This document discusses map operations, which convert one subdivision of a surface into another subdivision with certain properties. It provides examples of map operations that yield bipartite, chess-colorable, quadrilateral-faced, triangle-faced, and other properties. It shows how map operations can be represented by Truchet tiles and correspond to unit weaving patterns, tensegrities, and plain weaving. Finally, it notes that spherical designs may work on other surfaces if they tolerate conformal distortions and variations in extrinsic and intrinsic curvature.

1. Extra Ways to See:
An Artist’s Guide to Map Operations
James Mallos
ISAMA 2011
Chicago, Illinois

2. In 1954, Buckminster Fuller disclosed the
geodesic dome and ignited the topic of
symmetrical designs on the sphere in
America and Europe.

3. Elsewhere, symmetrical designs on the sphere
had never gone out of style.
Japanese temari balls. www.japan-cc.com

4. It’s now 57 years later, and
we’re rich with designs on the
57 years later, we are rich in inventions on
the sphere.
sphere.

5. We have: tensegrities...
Diamond pattern Zig-zag pattern Circuit pattern
Anthony Pugh, An Introduction to Tensegrity
Star pattern Lawrence Pendred, www.pendred.net

6. ...tensegrity membranes...
Shigematsu, Tanaka, and Noguchi, IASS-IACM 2008

7. ...unit-weavers...
Iq’s, Holger Strom Twogs, James Mallos

8. ...unit origami...
John Horigan, www.ozonehouse.com/john/origami

9. ...nexorades...
TaffGoch sketchup.google.com/3dwarehouse

10. ...weave patterns...
Akleman, Chen, Xing, and Gross, Siggraph 09

11. ...puzzles...
George W. Hart
Goldberg Puzzle, George W. Hart, www.georgehart.com

12. ...and mechanisms.
Juno’s Spinner, Junichi Yananose, www.polyhedra.jp

13. There are plenty of other
surfaces...

14. Gyroid surface, mathworld.wolfram.com
...will our designs work there?

15. Map operations
• A map operation converts one subdivision
of a surface (the base map) into another
(the resultant map.)
Ra()
Base Map Resultant Map

16. What is a map?

17. A map is:
A graph drawn on a closed surface in such a way that:
• the vertices are represented as distinct points
on the surface,
• the edges are represented as curves on the
surface intersecting only at the vertices,
• if we cut the surface along the graph, what
remains is a disjoint union of connected
components, called faces, each topologically
equivalent to an open disk.
Abstracted from Lando and Zvonkin, Graphs on Surfaces and Their Applications

18. Which of these are maps?

19. Maps are embedded general graphs.
They can have vertices and faces of valence {1, 2, 3, ...}

20. These maps all have one or more triangle faces:

21. Often the base map is a computer surface
model and the goal is to achieve certain
characteristics in the resultant map.
3D models courtesy of INRIA via the Aim@Shape Shape Repository

22. bip artite ches
s-co
lorab
le
The most important map operations
yield maps with guaranteed properties
—no matter the original map.
3-va
lent
-faced
quadrangle

23. Operation Guaranteed Property
Su() Bipartite
Pa() Chess-colorable
Ra() Bipartite and quadrilateral-faced
Me() Chess-colorable and 4-valent
Ki() Triangle-faced
Tr() 3-valent
Le() 3-valent
Or() Bipartite and quadrilateral-faced
Ex() Chess-colorable and 4-valent
Gy() Pentagon-faced
Sn() 5-valent
Mt() Triangle-faced and chirally chess-colorable
Be() 3-valent

24. Ki() Performed Algorithmically
M Ki(M)

25. Ki(M) yields a map where each original edge
is the diagonal of its own quadrilateral.

26. The same map that cuts the surface into
disks, also implicitly chops the surface
into quadrilaterals.
Did I mention they are all quadrilaterals?
This sounds like job for truchet tiles!

27. Truchet Tiles for Map Operations
Id Du
Su Pa
Ra Me
Ki Tr

28. Id(M)

29. Du(M)

30. Su(M)

31. Pa(M)

32. Ra(M)

33. Me(M)

34. Ki(M)

35. Tr(M)

36. Truchet Tiles for some Chiral Map Operations
Pr
Ca
Gy
Sn

37. Gy(M)

38. Sn(M)

39. Some Practical
Applications

40. Unit-weaver / Map Operation
Correspondences
Iq’s Twogs
Ra() Tr()

41. Tensegrity / Map Operation Correspondences
Diamond pattern
Sn()
Chiral edge = strut
Circuit pattern
Me()
Zig-zag pattern
Tr()
Chiralized by strut

42. Zig-Zag
Tensegrity

43. Circuit
Tensegrity

44. Diamond
Tensegrity

45. Star
Tensegrity

46. Elementary Tensegrities From Sn()
Sn( ) =
Sn( ) =
Sn( ) =
Tensegrity simulations with Springie

47. Weaving / Map Operation Correspondence
Chess coloring of Me()
Rinus Roelofs, Bridges 2010

48. Truchet Tile for Plain Weaving
Chess Coloring (Me())

49. Plain
Weaving

50. Will our spherical designs really work on
other surfaces?
• conformal distortions must be tolerated (i.e.,
in length and area, but not angle)
• variations in both extrinsic and intrinsic
curvature must tolerated: convex/concave/flat
extrinsic curvature; and gaussian positive/
negative/zero intrinsic curvature.

51. Thanks!
James Mallos