By Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, Micahel Stein (ed.)

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ISBN-13: 9780821850541

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Extra info for Applications of algebraic K-theory to algebraic geometry and number theory, Part 2

Sample text

For sufficiency, we proceed by induction on >. The result is trivial for the base cases > ) ( (only one angle, and the two neighboring creases will either be M, V or V, M) and > ) d (two angles, and all three possible ways to assign 2 Ms and 1 V, or vice-versa, can be readily checked to be foldable). For arbitrary >, we will always be able to find two adjacent creases q7Li and q7Li Ld to which the MV assignment assigns opposite parity. Let q7Li be M and q7Li Ld be V. , removed. The value of - b ~ will not have changed for the remaining sequence q7 , 333, q7Li d , q7Li L1 , 333, q7L> of creases, which are flat-foldable by the induction hypothesis.

To summarize, let g be a crease pattern for a flat origami model, but for the moment we are considering the boundary of the paper as part of the graph. If o denotes the set of edges in g embedded in the plane, then we call woD the f-net, which is the image of all creases and boundary of the paper after the model has been folded. We then call d w woDD the s-net. This is equivalent to imagining that we fold carbon-sensitive paper, rub all the crease lines firmly, and then unfold. The result will be the X-net.

While Kawasaki, Maekawa, and Justin undoubtedly had proofs of their own, the proofs presented below appear in [3]. 1 (Kawasaki [10], Justin [5], [6]). Let 8 be a vertex of degree 1z in a single vertex fold and let d , 333, 1z be the consecutive angles between the creases. Then 8 is a flat vertex fold if and only if d b 1 L n b c c c b 1z ) (3 (1) 7KH &RPELQDWRULFV RI )ODW )ROGV  Proof. Consider a simple closed curve which winds around the vertex. This curve mimics the path of an ant walking around the vertex on the surface of the paper after it is folded.