By C. Davis, B. Grünbaum, F.A. Sherk

Geometry has been outlined as that a part of arithmetic which makes attract the experience of sight; yet this definition is thrown doubtful through the lifestyles of serious geometers who have been blind or approximately so, reminiscent of Leonhard Euler. occasionally apparently geometric tools in research, so-called, consist in having recourse to notions outdoors these it sounds as if appropriate, in order that geometry needs to be the becoming a member of of in contrast to strands; yet then what lets say of the significance of axiomatic programmes in geometry, the place connection with notions outdoors a limited reper tory is banned? no matter what its definition, geometry sincerely has been greater than the sum of its effects, greater than the results of a few few axiom units. it's been a tremendous present in arithmetic, with a particular technique and a distinc ti v e spirit. A present, moreover, which has no longer been consistent. within the Thirties, after a interval of pervasive prominence, it in decline, even passe. those similar years have been these during which H. S. M. Coxeter was once starting his clinical paintings. Undeterred by means of the unfashionability of geometry, Coxeter pursued it with devotion and concept. through the Nineteen Fifties he seemed to the wider mathematical international as a consummate practitioner of a unusual, out-of-the-way paintings. this day there isn't any longer something that out-of-the-way approximately it. Coxeter has contributed to, exemplified, shall we nearly say presided over an unanticipated and dra matic revival of geometry.

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Odd symmetry. 3/4 square symmetry. Crease Patterns After experimenting with these crease patterns, this one was chosen. It yields the largest model that has the fewest steps and holds together well. If it does not maximize the size of the model, then it is very close to it and appears to maximize with respect to its simplicity. It uses 3/4 square symmetry. 42 Part I: Designing Origami Polyhedra 3/4 square symmetry. Landmarks Given the crease pattern, the landmarks are to be found. 1. Find α. Triangular face.

Find c. 5 d c Design Method Examples 43 4. Find e. 1464466 = (2 − 2) 4 α α f e 5. Find g, first find k. 2679491 = tan(15°). 2679491 = tan(15°) Of these, a and g are easy to find and sufficient for completing the folding pattern. 5°). 2679491 = tan(15°). 44 Part I: Designing Origami Polyhedra 1 2 15° Fold and unfold. 2679491 = tan(15°) Polygons Since polygons are the faces of polyhedra, we present how to fold certain polygons before we move into folding directions for polyhedra. Many of the polyhedra in this collection have faces that are regular polygons, which are polygons with congruent angles and sides.

1 2 Fold and unfold along the diagonal. Fold and unfold to the diagonal. 41421356 24 Part I: Designing Origami Polyhedra Dividing a Square into nths First, some specific cases will be shown. Then some general methods will be given. Divisions of 1/3 1 2 3 4 5 1/3 Fold and unfold in half on the left. Unfold. Crease at the bottom. Fold and unfold on the right. The 1/3 mark. 198912. 5%). 1 2 3 4 5 1/5 Fold and unfold creasing along part of the diagonal. Crease on the left. Unfold. Fold to the crease and unfold.