The elements of solid geometry; with numerous exercises - Softcover

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 Excerpt: ...So with the other edges around the top. Hence the upper edges form a regular pentagon, equal to the other faces. By § 93, all the trihedral /_$ at these vertices = the trihedral /_ at A..: the trihedral /$ are equal, the faces are equal regular pentagons, and the solid is a regular dodecahedron. 5. The regular icosahedron. Upon the given edge AB construct a regular pentagon ABCDE. Upon this as a base construct the regular pyramid F--ABCDE, with its lateral edges = AB, giving figure a. Upon the base edges of this pyramid erect equilateral triangles ABG, BCH, etc., so inclined that the distances between their vertices KG, GH, HI, etc., shall equal AB, i. e., so that the inserted triangles AGK, BGH, etc., shall be equilateral triangles. This gives figure b, with the open base GIIIJK. Since the = legged isosceles As AFC and ABC are upon the same base AC they are =, and Z AFC = Z ABC = 108..-. the pyramid B--AFCHG, having = lateral edges, and standing upon an equilateral base, one of whose angles is 108, must be a regular pyramid = pyramid F--ABCDE..-. the polyhedral Z B = the polyhedral Z F. Similarly all the other completed polyhedral Zs A, C, etc., can be shown to be =. Upon the edges GH, HI of the open base, erect the equilateral As GHL, IIIL, giving us figure c. The pentagon BCILG is regular, since it is equilateral, with two /$ = 108 each. The = legged isosceles As GUI and GLI upon the same base are =, and /_ GH1= /_ GLI= 108. So with the /j, between the other edges of the open base..-. the open base GHIJK is a regular pentagon, and can be taken as the base of a regular pyramid = pyramid F--ABCDE, giving figure d. The polyhedral /$ G, H, etc., can be proved = as were the /3 A, B, etc. Therefore, the figure, being formed from equilateral As, and having...