# Julian Lowell Coolidge's A History of Geometrical Methods PDF

By Julian Lowell Coolidge

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It may be noted that the indented faces of the pentagonal depressions exactly correspond to the inverted pyramids of f 2g2 which fit into them; and that the rhombic depressions similarly correspond to the under sides of the narrow wedges gl. Fgl is most easily visualized as the result of fitting the pentagonal spikes f2 into the pentagonal depressions of Eflgl (Plate IX). PLATE IX. Of these three figures Eflgl is the simplest; the rhombic depressions of Efl are just filled in level with the adjacent faces by the wedges gl, producing a figure which consists of a dodecahedron having a large pentagonal depression (with equilateral triangular sides) in each face.

4. NOTES ON THE PLATES PLATE I. The Icosahedron A needs no comment. B is obtained by erecting low triangular pyramids (b) on each face of 19 THE FIFTY-NINE IcOSAHEDRA A, and is thus what we may call (by analogy with cristallographic terminology) a triakisicosahedron, save that the edges of A are concave, not convex, edges of B. C is the figure of five octahedra, dual to that of five cubes; each of the twenty planes contains a face each of two of these octahedra (the face of C is in fact clearly a pair of crossed triangles); and for each of the ten possible pairs out of the five octahedra a pair of opposite faces of one are coplanar respectively with a pair of opposite faces of the other, this accounting just for the ten pairs of opposite faces of the icosahedron.

The accompanying sketch of the section of el, ft. ~l' by a plane perpendicular to a circumradius of this dodecahedron, and near its vertex, makes clear the relation between them. ) PLATE V. The combinations of these three figures are easily grasped from the plate. elfl and f)~) are both edge-connected, a 21 THE FIFTY-NINE ICOSAHEDRA piece of the former consisting of a piece of el with six pieces of fIr a piece of the latter of a piece of ~l with four pieces of fl. elfl~l on the other hand is body connected throughout.