CHSCHSUCBUCB 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

CHSCHSUCBUCB 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 -”

CHSCHSUCBUCB 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 ?

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

We don’t just want an optical illusion.

CHSCHSUCBUCB Our Real GoalOur Real Goal

We want a realizable 3D structure:

a bridge that we can walk across;

a building that accommodates usable rooms.

CHSCHSUCBUCB Inspiration !Inspiration !

M.C. Escher: “Möbius Strip II”

CHSCHSUCBUCB A Twisted Slab ...A Twisted Slab ...

CHSCHSUCBUCB A Twisted Slab ...A Twisted Slab ...

… is difficult to walk on !

CHSCHSUCBUCB Bézier PatchBézier Patch

CHSCHSUCBUCB Bézier PatchBézier Patch

CHSCHSUCBUCB Twisted C-SectionTwisted C-Section

Inspired by Brent Collins’ Sculptures

CHSCHSUCBUCB 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 !

CHSCHSUCBUCB Supported BridgeSupported Bridge

Return path lies underneath the walk-way.

CHSCHSUCBUCB Möbius Suspension BridgeMöbius Suspension Bridge

CHSCHSUCBUCB Another Suspension BridgeAnother Suspension Bridge

Closes the loop through a non-planar space curve

CHSCHSUCBUCB Emulating M.C. EscherEmulating M.C. Escher

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

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

Inspired by Escher’s “Möbius Strip II”

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

Use edge-flange as walk-way

CHSCHSUCBUCB Möbius BridgeMöbius Bridge

CHSCHSUCBUCB Möbius BridgeMöbius Bridge

CHSCHSUCBUCB Möbius BridgeMöbius Bridge

CHSCHSUCBUCB Another ApproachAnother Approach

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

Recycling Symbol with 3-fold symmetry.

CHSCHSUCBUCB ““Japanese” Möbius BridgeJapanese” Möbius Bridge

Asymmetric recycling symbol

Walk on edges of Möbius band

CHSCHSUCBUCB Other Möbius Constructions ?Other Möbius Constructions ?

There are plenty of possibilities forfunctional Möbius bridges.

What about Möbius buildings ?

CHSCHSUCBUCB Möbius Building DesignsMöbius Building Designs

Peter Eisenman Van Berkel & Bos

CHSCHSUCBUCB Deforming the Basic Möbius LoopDeforming the Basic Möbius Loop

CHSCHSUCBUCB Form Follows FunctionForm Follows Function

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

CHSCHSUCBUCB Möbius StructuresMöbius Structures

90° 180°

CHSCHSUCBUCB 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.

CHSCHSUCBUCB Practical Möbius BuildingsPractical Möbius Buildings

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

CHSCHSUCBUCB A Practical Möbius BuildingA Practical Möbius Building

Glass windows

Mostly opaque

Office Tower(view windows)

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

CHSCHSUCBUCB Experiments with Vertical LoopsExperiments with Vertical Loops

Reducing the flat area byunwindingthe spiral

CHSCHSUCBUCB ““Lambda” Möbius HouseLambda” Möbius House

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

CHSCHSUCBUCB ““Lambda” Möbius HouseLambda” Möbius House

CHSCHSUCBUCB Lambda Möbius HouseLambda Möbius House

CHSCHSUCBUCB Möbius House and BridgeMöbius House and Bridge

for comparison

Non-rectangular profile

CHSCHSUCBUCB Möbius Houses and BridgesMöbius Houses and Bridges

Functional realizations exist for both. Bridge constructions seem quite feasible

and affordable (depending on scale). Möbius buildings tend to be rather large

in order to allow a usable inner structure.

What if the funds are not sufficient for either one ?

CHSCHSUCBUCB Möbius Sculpture by Max BillMöbius Sculpture by Max Bill

CHSCHSUCBUCB Möbius Sculptures by Keizo UshioMöbius Sculptures by Keizo Ushio

CHSCHSUCBUCB More Split Möbius BandsMore Split Möbius Bands

Typical lateral splitby M.C. Escher

And a maquette made by Solid Free-form Fabrication

CHSCHSUCBUCB Another Möbius SplitAnother Möbius Split

Typical lateral splitby M.C. Escher

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

CHSCHSUCBUCB ““Möbius Space”Möbius Space”

Interior space has the shape of a Möbius band.

CHSCHSUCBUCB Maquette of “Möbius Space”Maquette of “Möbius Space”

CHSCHSUCBUCB ConclusionsConclusions

Möbius topology is mysterious, intriguing. It constitutes a good symbol for the annual

Bridges Conferences. A commemorative construction might take

the 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.

CHSCHSUCBUCB Questions ?Questions ?