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.
The Plain-Weaving TheoremEvery polygonal surface meshdescribes a plain-weaving. Akleman, E., Chen, J., Xing, Q., and, Gross, J. ’2009.
A consequence of PWT: every tessellation ofthe plane describes a plain-woven fabric.
The PWT can be demonstrated with a specialset of Truchet tiles.
A virtual Truchet tiling can be done using thecomputer graphics technique of texture mapping.
Does a small set of polygons sufﬁce (justtriangles, say)?
No...not if we want the boundaries of thebasket to have selvaged edges. Boundariesare, in effect, large n-gons that need to be tiledlike all the other polygons.
Model courtesy INRIA via the Aim@Shape Shape Repository.A 3D model decorated with virtual Truchet tiles.
Model courtesy INRIA via the Aim@Shape Shape Repository. Offering, James Mallos, 2008A woven sculpture derived from a surface mesh.
Model courtesy INRIA via the Aim@Shape Shape Repository. Olivier’s Fingertip, James Mallos, 2008A woven sculpture derived from a surface mesh.
Model courtesy INRIA via the Aim@Shape Shape Repository. Big Little, James Mallos, 2010A woven sculpture derived from a surface mesh.
Can knitting and crochet also make anysurface?What’s the difference, K & C vs. W?
• Weaving is a multicomponent link, or sometimes a single-component link (a knot).• Knitting and crochet are manipulations of the unknot.
Since they are manipulations of the unknot, K& C can be done with the ends of the yarntied together.In practice, this adds no difﬁculty.
Because they are manipulations of theunknot, C & K unravel. W does not.
• 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.)
Finding a linear order of working that covers the surface: how would you mow the grass on this planet?
Three Ways to Mow Grass Serpentine LoopBoustrophedonic Spiral (Traveling Salesman)
They all do!Any compact surface can be mapped onto theinterior of a plane polygon—the topologicalcomplexities are conﬁned to the way thepolygon edges identify in pairs.A method of cutting grass in the interior of aplane polygon (without crossing the perimeter)will map onto any surface.
Of the three mowing schemes, only theSerpentine Loop is versatile. Serpentine LoopBoustrophedonic Spiral (Traveling Salesman)Got an obstacle? Take cities in that region off thesalesman’s list. Need more reﬁnement somewhere?Add more cities there.
Gopi and Eppstein 2004A triangle strip corresponds to a Hamiltonian Cycle(TSP solution) on the dual graph of the triangulation.
Example of a Hamiltonian cycle on thedodecahedron (dual to the icosahedron.)
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.)
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.
• The Single Strip Algorithm can be made to respect constraints such as preferred directions. Gopi and Eppstein 2004
The Single-Strip Algoritm gives us a strip(or loop) of triangles, how do we knit andassemble a strip of triangles?
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.
• 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
Four “emoticons” can naturally represent the four labels: open left open right close left close right undp
Some Undip Codewords for Deltahedra• Tetrahedron: undp and nupd• Octahedron: unnduppd and nuupnddp• Icosahedron: nnununuuupppdpdndddp
Note:Codewords sufﬁce for genus 0 surfaces only.Higher genus surfaces need more information.
TRIANGLE CONTEXT CHART CAST OFF d p CLOSE LEFT CLOSE RIGHT u n CAST ON OPEN LEFT OPEN RIGHT
Caveat:• We want correctly imbedded surfaces.• Correct Gaussian curvature (intrinsic curvature) is necessary but not sufﬁcient.• Correct topology is necessary but not sufﬁcient.