By John Montroll
This e-book unravels the secret of Geometry in Origami with a special procedure: sixty four Polyhedra designs, each one made of a unmarried sq. sheet of paper, no cuts, no glue; every one polyhedron the biggest attainable from the beginning measurement of sq. and every having an inventive locking mechanism to carry its form.
the writer covers the 5 Platonic solids (cube, tetrahedron, octahedron, icosahedron and dodecahedron). There are considerable diversifications with varied colour styles and sunken facets. Dipyramids and Dimpled Dipyramids, unexplored sooner than this in Origami, also are lined. There are a complete of sixty four versions within the e-book. all of the designs have an enticing glance and a satisfying folding series and are in response to certain mathematical equations
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Additional resources for Origami Polyhedra Design
Sample text
618034 3 1 Bronze Rectangle 1 Golden Rectangle Polygons 45 Equilateral Triangle Equilateral Triangle Tetrahedron of Triangles Octahemioctahedron Icosahedron Equilateral triangles are used in several polyhedra. The tetrahedron of triangles, octahemioctahedron, and icosahedron are composed entirely of them. 8660254 (1/3)h 1 This shape has three sides with interior angles of 60°. For sides of length 1, the height is 3 2. The bisectors—lines that divide each angle in half— divide the height into thirds where they intersect.
Let the length of each side = 1. Find α. 618034 = (1+ 5)/2 = φ, the golden mean. 1 The angle at each vertex = 108°. 2. Find height h. First find angle β. α + β = 108°/2 Since α = 36° then β = 18°. 5388418 3. Find the ratio of hb to h, where hb is the height of the bottom section. 618034 =φ −1 48 Part I: Designing Origami Polyhedra h hb β α 4. Scale the pentagon so h b = 1. 618034 1 where h t is the height of the top section. The pentagon can be folded with book-fold or diagonal symmetry. Book-fold Symmetry The pentagon can be oriented at different heights in the square.
Fold and unfold. Math and Design 21 Calculation of Landmark Using Trigonometry Trigonometry is also used to calculate landmarks. Here are some basic trigonometric formulas that we use often in such calculations. Any triangle. Right triangle. H O x c B Adjacent Opposite Hypotenuse a b C A To find length of a line: 2 2 A +O =H A 2 sin(x) = O/H cos(x) = A/H tan(x) = O/A Law of Sines To find angle: sin(a) arcsin(O/H) = x arccos(A/H) = x arctan(O/A) = x A = sin(b) B = sin(c) C or A sin(a) = B sin(b) = C sin(c) Example: 2 P 2 3 Crease pattern and one of the triangular faces.