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

2. 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 -”

3. 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 ?

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

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

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

7. A Twisted Slab ...

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

9. Bézier Patch

10. Bézier Patch

11. Twisted C-Section
Inspired by Brent Collins’ Sculptures

12. 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 !

13. Supported Bridge
Return path lies underneath the walk-way.

14. Möbius Suspension Bridge

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

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

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

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

19. Möbius Bridge

20. Möbius Bridge

21. Möbius Bridge

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

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

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

25. Möbius Building Designs
Peter Eisenman Van Berkel & Bos

26. Deforming the Basic Möbius Loop

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

28. Möbius Structures
90°
180°

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

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

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

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

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

34. “Lambda” Möbius House

35. Lambda Möbius House

36. Möbius House and Bridge
Non-rectangular profile
for comparison

37. 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 ?

38. Möbius Sculpture by Max Bill

39. Möbius Sculptures by Keizo Ushio

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

41. 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.

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

43. Maquette of “Möbius Space”

44. 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.

45. Questions ?