By John Montroll

N this attention-grabbing consultant for paperfolders, origami professional John Montroll presents basic instructions and obviously precise diagrams for growing remarkable polyhedra. step by step directions exhibit tips on how to create 34 diverse versions. Grouped in accordance with point of trouble, the types variety from the straightforward Triangular Diamond and the Pyramid, to the extra complicated Icosahedron and the hugely tough Dimpled Snub dice and the incredible Stella Octangula.

A problem to devotees of the traditional eastern artwork of paperfolding, those multifaceted marvels also will attract scholars and someone drawn to geometrical configurations.

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Additional resources for A Constellation of Origami Polyhedra

Sample text

To determine if a is greater than h, the value of h is found. Figure 2-13 Drawing for Example 12. b a c' h a c sin a = h b h sin 35% = 20 h . 5736h h . 47 Therefore, a > h and the diagram shows two solutions for the triangle. 5736h sin b . 8825 sin a = a sin 35% = 13 b . 8825 b . 95% Because sin β is positive in both quadrant I and quadrant II, β can have two values and therefore two different solutions for the triangle. b . 95% b . 95% . 05% Solution 1: The sum of the angles of a triangle is 180°.

9744h f. 5299 f . 07 Example 11: Solve the triangle in Figure 2-12 given a = 125%, b = 35%, and b, = 42. Figure 2-12 Drawing for Example 11. a c b From the fact that there are 180° in any triangle, then c = 180% - a - b c = 180% - 125% - 35% c = 20% Again, using the Law of Sines, 42 = c sin 35% sin 20% 42 . 3420h c. 5736 c . F 7/27/01 8:46 AM Page 35 Chapter 2: Trigonometry of Triangles 35 42 = a sin 35% sin 125% 42 . 8192h a. 5736 a . 98 The second use of the Law of Sines is for solving a triangle given the lengths of two sides and the measure of the angle opposite one of them.

Figure 2-5 Drawing for Example 5. 06' tall. Example 6: Using Figure 2-6, find the length of sides x and y and the area of the large triangle. F 7/27/01 8:45 AM Page 27 Chapter 2: Trigonometry of Triangles 27 Figure 2-6 Drawing for Example 6. x y h 70° 70° 12 Because this is an isosceles triangle, and equal sides are opposite equal angles, the values of x and y are the same. If the triangle is divided into two right triangles, the base of each will be 6. 88 sq units Law of Cosines The previous section covered the solving of right triangles.