Totally rational polytopes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:53:00Z http://mathoverflow.net/feeds/question/71949 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71949/totally-rational-polytopes Totally rational polytopes Joseph O'Rourke 2011-08-03T01:29:14Z 2011-08-14T15:11:34Z <p>Define a convex polytope in \$\mathbb{R}^d\$ as <em>totally rational</em> (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, etc., and finally its (positive) volume is rational. So: rational coordinates, and the measure of every \$k\$-dimensional face, \$1 \le k \le d{-}1\$, is rational, and the \$d\$-dimensional volume is positive and rational. (Scaling could then convert all these rationals to integers.) For example, the hypercube with vertex coordinates <code>\$\{0,1\}^d\$</code> is totally rational. Similarly an axis-aligned box with integral vertex coordinates is totally rational.</p> <blockquote> <p><b>Q1.</b> Are there other classes of totally rational polytopes, classes defined for all \$d\$?</p> </blockquote> <p>In particular,</p> <blockquote> <p><b>Q2.</b> Do there exist totally rational simplices in \$\mathbb{R}^d\$ for arbitrarily large \$d\$?</p> </blockquote> <p>Pythagorean triples yield totally rational triangles. I am not even certain that the <em>Heronian tetrahedra</em> described in <a href="http://mathworld.wolfram.com/HeronianTetrahedron.html" rel="nofollow">this MathWorld article</a> are totally rational, because it is unclear (to me) if they can be realized with rational vertex coordinates.</p> <p>All this is likely known, in which case key search phrases or other pointers would be welcomed. Thanks! <hr /> <b>Addendum.</b> Gerry Myerson's useful summary of Problem D22 in <em>Unsolved Problems In Number Theory</em> answers <b>Q2</b>: The problem is open! <b>Q1</b> remains (apparently) interesting; see the comments by Steve Huntsman and Gerhard Paseman.</p> http://mathoverflow.net/questions/71949/totally-rational-polytopes/71968#71968 Answer by Gerry Myerson for Totally rational polytopes Gerry Myerson 2011-08-03T06:00:57Z 2011-08-03T06:00:57Z <p>Guy, Unsolved Problems In Number Theory, problem D22: Simplexes with rational content. "Are there simplexes in any number of dimensions, all of whose contents (lengths, areas, volumes, hypewrvolumes) are rational?" </p> <p>Guy notes the answer is "yes" in 2 dimensions, by Heron triangles. Also "yes" in three dimensions: "John Leech notes that four copies of an acute-angled Heron triangle will fit together to form such a tetrahedron, provided that the volume is made rational, and this is not difficult." The smallest example has three pairs of opposite edges of lengths 148, 195, and 203. </p> <p>There is much more discussion, more examples, and several references. So far as I can see, there is no discussion of dimensions exceeding 3. </p>