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 ?
