Paper by Erik D. Demaine

Reference:

Tomohiro Tachi and Erik D. Demaine, “Degenerative Coordinates in 22.5° Grid System”, in Origami⁵: Proceedings of the 5th International Conference on Origami in Science, Mathematics and Education (OSME 2010), Singapore, July 13–17, 2010, to appear, A K Peters.

BibTeX

@InCollection{AngularGrid_Origami5,
  AUTHOR        = {Tomohiro Tachi and Erik D. Demaine},
  TITLE         = {Degenerative Coordinates in 22.5$^\circ$ Grid System},
  BOOKTITLE     = {Origami$^5$: Proceedings of the 5th International Conference on Origami in Science, Mathematics and Education (OSME 2010)},
  bookurl       = {http://www.origami-usa.org/5osme},
  ADDRESS       = {Singapore},
  MONTH         = {July 13--17},
  YEAR          = 2010,
  PAGES         = {to appear},
  PUBLISHER     = {A K Peters},

  replaces      = {AngularGrid_OSME2010},
  papers        = {AngularGrid_OSME2010; AngularGrid_IJCGA},
}

Abstract:

We consider the construction of points within a square of paper by drawing a line (crease) through an existing point with angle equal to an integer multiple of 22.5°, which is a very restricted form of the Huzita–Justin origami construction axioms. We show that a point can be constructed by a sequence of such operations if and only if its coordinates are both of the form (m + n √2)/2^ℓ for integers m, n, and ℓ ≥ 0, and that all such points can be constructed efficiently. This theorem explains how the restriction of angles to integer multiples of 22.5° forces point coordinates to degenerate into a reasonably controlled grid.

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AngularGrid_OSME2010 (Degenerative Coordinates in 22.5° Grid System)