BID Lunch, February 25, 2014BID Lunch, February 25, 2014

“LEGO®” Knots

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Carlo H. SéquinMichelle Galemmo

The Bridges ConferenceThe Bridges Conference

Mathematical Connections in Art, Music, and Science.

Annually held in July/August since 1998 (4-5 days).

Attracts: mathematicians, scientists, artists, educators, musicians, writers, dancers, weavers, model builders, and computer scientists.

Conference Venues: Formal paper presentations: regular, short, invited/plenary.

Also: workshops, art gallery, and informal show-and-tell/sell.

Evenings: concerts, theater, movies (with a math connection).

Bridges in Many Wonderful PlacesBridges in Many Wonderful Places

Cities: Winfield, Towson, London, Coimbra, Granada, Donostia, Leeuwarden, Enschede, Pecs, …

My Favorite Annual Conference: 2014My Favorite Annual Conference: 2014

BRIDGES Art …

My First 20 Bridges PapersMy First 20 Bridges Papers

Inspiration: Henk van PuttenInspiration: Henk van Putten

“Borsalino” “Interaction”

Sculptural forms put together from a few modular shapes

Geometry of the Geometry of the BorsalinoBorsalino

Just 2 geometrical components: 3 semi-circular end-caps (orange) 6 curved connectors, bending through 45º== a square cross section swept along 9 circular arcs.

The Wonders of Rapid-PrototypingThe Wonders of Rapid-Prototyping

Two modular components can form the Borsalino

C R=2.4142

E R=1.0

Playing with those Two ComponentsPlaying with those Two Components

The The ““LooseLoose”” Borsalino Borsalino

Two larger types of parts required: 3 end-caps; 6 connectors.

Rapid Prototyping with FDMRapid Prototyping with FDM

A Look Into the FDM MachineA Look Into the FDM Machine

Galapagos 6 sculpture in progress

2 NOZZLES

Inexpensive FDM MachineInexpensive FDM Machine

Afinia-H-Series 3D printer

Has only ONE type of filament.

Support structures: not desirable: are hard to remove,

leave part surface scarred.

Limited cantilevering is possibleto form overhangs and bridges:( 45º max).

SUPPORT

Fabrication Issues:Fabrication Issues:Optimize Build OrientationOptimize Build Orientation

Horizontal tube (flat):filled with support

Tube on edge:may tip over,gets squashed

Vertical tube:support at flanges

Can we orient part so no supports are needed? (All surfaces must be steeper than 45º)

SUPPORT

Square flanges Tapered flanges Upside-down

support no support no support

Fabrication Issues: Fabrication Issues: FlangesFlanges

Fabrication Issues: Fabrication Issues: Curved ConnectorsCurved Connectors

Ignore Problem Built-in Wall Extended Taper

Chosen Solution!

No face steeper than 45º !

Fabrication Issues: Fabrication Issues: End-CapsEnd-Caps

Form a 45º slanted “cathedral ceiling” internally:

needs support needs no support

Needs some support for central arch !

We wanted to keep inner tube open: Use scaffolding and take the trouble of doing some clean-up!

Fabrication Issues: Fabrication Issues:

Enlarged End-CapsEnlarged End-Caps

Parts Catalog So FarParts Catalog So Far

2 types of end-caps; 3 curved connectors

What can we do with those 5 Parts?What can we do with those 5 Parts?

Wild and crazy “snakes”, e.g., a Hilbert curve;

Emulation of other Henk van Putten sculptures …

EmulationsEmulations

Interaction

Many More Possibilities …Many More Possibilities …

Mix and match …

D

CB

A

2 End-Caps plus 3 Connector Types ...2 End-Caps plus 3 Connector Types ...

allow us to build these allow us to build these Twisted 2-Lobe BorsalinosTwisted 2-Lobe Borsalinos

The mathematics does not really work out quite right;they are off by 6%, 15%, and 6% in the radius ratios.

Flipped-end BorsalinoFlipped-end Borsalino

Put extension between connectors

movebackward

stretched

move up,forward

needsdiagonalend-cap

moveforward

movebackward

Flipped-end BorsalinoFlipped-end Borsalino

The New Rhombic End-cap …The New Rhombic End-cap …… yields new possibilities:

Rhombic BorsalinosRhombic Borsalinos We just need to make a

new connector part: bending again through 45º, but in diagonal direction!

This Borsalino now has a loose enough geometry,so that there is roomto introduce twisted legs:

Twisted Connector PiecesTwisted Connector Pieces

Enlarged connector pieces with 45º twist:

results in two different pieces (azimuth!)

4 pairs make a nice twisted ring (360º )

Twisted BorsalinosTwisted Borsalinos

1 twisted branch; 3 twisted branches.

Triply Twisted Rhombic BorsalinoTriply Twisted Rhombic Borsalino

Sculpture!

Inspiration: Paul BlochInspiration: Paul Bloch

“After Wright” (Guggenheim, NYC)

Helical PiecesHelical Pieces

Another useful component!

Spiral SculpturesSpiral Sculptures

Using the helical pieces

A Look Behind the SceneA Look Behind the Scene

It does not really close smoothly!

Inspiration: Paul BlochInspiration: Paul Bloch

“After Wright” (Guggenheim, NYC)

““CoccoonCoccoon””

Return path through the center of the helix.

Inspiration: Bruce BeasleyInspiration: Bruce Beasley

Autodesk exhibition, December 2013

Inspiration: Jon KrawczykInspiration: Jon Krawczyk

303 2nd Street, San Francisco

““Pas de DeuxPas de Deux””

Real LEGOReal LEGO®® Knots ? Knots ?

Beginning of the table of knots …

This is all you have seen so far. It is not really a knot!

Unknot

Trefoil

Figure-8

Real Knots: Trefoil (3_1)Real Knots: Trefoil (3_1)

One new piece (magenta) for smooth closure

Trefoil KnotTrefoil Knot

Real Knots: Figure-8 Knot (4_1)Real Knots: Figure-8 Knot (4_1)

Two new pieces (magenta, red) for smooth closure

Figure-8 KnotFigure-8 Knot

LEGOLEGO®® DUPLO DUPLO

Match interface

LEGOLEGO®® DUPLO DUPLO

Just playing around . . .

LEGOLEGO®® DUPLO DUPLO

Borromean Link Hopf Link

What Is the Right Interface ?What Is the Right Interface ?

Open tubes

or

LEGO nibs ?

Making Sculptures Glow …Making Sculptures Glow …

Making Sculptures Glow …Making Sculptures Glow …

testing

Making Sculptures Glow …Making Sculptures Glow …

Hands-on SculptingHands-on Sculpting

Xmas break 2013

Branching Out ?Branching Out ?

Junction pieces arbitrary graphs

““OrganicOrganic”” Looking Objects Looking Objects

Trees or corrals ?

Making Graph StructuresMaking Graph Structures

Tetrahedral graph:

4 valence-3 vertices

Next Attempt: More compact, but . . .Next Attempt: More compact, but . . .

This is not the tetrahedral graph!

Not this:

But this:

Tetra GraphTetra Graph

OK tetrahedral graph.

But needed one extra custom-made piece!

Other Attempts ?Other Attempts ?

What about the cubical edge graph ?

Cubical Edge Graph: 2Cubical Edge Graph: 2ndnd Attempt Attempt New stub placement in square frame:

Place 2 Y-component close together, pointing in opposite directions.

Put two such frames on top of one another and connect pairs of stubs with the rhombic end-caps.

Cubical Edge Graph -- SolutionCubical Edge Graph -- Solution

Stack the two frames in an offset manner,so that the connecting arcs run at a 45º angle.

Cubical Edge Graph as a SculptureCubical Edge Graph as a Sculpture

Max Bill Sculpture, and beyond . . .Max Bill Sculpture, and beyond . . .

What is needed to emulate this?

But there are other ways But there are other ways

of of ““branching outbranching out”” . . . . . .

BID Lunch, February 25, 2014BID Lunch, February 25, 2014

Tria-Tubes

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Michelle GalemmoCarlo H. Séquin

A New ProfileA New Profile

All sculptures shown were based on a square cross section:

What will happen if we try a different profile,e.g. a triangular one ?

Designing a Designing a ““Tri-BorsalinoTri-Borsalino”” This is the tightest turn without self-intersections of

a circular tube containing a triangular cross section in an arbitrary azimuth (angle*) orientation.

Two different alignments result in two different end-caps:

*

Dimensioning the TRIA-TUBESDimensioning the TRIA-TUBES

Two Triangular End-CapsTwo Triangular End-Caps

Type #1 Type #2

Two Two ““Tri-BorsalinosTri-Borsalinos”” We keep the 3D space curve of the classical Borsalino

as the sweep path for our new triangular cross section.

The two end-caps produce two different Tri-Borsalinos.

The connector pieces bending through 45º need to have a twist of 15º to make smooth connections.

Our Initial Parts ListOur Initial Parts List 2 types of end-caps:

Curved connectors: for type #1 for type #2

Connector sleeve

Some Possible Assemblies:Some Possible Assemblies:Using just End-CapsUsing just End-Caps

two hexagonal rings and a triangular loop

Some Possible Assemblies:Some Possible Assemblies:from only curved connector piecesfrom only curved connector pieces

A variety of twisted Möbius rings

What Next ?What Next ?

More twisted connectors?

Twisted end-caps?

Straight extension pieces for flip-over Borsalino?

Tri-Borsalino DerivativesTri-Borsalino Derivatives

r = 0.57735

Extend the link (by √8) between the connectors.

R(endcap) = sqrt(4/3) = 2r = 1.1547 = cos(30)*4/3

ConclusionsConclusions

LEGO®-Knots plus Tria-Tubes:

An ever expanding modular system to do hands-on geometrical sculpture for people who do not want to do math or touch a computer.

The result is a mixture of:-- creative design decisions and-- practical fabrication considerations.