using only a spherical ruler (to construct great lines) and a pair of compasses, how can you construct a regular dodecahedron on the surface of the sphere? thank you very much.
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Are you able to make a cube? If so, the vertices of a cube are included in the vertices of a regular dodecahedron. Then, you just need to add the remaining points locating them by the compass. Now, I shall not add anything else not to spoil the pleasure to solve this incredibly easy task... (PS: and of course, a cube is easily obtained starting from a tetrahedron, locating the laking vertices by a ruler. So, everything is reduced to the construction of a tetrahedron) Summary: the details of the construction are described in various people's comments below. One needs a "spherical ruler" (able to draw the great circle passing for two non-antipodal given points) and a compass (C) (able to draw a circle with given center passing for a given point). One first draws three orthogonal great circles (the third one is tricky: see below), that is an octahedron; then one makes a cube, and then the dodecahedron. |
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I'm not certain if this quite fits the ruler and compasses rules, but perhaps it can be modified: Carolyn A. Yackel has a paper in the proceedings of the Bridges 2009 conference, called "Marking a Physical Sphere with a Projected Platonic Solid". She gives a method to construct the regular dodecahedron following the "rules" of temari (a Japanese art form involving thread wrapped around balls in various symmetries), which involves measuring with paper tape and using traditional paper folding techniques. |
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Not an answer to the question as stated but relevant: I don't seem to have a reference, but the quickest description of possible constructions on, say, the unit sphere, is by the angles made by intersecting curves. The constructible angles are the same as the constructible angles in the Euclidean plane. Thus the constructible lengths are those arclengths $\alpha$ for which $\cos \alpha$ or $ \sin \alpha$ or $\tan \alpha$ (the conditions are equivalent) are in the "constructible field," the smallest extension of the rationals in which the square root of any positive element is still in the field. One might also wish to require $\alpha \leq \pi.$ Actually, let me make that a request. If anybody knows of a reference on the constructible lengths and angles on the surface of the sphere, please let me know. This is strictly analogous to (and presumably far, far older than) the situation in the hyperbolic plane, I will try to make a working link: http://zakuski.math.utsa.edu/~jagy/papers/Intelligencer_1995.pdf See also Marvin Jay Greenberg, "Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries," The American Mathematical Monthly (an M.A.A. journal), Volume 117, number 3, March 2010, pages 198-219. I have a pdf of that as well if anyone cannot find it. There is a bit of a story. The results on constructibility in $H^2$ were in a string of papers in Russian and Ukrainian in the 1930's and 1940's. I found, and used, the simple conclusions. I later sent my paper to Greenberg, so that material is in the M.A.A. paper mentioned and in the fourth edition of his book. Meanwhile, Robin Hartshorne (yes, that Hartshorne) heard of this result from Marvin and came up with his own proof using the Hilbert Field of Ends formalism, expressing regret that such a pretty result did not make it into his own book on the subject, "Geometry: Euclid and Beyond." |
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