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