Waterloo, March 2017
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Waterloo, March 2017

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Waterloo, March 2017




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Presentation on theme: "Waterloo, March 2017"— Presentation transcript:

Slide1

Waterloo, March 2017

Artistic Geometry

Carlo H.

Séquin

EECS Computer Sciences University of California, Berkeley

Slide2

Basel, Switzerland

M N G

Slide3

Jakob Bernoulli (1654‒1705)

Logarithmic Spiral

Slide4

Leonhard Euler (1707‒1783)

Imaginary Numbers

Slide5

Descriptive Geometry

Slide6

“Re-creations” of Inspiring Art

Alexander Calder: Mobiles

C. H. Séquin, circa 1957

Slide7

String Sculptures

?:

Naum Gabo String Sculptures

C.

H. Séquin, circa 1977

Slide8

Orderly Tangles

Alan Holden

“Glowing TetraTangle”C. H. Séquin (1983)

Slide9

Knots

Chinese Button Knot

C

. H. Séquin (1994)

Slide10

Simple Geometry

FDM Models by C. H. Séquin (2000)

Max Bill

Slide11

Ribbon Sculptures

Brent Collins: Pax Mundi (1994)

Stelvio

Altamont

Slide12

Saddle-Toroids

TwistedHexagon

Brent

Collins (1993) Hyperbolic Hexagon

Monkey

Trefoil

Slide13

Brent Collins (1997)

Hyperbolic Hexagon II

Slide14

Brent Collins: Stacked Saddles

All photos by Phillip Geller

Slide15

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!”

Slide16

Scherk’s 2nd Minimal Surface (1834)

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

Slide17

Generalizing the “Scherk

Tower”

Normal

biped

saddles

Generalization to

higher-order saddles(“Monkey saddle”)

Scherk Tower

Slide18

Closing the Loop

straight

or

twisted

Scherk Tower”

Scherk-Collins Toroids

Slide19

Brent Collins: Hyperbolic Hexagon(1993)

Six balanced saddles

in a circular ring.

Inspired by the shape

of a soap film

suspended in a wire frame.

=

Warped

Scherk

Tower

(with 6 stories).

Slide20

Brent Collins’ Prototyping Process

Armature for the "Hyperbolic Heptagon"

Mockup for the

"Saddle Trefoil"

Time-consuming ! (1-3 weeks)

Slide21

Sculpture Generator 1

, GUI

Slide22

Some of the Parameters in “SC1”

Slide23

Generated Scherk-Collins Shapes

Slide24

Base Geometry: One “Scherk Story”

Taylored

hyperbolas

, hugging a

circle

Hyperbolic Slices

Triangle

Strips

Pre-computed

-- then warped into toroid

Slide25

Shapes from Sculpture Generator 1

Slide26

Collins’ Fabrication Process

Example: “Vox Solis”

Layered laminated main shape

Wood master pattern

for sculpture

Slide27

Slices through “Minimal Trefoil”

50%

10%

23%

30%

45%

5%

20%

27%

35%

2%

15%

25%

Slide28

One thick slicethru 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, 3/4of one board.

Profiled Slice through

Heptoroid

Slide29

Emergence of the Heptoroid (1)

Assembly of the precut boards

Slide30

Emergence of the Heptoroid (2)

Forming a continuous smooth edge

Slide31

Emergence of the “Heptoroid” (3)

Smoothing the whole surface

Slide32

The Finished Heptoroid

at Fermi Lab Art Gallery (1998).

Slide33

Inauguration Sutardja Dai Hall 2/27/09

Slide34

“Scherk-Collins” Sculptures (FDM)

Slide35

Cohesion

SIGGRAPH’2003 Art Gallery

Slide36

Hypersculpture: Family of 12 Trefoils

W=2

W=1

B=1 B=2 B=3 B=4

Slide37

Going more than once around the loop

… results in an interwoven structure.

W

= 380°

W

= 560

°

W

= 720

°

Slide38

11 Stories, Monkey-Saddles, W=2:

cross–eye stereo picture

Slide39

9-story Intertwined Double Toroid

Bronzeinvestment casting from wax original made on 3D Systems’“Thermojet”

Slide40

Extension of Concept

Allow different kinds of “stretching” …

Slide41

Extending the Paradigm: Totem 3

Bronze Investment Cast

Slide42

Stepwise Expansion of Horizon

Playing with many different shapes andexperimenting at the limit of the domain of the sculpture generator,stimulates new ideas for alternative shapes and generating paradigms.

Swiss Mountains

Slide43

Sculpture Generator 1as a Playground

The computer becomes

an amplifier / accelerator

for the creative process.

Slide44

V-art

Virtual

Glass

Scherk

Tower

with

Monkey

Saddles

(Radiance

40 hours)

Jane Yen

Slide45

Slide46

Yet Another Medium: Stone

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

Slide47

Spring, 2012

Slide48

Slide49

Slide50

Slide51

Slide52

Slide53

The Viae Globi Series

Another example how one special piece of artled to a computer program, which then allowed me to make a whole series of sculpture designs that all seem to belong to the same family.

(Roads on a Sphere)

Slide54

Brent Collins’ Pax Mundi1997: Wood, 30”diam.

2006: Commission from

H&R Block, Kansas City

to make a 70”diameter

version in bronze.

My task:

Define the master geometry.

CAD tools play important role!

Slide55

How to Model Pax Mundi ...

Already addressed that issue in 1998:

Pax

Mundi

could

not

be done with

Sculpture Generator I

Needed a more general program !

Used the “Berkeley SLIDE” environment.

First:

Needed to find the basic paradigm

  

Slide56

Sculptures by Naum Gabo

Pathway on a sphere:Edge of surface is like seam of tennis- or base-ball; “2-period Gabo curve.”

Slide57

2-period “Gabo Curve”

Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used (symmetry!).

Slide58

4-period “Gabo Curve”

Same construction as for as for 2-period curve

Slide59

Pax Mundi Revisited

Can be seen as: “Amplitude modulated, 4-period Gabo curve”

Slide60

Progressive Sweeps

Sculpture is not just a mathematical curve.

There is some substance; it has volume.

Define shape by sweeping a cross section

along a given 3D space curve.

Slide61

SLIDE-GUI for “Pax Mundi” Shapes

Good combination of interactive 3D graphics

and parameterizable procedural constructs.

Slide62

Modularity of Gabo Sweep Generator

Sweep Curve Generator:Gabo Curves as B-splines:Cross Section Fine Tuner:Paramererized shapes:Sweep / Twist Controller:How is cross section applied?

Slide63

Intrinsic Sweep Mode

Keep cross section perpendicular to tangent

.

Place

cross section into the

x-y

-plane

of the

Frenet

frame.

Keep orientation / rotation as it was

in the defining

x-y

-coordinate system.

Add any additional azimuth angle

as a rotation around the

z

-axis (tangent).

Slide64

Intrinsic Sweep Mode

Problems at inflection points (in plane): Pinched-off “hour-glass” shapes.

“Natural” orientationwith Frenet frame

Slide65

Minimum-Torsion (-Rotation) Sweep

Project orientation of cross section forward,from one vertex of the sweep polyline to the next.Neutralize rotation of Frenet frame

intrinsic minimum torsion

Slide66

Azimuth / Twist Control

Starting with torsion/rotation minimization:

azimuth = 0, azimuth = 90, azimuth = 90, twist = 0; twist = 0; twist =180;

Slide67

Local Azimuth Control = “Warp”

Starting with torsion/rotation minimization:

azimuth = 0, azimuth = 0, twist = 0; warp = -90;

Slide68

Azimuth

/ Twist Control

Controls applied to the 2-period Gabo curve:

Natural orientation

with

Frenet

frame

Torsion Minimization:Azimuth: tangential / normal

900

° of twist

added.

Slide69

Target Geometry(2007)

Constraints:

Bronze, 70

diameter

Less than 1500 pounds

Less than $50

000

Maintain beauty, strength

Minimize master geometry

Slide70

Emulation; Define Master Pattern

Use 4 copies.

Master to make

a mold from.

Alignment tab

Slide71

Joe Valasek’s CNC Milling Machine

Styrofoam milling machine

Slide72

Machined Master Pattern #2

Slide73

(Cut) Master  Silicone Rubber Mold

Slide74

Mold  Several (4) Wax Copies

Slide75

Spruing the Wax Parts for Casting

Slide76

Ceramic Slurry Shell Around Wax Part

Slide77

Taking the Shell out of the Kiln

Slide78

Shell Ready for Casting

Slide79

The Pour

Slide80

Casting with Liquid Bronze

Slide81

Freeing the Bronze Cast

Slide82

Assembling the Segments

Slide83

The “Growing” Ribbon

Slide84

Assembly Completed

Slide85

Front Door of the ...

H&R Block Building

Slide86

Steve Reinmuth, Bronze Studio, Eugene OR

http://www.reinmuth.com/

Slide87

Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin

Slide88

Extension: Free-form Curve on a Sphere

Spherical Spline Path Editor (Jane Yen)

Smooth

interpolating curve

through sparse data points

Slide89

Many Different Viae Globi Models

Slide90

Paradigm Extension: Sweep Path is no longer confined to a sphere!

Chinese Button Knot

Slide91

ChineseButton Knot(Knot 940)Bronze, Dec. 2007Carlo Séquincast & patina bySteve Reinmuth

Slide92

Design Requires Paradigm Extension

Music of the Spheres

(Brent Collins)

Slide93

Music of the Spheres (Brent Collins)

Definition of Sweep Path

(hugging 4 different spheres)

Slide94

Partitioning; Joint Design

18 pieces:

fit in kiln!

1/3 = unique

geometry

Alignment stubs

Slide95

Some Segments Were Made Hollow

This needs a

double-walled tube mold!

Slide96

Some of the Hollow Metal Parts

Slide97

Assembly of Music of the Spheres

Slide98

Installation at MWSU, Feb. 2013

Steve Reinmuth Brent Collins

Slide99

Illuminated Music of the Spheres

Photo by Phillip Geller

Slide100

=== More Recent Endeavors ===

Modeling two different classes of sculpturesby Charles Perry and by Eva Hild.

Slide101

“Tetra”, Waterfront Park, Louisville, KYCharles Perry, 1999, bronze

Multiple views from the Web: Identify corresponding branches.

Slide102

Modeling “Tetra” by Charles Perry (2)

Crude Paper Models:

Assembling

labeled ribbons

Un-twisted tetrahedral frame

Twisted tetra frame

as in Perry’s

“Tetra”

Slide103

Modeling “Tetra” by Charles Perry (3)

Annotated sculpture image

Metal-rings plusscotch-tape model

CAD model ofPerry’s “Tetra”

Maquette

of

Perry’s “

Tetra

Slide104

Modification of Perry’s “Tetra” Sculpture

Using my generator for tetrahedral ribbon frames,individually adjusting the twist of all six ribbons:

Untwisted tetra frame – Emulating Perry’s “

Tetra” and “D2d”

4 ribbons have a ±360

 twist

Slide105

“Tetra_4M”Modification of Perry’s Tetra Sculpture

The four twisted tetra-edges rotate through only 180.This keeps the surface double-sided, but only 2 (different) borders.

Double-sided (

orientable)Number of borders b = 2Euler characteristic χ = –2Genus g = (2 – χ – b)/2 = 1It is a torus with 2 punctures.

Original -- Modified

3D-Print, painted

Slide106

“Tetra_6M”Modification of Perry’s Tetra Sculpture

All SIX tetra-edges are twisted through 180.This also makes it single-sided!This one has 3 identical borders,forming a Borromean link !

Slide107

2-Manifold Sculptures

“Wholly” by Eva Hild (Sweden)

Slide108

Parameterized Control Mesh

Coarse control mesh attached to 9 panelsthat can be moved andscaled individually Surface after 3 levels of CC - subdivision

Slide109

QUESTIONS ?

?

Slide110

== SPARES ==

Slide111

12-Story Scherk-Collins Toroid

branches = 4storeys = 11height = 1.55flange = 1.00thickness = 0.06rim_bulge = 1.00warp = 330.00twist = 247.50azimuth = 56.25mesh_tiles = 0textr_tiles = 1detail = 8bounding box: xmax= 6.01, ymax= 1.14, zmax= 5.55, xmin= -7.93, ymin= -1.14, zmin= -8.41

Slide112

David Lynn, Nova Blue Studio Arts

http://sites.google.com/site/novabluestudioarts/

Slide113

Master Module for “Millennium Arch”

Slide114

Fabrication of “Millennium Arch”

The mold for the key module

A polyester segment cast

Slide115

Two Times Three Modules

Slide116

Merging the Two Half-Circles

Slide117

Slide118

Brent Collins and David Lynn

Slide119

“Millennium” Arch by Night