sculptor Wu

• Square cross sections

• Simple lap joint• No glue• Trial and error to

determine length12 Pentagons

Second Puzzle from Lin and Wu

10 Triangles

Many Analogous Puzzles

Possible• Each regular polylink gives a puzzle• Also can combine several together:

– Different ones interweaved– Same one nested

• Need critical dimensions to cut lengths• No closed-form formulas for lengths• Wrote program to:

– Determine dimensions– Output templates to print, cut, assemble– Output STL files for solid freeform

fabrication

Generates Template to Print and Cut

4 Triangles

Paper or Wood Models

6 Squares

Solid Freeform Fabrication

6 Squares

Square Cross Section

6 Pentagons

Paper or Wood Models

Orderly Tangle Puzzle

aluminum

30 parts

Selective Laser Sintering (SLS)

Selective Laser Sintering

Curved ComponentsCentral Inversion

4 Triangles Inverted

Curved ComponentsCentral Inversion

20 Triangles Inverted

SLS Models — Inverted Tangles

4 triangles 5 tetrahedra

Tangle Software

Available free at

http://www.georgehart.com

5. Puzzles

Sculpture Puzzles

aluminum

12 parts

Sculpture Puzzles

SLS

3 inch

30-Piece Puzzle

6. Student Projects

CSE391 Special TopicsAlgorithms for Solid Modeling

Student Projects