A theoretical consideration of a method that can be used to knit or crochet an arbitrary surface. The method is based on the triangle-strip method in computer graphics. A talk presented at ISAMA 2010 in Chicago, IL.

1. Triangle Strip Knitting
James Mallos
ISAMA 2010
Chicago, Illinois

2. Can we knit (or crochet) any surface?

3. Rules
• One yarn.
• One piece (work in progress must remain a
single, well-connected piece.)
• One technique (sewing up seams is not
allowed.)

4. We already know weaving can do this.

5. The Plain-Weaving Theorem
Every polygonal surface mesh
describes a plain-weaving.
Akleman, E., Chen, J., Xing, Q., and, Gross, J. ’2009.

6. A consequence of PWT: every tessellation of
the plane describes a plain-woven fabric.

7. The PWT can be demonstrated with a special
set of Truchet tiles.

8. A virtual Truchet tiling can be done using the
computer graphics technique of texture mapping.

9. Does a small set of polygons suffice (just
triangles, say)?

10. No...not if we want the boundaries of the
basket to have selvaged edges. Boundaries
are, in effect, large n-gons that need to be tiled
like all the other polygons.

11. Model courtesy INRIA
via the Aim@Shape
Shape Repository.
A 3D model decorated with virtual Truchet tiles.

12. Model courtesy
INRIA via the
Aim@Shape Shape
Repository.
Offering, James Mallos, 2008
A woven sculpture derived from a
surface mesh.

13. Model courtesy
INRIA via the
Aim@Shape Shape
Repository.
Olivier’s Fingertip, James Mallos, 2008
A woven sculpture derived from a
surface mesh.

14. Model courtesy
INRIA via the
Aim@Shape Shape
Repository.
Big Little, James Mallos, 2010
A woven sculpture derived from a
surface mesh.

15. Can knitting and crochet also make any
surface?
What’s the difference, K & C vs. W?

16. • Weaving is a multicomponent link, or
sometimes a single-component link (a knot).
• Knitting and crochet are manipulations of
the unknot.

17. Since they are manipulations of the unknot, K
& C can be done with the ends of the yarn
tied together.
In practice, this adds no difficulty.

18. Because they are manipulations of the
unknot, C & K unravel. W does not.

19. • W has rotational symmetry around its
openings (a fact which makes Truchet tiles
easy use)
• K and C do not have rotational symmetry:
every K-tile or C-tile must be properly
oriented inside its n-gon.
• W does not reveal its order of working, but
K and C do (K-tiles and C-tiles must align in
a linear pattern that covers the surface.)

20. Finding a linear order of working that covers the
surface:
how would you mow the grass
on this planet?

21. Three Ways to Mow Grass
Serpentine Loop
Boustrophedonic Spiral
(Traveling Salesman)

22. Which way works on a general surface?

23. They all do!
Any compact surface can be mapped onto the
interior of a plane polygon—the topological
complexities are confined to the way the
polygon edges identify in pairs.
A method of cutting grass in the interior of a
plane polygon (without crossing the perimeter)
will map onto any surface.

24. Of the three mowing schemes, only the
Serpentine Loop is versatile.
Serpentine Loop
Boustrophedonic Spiral
(Traveling Salesman)
Got an obstacle? Take cities in that region off the
salesman’s list. Need more refinement somewhere?
Add more cities there.

25. Gopi and Eppstein 2004
A triangle strip corresponds to a Hamiltonian Cycle
(TSP solution) on the dual graph of the triangulation.

26. Example of a Hamiltonian cycle on the
dodecahedron (dual to the icosahedron.)

27. Hamiltonian Facts of Life
• Nearly all triangulations without boundaries
have Hamiltonian duals.
• If more than 15% of the triangles are on
boundaries (and therefore 2-valent), the dual
is unlikely to be Hamiltonian.
• Searching for a Hamilton path or circuit in
the dual cubic graph becomes intractable for
large triangulations. (NP complete.)

28. Good news: If we don’t find a Hamiltonian cycle,
we can make one!
The Single Strip Algorithm (Gopi and
Eppstein, 2004)
• Don’t try to find a Hamilton circuit, make one
by gently editing the triangulation at a few
points.

29. • The Single Strip Algorithm can be made to
respect constraints such as preferred
directions.
Gopi and Eppstein 2004

30. The Single-Strip Algoritm gives us a strip
(or loop) of triangles, how do we knit and
assemble a strip of triangles?

31. There are four kinds of vertex in the hamiltonian
cycle that can be labelled in this way:
• Arbitrarily choose a mid-edge in the
Hamiltonian Cycle as a starting point.
• Arbitrarily choose a side of the surface at
the starting point.
• Arbitrarily choose a direction of travel.

32. • Label each vertex according to whether the
non-Hamiltonian edge extends to the left or
the right, and...
• whether the adjacent vertex on the non-
Hamiltonian edge has already been labelled
(close) or not (open.)
• Finish when the starting point is
encountered

33. Four “emoticons” can naturally represent the four labels:
open left
open right
close left
close right
undp

34. Some Undip Codewords for Deltahedra
• Tetrahedron: undp and nupd
• Octahedron: unnduppd and nuupnddp
• Icosahedron: nnununuuupppdpdndddp

35. Note:
Codewords suffice for genus 0 surfaces only.
Higher genus surfaces need more information.

36. TRIANGLE CONTEXT CHART
CAST OFF
d p
CLOSE LEFT CLOSE RIGHT
u n
CAST ON
OPEN LEFT OPEN RIGHT

37. Caveat:
• We want correctly imbedded surfaces.
• Correct Gaussian curvature (intrinsic
curvature) is necessary but not sufficient.
• Correct topology is necessary but not
sufficient.