Just taking rational points near the 20 vertices doesn't work, because the result won't be a dodecahedron at all. Five of the vertices that are near 5 vertices of the original dodecahedron will be near to lying in a plane, but they won't be exactly in a plane. So instead of approximating the vertices, approximate the faces. The regular dodecahedron's 12 faces lie on 12 planes, and each of the vertices is the intersection of three of those planes. Each of the planes has an equation of the form Ax + By + Cz = D. Taking close rational approximations to A, B, C, and D will yield a combinatorial dodecahedron with all its vertices close to those of the regular dodecahedron. Andy On Mon, Apr 27, 2020 at 2:05 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
To construct a family of pentagonal dodecahedra P_k , all their vertices with rational Cartesian components, which successively approximate a regular specimen P .
More pedantically, for each real error bound e > 0 , there should exist some h such that dist(P, P_k) < e for all k >= h ; where dist(P, Q) is defined as (say) square root of sum of squared distances between corresponding vertices of (combinatorially isomorphic) polyhedra P, Q .
See
Adam Goucher, James Buddenhagen https://cp4space.wordpress.com/2013/09/22/rational-approximations-to-platoni... (Have I come across those fellows somewhere before?)
Günter M. Ziegler (2007) "Non-rational configurations, polytopes, and surfaces" https://arxiv.org/abs/0710.4453 (Undefined terms in the latter may not mean quite what one might expect: like "simple", "polyhedron", "polytope"!)
Fred Lunnon
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