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UCBUCB BRIDGES, Winfield KS, July 2000BRIDGES, Winfield KS, July 2000

“- To Build a Twisted Bridge -”

Carlo H. Séquin

University of California, Berkeley

MATHEMATICAL CONNECTIONSIN ART, MUSIC, AND SCIENCE

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UCBUCB Talk ObjectivesTalk Objectives

Explore the feasibility of buildings or bridges in the shape of Möbius bands.

Title is an allusion to Robert Heinlein’s delightful short story“- And He Built a Crooked House -”

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UCBUCB MotivationMotivation

Annual series of BRIDGES conferenceswould like to have a commemorative entityon the campus of Southwestern College.

During the 1999 BRIDGES conference,there was a brain-storming session in which various (crazy?) ideas were brought forward.

Escher, Möbius, Klein,… are the heroesof this ART-MATH community.

So why not an Escher Garden, or a Klein-bottle house, or a Möbius bridge ?

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UCBUCB Escher Illustration by Sean O'MalleyEscher Illustration by Sean O'Malley

We don’t just want an optical illusion.

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UCBUCB Our Real GoalOur Real Goal

We want a realizable 3D structure:

a bridge that we can walk across;

a building that accommodates usable rooms.

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UCBUCB Inspiration !Inspiration !

M.C. Escher: “Möbius Strip II”

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UCBUCB A Twisted Slab ...A Twisted Slab ...

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UCBUCB A Twisted Slab ...A Twisted Slab ...

… is difficult to walk on !

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UCBUCB Bézier PatchBézier Patch

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UCBUCB Bézier PatchBézier Patch

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UCBUCB Twisted C-SectionTwisted C-Section

Inspired by Brent Collins’ Sculptures

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UCBUCB Close the Loop !Close the Loop !

A twisted band is not a Möbius strip !

It is only complete when the loop is closed.

It is not so obvious what to do with the

return path !

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UCBUCB Supported BridgeSupported Bridge

Return path lies underneath the walk-way.

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UCBUCB Möbius Suspension BridgeMöbius Suspension Bridge

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UCBUCB Another Suspension BridgeAnother Suspension Bridge

Closes the loop through a non-planar space curve

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UCBUCB Emulating M.C. EscherEmulating M.C. Escher

Can we turn this shape into a usable bridge for humans ?

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UCBUCB Figure-8 Möbius Bridge, Type IFigure-8 Möbius Bridge, Type I

Inspired by Escher’s “Möbius Strip II”

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UCBUCB Figure-8 Möbius Bridge, Type IIFigure-8 Möbius Bridge, Type II

Use edge-flange as walk-way

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UCBUCB Möbius BridgeMöbius Bridge

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UCBUCB Möbius BridgeMöbius Bridge

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UCBUCB Möbius BridgeMöbius Bridge

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UCBUCB Another ApproachAnother Approach

Starting from M.C. Escher’s “Möbius Strip I”

Recycling Symbol with 3-fold symmetry.

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UCBUCB ““Japanese” Möbius BridgeJapanese” Möbius Bridge

Asymmetric recycling symbol

Walk on edges of Möbius band

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UCBUCB Other Möbius Constructions ?Other Möbius Constructions ?

There are plenty of possibilities forfunctional Möbius bridges.

What about Möbius buildings ?

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UCBUCB Möbius Building DesignsMöbius Building Designs

Peter Eisenman Van Berkel & Bos

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UCBUCB Deforming the Basic Möbius LoopDeforming the Basic Möbius Loop

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UCBUCB Form Follows FunctionForm Follows Function

Start with a practial building module, say, 30’ by 30’ by 30’.

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UCBUCB Möbius StructuresMöbius Structures

90° 180°

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UCBUCB Towards Real Möbius BuildingsTowards Real Möbius Buildings

Flatten cross section to 2:1(4 stories tall in upper arch).

Soften the corners for more aesthetic appeal.

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UCBUCB Practical Möbius BuildingsPractical Möbius Buildings

Reduce the span of the arch by closing loop on the outside.

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UCBUCB A Practical Möbius BuildingA Practical Möbius Building

Glass windows

Mostly opaque

Office Tower(view windows)

Entrance atrium,Cafeteria,Lounges,Library(glass ceilings)

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UCBUCB Experiments with Vertical LoopsExperiments with Vertical Loops

Reducing the flat area byunwindingthe spiral

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UCBUCB ““Lambda” Möbius HouseLambda” Möbius House

The shortest way to connect “front” to “back”

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UCBUCB ““Lambda” Möbius HouseLambda” Möbius House

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UCBUCB Lambda Möbius HouseLambda Möbius House

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UCBUCB Möbius House and BridgeMöbius House and Bridge

for comparison

Non-rectangular profile

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UCBUCB Möbius Houses and BridgesMöbius Houses and Bridges

Functional realizations exist for both.

Bridge constructions seem quite feasibleand affordable (depending on scale).

Möbius buildings tend to be rather largein order to allow a usable inner structure.

What if the funds are not sufficient for either one ?

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UCBUCB Möbius Sculpture by Max BillMöbius Sculpture by Max Bill

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UCBUCB Möbius Sculptures by Keizo UshioMöbius Sculptures by Keizo Ushio

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UCBUCB More Split Möbius BandsMore Split Möbius Bands

Typical lateral splitby M.C. Escher

And a maquette made by Solid Free-form Fabrication

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UCBUCB Another Möbius SplitAnother Möbius Split

Typical lateral splitby M.C. Escher

Splitting the band in the thickness direction --creates a Möbius space.

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UCBUCB ““Möbius Space”Möbius Space”

Interior space has the shape of a Möbius band.

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UCBUCB Maquette of “Möbius Space”Maquette of “Möbius Space”

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UCBUCB ConclusionsConclusions

Möbius topology is mysterious, intriguing.

It constitutes a good symbol for the annual Bridges Conferences.

A commemorative construction might takethe form of a Bridge, a House, a Sculpture.

Various conceptual possibilities have been introduced in this talk --more development and refinement is needed.

Hopefully, there will be an actual physical construction on Campus before too long.

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UCBUCB Questions ?Questions ?