By Katsuhiro Shiohama, Takashi Sakai, Toshikazu Sunada
Read or Download Curvature and Topology of Riemannian Manifolds. Proc. Taniguchi Symposium, Katata, 1985 PDF
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Additional resources for Curvature and Topology of Riemannian Manifolds. Proc. Taniguchi Symposium, Katata, 1985
There may have been an initial attempt to work with central triangles as in IV,13. 30 A MATHEMATICAL HISTORY OF THE GOLDEN NUMBER (3) (Fig. ) The equality of angles DAB and CDB suggests the set-up oflll,32 and leads to the circumscription of a circle about ~ACD. Note that, because of the repetition of angles in the frame of Figure 1-42a, the circumscribed circle will have the same diameter as the original circle. (4) (Fig. ) The equality of the angles determines isosceles triangles and shows that AC = DC = DB.
3. Start with a fixed line segment F and call it rational. Then another line segment A is also called rational if either A is commensurable with F and/or if S(A) is commensurable with S(F). Otherwise A is called irrational .... Notes: (I) Euclid shows as part of X,9 that if S(A) is not commensurable with S(F) then A will not be commensurable with F. 3 a line has, so to speak, two chances to be rational; even ifthe line A is not commensurable with the fixed line F, A will still be called rational if S(A) is commensurable with S(F).
In this case I have given an isometric drawing together with top and front views (Fig. 1-36). The important thing in considering Euclid's construction is to orient the dodecahedron so that it is standing vertically and balanced on an edge (as opposed to resting on a face, as is the case with the construction of Pappus, Section 27, or balanced on a vertex, as is the case with the construction of Bombelli, Section 31,F). In this position the opposite top edge UV will also be horizontal. In the top view we will then see the two faces I and II which share the edge UV, as well as the faces III and IV which have U and V as vertices.
Curvature and Topology of Riemannian Manifolds. Proc. Taniguchi Symposium, Katata, 1985 by Katsuhiro Shiohama, Takashi Sakai, Toshikazu Sunada