Two-Year College Mathematics Readings by Warren Page

By Warren Page

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That means that the paper is constrained by the properties of Euclidean geometry. 5-dimensional and 3-dimensional forms using algorithms such as circle-river packing [Lang 03] and Origamizer [Tachi 09] that rely on Euclidean metrics. But what if the paper were not Euclidean? 1007/s00283-012-9274-3 AND ROBERT J. LANG In this work, we explore non-Euclidean origami, specifically, origami carried out with paper that possesses a uniform negative Gaussian curvature, with particular emphasis on the most iconic origami figure, the traditional tsuru, or origami crane.

The amount of bunching is critical, though; it needs to be just the right amount at every point of the paper. Too much, and the curvature is too large at that point; too little, and the curvature is too small. And if the curvature is nonuniform, then one loses the ability to do metrically flat folding. We need to introduce just the right amount of curvature at every point so that the result is congruent to the hyperbolic desk that we are folding against. So, let’s use the ‘‘desk’’ as a mold; we will take a Euclidean sheet of paper and mold it against a hyperbolic form; the result will be a pseudospherical sheet of paper from which we can do true hyperbolic origami.

The mold and resulting sheet are shown together in Figure 3. We now have hyperbolic paper.

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