Define a convex polytope in $\mathbb{R}^d$ as totally rational (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 $\{0,1\}^d$ is totally rational. Similarly an axis-aligned box with integral vertex coordinates is totally rational.

Q1. Are there other classes of totally rational polytopes, classes defined for all $d$?

In particular,

Q2. Do there exist totally rational simplices in $\mathbb{R}^d$ for arbitrarily large $d$?

Pythagorean triples yield totally rational triangles. I am not even certain that the Heronian tetrahedra described in this MathWorld article are totally rational, because it is unclear (to me) if they can be realized with rational vertex coordinates.

All this is likely known, in which case key search phrases or other pointers would be welcomed. Thanks!

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Addendum. Gerry Myerson's useful summary of Problem D22 in Unsolved Problems In Number Theory answers Q2: The problem is open. ! Q1 remains (apparently) interesting; see the comments by Steve Huntsman and Gerhard Paseman.

Define a convex polytope in $\mathbb{R}^d$ as totally rational (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 $\{0,1\}^d$ is totally rational. Similarly an axis-aligned box with integral vertex coordinates is totally rational.

Q1. Are there other classes of totally rational polytopes, classes defined for all $d$?

In particular,

Q2. Do there exist totally rational simplices in $\mathbb{R}^d$ for arbitrarily large $d$?

Pythagorean triples yield totally rational triangles. I am not even certain that the Heronian tetrahedra described in this MathWorld article are totally rational, because it is unclear (to me) if they can be realized with rational vertex coordinates.

All this is likely known, in which case key search phrases or other pointers would be welcomed. Thanks!

---

Addendum. Gerry Myerson's useful summary of Problem D22 in Unsolved Problems In Number Theory answers Q2: The problem is open. Q1 remains interesting; see the comments by Steve Huntsman and Gerhard Paseman.

Define a convex polytope in $\mathbb{R}^d$ as totally rational (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 $\{0,1\}^d$ is totally rational. Similarly an axis-aligned box with integral vertex coordinates is totally rational.

Q1. Are there other classes of totally rational polytopes, classes defined for all $d$?

In particular,

Q2. Do there exist totally rational simplices in $\mathbb{R}^d$ for arbitrarily large $d$?

Pythagorean triples yield totally rational triangles. I am not even certain that the Heronian tetrahedra described in this MathWorld article are totally rational, because it is unclear (to me) if they can be realized with rational vertex coordinates.

All this is likely known, in which case key search phrases or other pointers would be welcomed. Thanks!