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](http://mathworld.wolfram.com/HeronianTetrahedron.html)
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 (apparently) interesting; see the comments by Steve Huntsman and Gerhard Paseman.



Totally rational polytopes
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?" 


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. 


There is much more discussion, more examples, and several references. So far as I can see, there is no discussion of dimensions exceeding 3. 


