CS Scholars Program, April 8, 2015

The Power & Beauty of Geometry and the Secret of a Happy Life

Carlo H. Séquin University of California, Berkeley

Important Questions



Why are you in College ?

Important Questions



Why in EECS ?

Important Questions



What do you hope to get out of your four college years ?

Important Questions

 What is the secret of a happy life ?

Another Important Task

 Broaden your horizon !

 Find out what you really like to do.

M N G

Basel, Switzerland

Jakob Bernoulli (1654‒1705)

Logarithmic Spiral

Leonhard Euler (1707‒1783)

Imaginary Numbers

Descriptive Geometry

Geometry in every assignment . . .

CCD TV Camera (1973) Soda Hall (1992) RISC 1 MicroChip (1982) 3D-Yin-Yang (2000)

Recent Designs and Models

Brent Collins (1997)

“ Hyperbolic Hexagon II ”

Brent Collins: Stacked Saddles

All photos by Phillip Geller

The Math in Collins’ Sculptures

 Collins works with rulers and compasses; any math in his early work is intuitive.

 He is inspired by nature, e.g. soap films (= minimal area surfaces).

 Prof. George Francis: “Connection to math.

Minimal Surfaces!”

Scherk’s 2 nd Minimal Surface (1834)

 The central part of this is a “Scherk Tower.”

Generalizing the “Scherk Tower”

Normal “ biped ” saddles “ Scherk Tower ” Generalization to higher-order saddles (“Monkey saddle”)

Closing the Loop

straight or twisted “ Scherk Tower ” “ Scherk-Collins Toroids ”

Brent Collins:

Hyperbolic Hexagon

Six balanced saddles in a circular ring.

Inspired by the shape of a soap film suspended in a wire frame.

= Deformed “Scherk Tower”.

Sculpture Generator 1, GUI

Shapes from Sculpture Generator 1

Profiled Slice through

“

Heptoroid

”  One thick slice thru sculpture, from which Brent can cut boards and assemble a rough shape.

 Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board.

Emergence of the Heptoroid (1)

Assembly of the precut boards

Emergence of the Heptoroid (2)

Forming a continuous smooth edge

The Finished

Heptoroid

 at Fermi Lab Art Gallery (1998).

Snowsculpting Championships 2003

“Whirled White Web” (C. S équin, S. Wagon, D. Schwalbe, B. Collins, S. Reinmuth)

12:40 pm -- 42

°

F

12:41 pm -- 42

°

F

“

WWW

”

Wins Silver Medal

V-art

Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

Yet Another Medium: Stone

Progress picture from Dingli Stone Carving Art Co., SE China

Spring, 2012

12-Story Scherk-Collins Toroid

                   branches = 4 storeys = 11 height = 1.55

flange = 1.00

thickness = 0.06

rim_bulge = 1.00

warp = 330.00

twist = 247.50

azimuth = 56.25

mesh_tiles = 0 textr_tiles = 1 detail = 8 bounding box: xmax= 6.01, ymax= 1.14, zmax= 5.55, xmin= -7.93, ymin= -1.14, zmin= -8.41

12 Signs of the Zodiac

David Lynn,

Nova Blue Studio Arts  http://sites.google.com/site/novabluestudioarts/

Master Module for

“

Millennium Arch

”

Fabrication of

“

Millennium Arch

” The mold for the key module A polyester segment cast

Two Times Three Modules

Merging the Two Half-Circles

Brent Collins and David Lynn

“

Millennium

”

Arch by Night

Vitruvian Man by Leonardo Millennium Man

ART



MATH

Inspiring Sculptures by Brent Collins Procedural Capture in Sculpture Generator

MATH



ART

Making a Single-Sided Surface

Twisting a ribbon into a Möbius band

Simple Möbius Bands

 A single-sided surface with a single edge : A closed ribbon with a 180 ° flip.

A closed ribbon with a 540 ° flip.

Twisted Möbius Bands in Art

Web Max Bill M.C. Escher M.C. Escher

Deformations of Möbius Bands

Sue-Dan-ese Boy-Cap Möbius band

Classical

“

Inverted-Sock

”

Klein Bottle

Felix Klein (1849-1925)

Fancy Klein Bottles

Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK

(1)

How to Construct a Klein Bottle

(2) (3) (4)

2 Möbius Bands Make a Klein Bottle

KOJ = MR + ML

Limerick

A mathematician named Klein thought Möbius bands are divine.

Said he: " If you glue the edges of two, you'll get a weird bottle like mine.

"

Split Klein-Bottle Model

 Made on an FDM machine

Klein Bottle made from two Boy-Caps

A Boy Cap is a Möbius band !

+ =

Two Möbius bands make a Klein bottle !

Klein Bottle from Mirrored Boy-Caps

Polyhedron Subdvision Gridded

Actual Sculpture Model

S 6 Klein Bottle Rendered by C. Mouradian

http://netcyborg.free.fr/

More Complex Single-sided Surfaces

 To make a surface of genus

h

, punch

h

holes into a sphere and close them up with Boy-Caps.

 A Klein bottle is of genus 2, it uses two Boy-Caps.

Construction of a Genus-4 Surface

4 Boy caps grafted onto a sphere with tetra symmetry

Octa-Boy Sculpture

The Bridges Conference

Mathematical Connections in Art, Music, and Science

the largest, best-established, annual Math / Art conference in the world www.BridgesMathArt.org

My Favorite Annual Conference: 2014

 BRIDGES Art …

“

LEGO-Knots

”  2 types of end-caps; 3 curved connectors

Inspiration for

“

LEGO-Knots

” Henk van Putten

“

LEGO-Knot

”

Realization of “Borsalino”

E R=1.0

C R=2.4142

Two modular components can form the

Borsalino

“

Pas de Deux

”

Making Sculptures Glow …

Glimpses of My Portfolio

 20 talks at the “ Bridges ” Math-Art conferences

Assets Beyond Course Credits

Stellar grades are not everything! (But keep GPA above 3.0).

What recruiters are looking for:  Demonstrable achievements  Strong recommendation letters  Get involved in research early !

 Do more than one project; get to know more than one professor.

Look for Posted URAP Projects

Some of my recent URAP projects (spanning Art and Science):  Building a “ LEGO-Knot ” system  "Ribbon/Surface Sculptures" – Generator for sculptures in the style of Charles Perry or Eva Hild.

 "7-Around" – Assembling a large hyperbolic "disk" from equilateral triangles.

 “ The Beauty of Knots ”

Ribbon-Surface Sculpture Generator

Example:

TETRA

by Charles Perry (1999) Original (bronze) CAD Model FDM Maquette

Large “7-Around” Hyperbolic Disks made from flat equilateral triangles

Crochet model Poincar é disk CAD model

“

The Beauty of Knots

”

Presenting Your Accomplishments

Build up a Portfolio:  Course project reports;  Demonstrations of creative work;  Photos of things you built.

Prepare your “Elevator-Speech”:  1 – 2 minute summary of your work interesting enough that a listener wants to hear more, and perhaps wants to see your portfolio!

Torus-Knot_5,3

Inspired by a well defined type of mathematical knot Torus-Knot_5,3

Chinese Button Knot (Knot 9

40

)

Bronze, Dec. 2007 Carlo S équin cast & patina by Steve Reinmuth

Figure-8 Knot Bronze, Dec. 2007 Carlo S équin

2 nd Prize, AMS Exhibit 2009

Granny-Knot-Lattice (S é quin, 1981)

Metal Sculpture at SIGGRAPH 2006

“Volution’s Evolution” (Patina ’d Bronze, 2013)

“Pax Mundi” (Bronze, 2007) Team effort: Brent Collins, Steve Reinmuth, Carlo S équin

Music of the Spheres, MWSU 2013

Photo by Phillip Geller

Evolving Trefoil (polyester resin, 2013)

Inauguration Sutardja Dai Hall 2/27/09

Pillar of Engineering (2012)

QUESTIONS ?

?

http://www.cs.berkeley.edu/~sequin/TALKS/

What came first: Art or Mathematics ?

 Question posed Nov. 16, 2006 by Dr. Ivan Sutherland “ father ” of computer graphics (

SKETCHPAD

, 1963).

Early “Free-Form” Art

Cave paintings, Lascaux Venus von Willendorf

Regular, Geometric Art

 Early art: Patterns on bones, pots, weavings...

 Mathematics (geometry) to help make things fit:

Another Question:

What came first: Art or Science?

What is Art ? -- What do artists do ? What is Science ? -- What do scientists do ?

ART



SCIENCE

Scientists are model-builders .

 They carefully observe a domain of interest.

 Then they cast their findings into a predictive model (which may be refined over time).

Artists also start with observations of the world ,  then they render it from their own perspective,  emphasizing certain aspects to make some statement,  or projecting an alternate vision of the world.