BRIDGES, Winfield KS, July 2000 MATHEMATICAL CONNECTIONS IN ART, MUSIC, AND SCIENCE “- To Build a Twisted Bridge -” Carlo H. Séquin University of California, Berkeley

Talk 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 -”

Motivation Annual series of BRIDGES conferences would like to have a commemorative entity on 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 heroes of this ART-MATH community. So why not an Escher Garden, or a Klein-bottle house, or a Möbius bridge ?

Escher Illustration by Sean O'Malley We don’t just want an optical illusion.

Our Real Goal We want a realizable 3D structure: a bridge that we can walk across; a building that accommodates usable rooms.

Inspiration ! M.C. Escher: “Möbius Strip II”

A Twisted Slab ...

A Twisted Slab ... … is difficult to walk on !

Bézier Patch

Bézier Patch

Twisted C-Section Inspired by Brent Collins’ Sculptures

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 !

Supported Bridge Return path lies underneath the walk-way.

Möbius Suspension Bridge

Another Suspension Bridge Closes the loop through a non-planar space curve

Emulating M.C. Escher Can we turn this shape into a usable bridge for humans ?

Figure-8 Möbius Bridge, Type I Inspired by Escher’s “Möbius Strip II”

Figure-8 Möbius Bridge, Type II Use edge-flange as walk-way

Möbius Bridge

Möbius Bridge

Möbius Bridge

Another Approach Starting from M.C. Escher’s “Möbius Strip I” Recycling Symbol with 3-fold symmetry.

“Japanese” Möbius Bridge Asymmetric recycling symbol Walk on edges of Möbius band

Other Möbius Constructions ? There are plenty of possibilities for functional Möbius bridges. What about Möbius buildings ?

Möbius Building Designs Peter Eisenman Van Berkel & Bos

Deforming the Basic Möbius Loop

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

Möbius Structures 90° 180°

Towards Real Möbius Buildings Flatten cross section to 2:1 (4 stories tall in upper arch). Soften the corners for more aesthetic appeal.

Practical Möbius Buildings Reduce the span of the arch by closing loop on the outside.

A Practical Möbius Building Glass windows Office Tower (view windows) Mostly opaque Entrance atrium, Cafeteria, Lounges, Library (glass ceilings)

Experiments with Vertical Loops Reducing the flat area by unwinding the spiral

“Lambda” Möbius House The shortest way to connect “front” to “back”

“Lambda” Möbius House

Lambda Möbius House

Möbius House and Bridge Non-rectangular profile for comparison

Möbius 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 ?

Möbius Sculpture by Max Bill

Möbius Sculptures by Keizo Ushio

More Split Möbius Bands Typical lateral split by M.C. Escher And a maquette made by Solid Free-form Fabrication

Typical lateral split by M.C. Escher Another Möbius Split Typical lateral split by M.C. Escher Splitting the band in the thickness direction -- creates a Möbius space.

“Möbius Space” Interior space has the shape of a Möbius band.

Maquette of “Möbius Space”

Conclusions 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.

Questions ?