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Given a finite set of convex $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.

Question: Is it true that there are only finitely many different convex $(d+1)$-dimensional polytopes whose facets are solely (scaled and rotated versions of) polytopes in $\mathcal P$?


Some clarifications

A face of a polytope is the intersection of the polytope with a touching hyperplane, so subdividing facets does not count here.

In general, two $(d+1)$-polytopes shall be considered as different if they differ not just in scale and orientation. By the usual rigidity arguments, given the shape of facets and their connections, the metric of the polytope is uniquely determined. Hence, if we can built different polytopes, they will be combinatorially different as well.

Example

There are only finitely many polyhedra that can be built from any finite set of regular polygons, but as far as I know, this result is by enumeration (see, e.g. Johnson solids).

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I think this is indeed true. This should follow from the rigidity of convex spherical polytopes and a few more lemmas. I'll explain how to deal with cases $d\ge 3$.

Lemma 1, rigidity. Two convex spherical cobminatorially equivalent polytopes with isomertric faces are isometric.

Lemma 2, bounded volume. The boundary of any convex subset of round $\mathbb S^n$ has volume bounded from above by the volume of round $\mathbb S^{n-1}$.

Lemma 3, gap. Suppose we have a spherical convex polytope in $S^n$ of diameter less than $\pi-\varepsilon$ then the volume of the dual polytope is bounded from below by $c(n,\varepsilon)>0$.

Lemma 4, sum of dual solid angles. Let $P$ be a convex $d+1$-polytope. For each of its vertices consider the dual cone. Then the sum of solid angles of such dual cones is equal to the volume of the round $\mathbb S^d$.

Proof of finitness. So, suppose we have a finite number of convex $d$-polytopes $P_1,\ldots P_k$. Denote by $S_1,\ldots, S_m$ the spherical $d-1$ polytopes corresponding to vertices of $P_1,\ldots, P_k$.

Suppose we have a $d+1$-polytope $P$ with faces homothetic to $P_1,\ldots P_k$. Then for each vertex of $P$ we get a convex spherical $d$-polytope whose faces are composed from $S_1,\ldots, S_m$. Note the the number of such spherical polytopes is finite up to isometry, by Lemmas 1 and 2. Moreover, all of them have diameter less than $\pi-\varepsilon$ for some $\varepsilon$ . So by Lemma 3 the volumes of spherical polytopes dual to those (coming from vertices) are at least $c(d,\varepsilon)$. Hence by Lemma 4 the total number of vertices of $P$ is at most $vol(S^d)/c(d,\varepsilon)$. Hence we have only finite number of combinatorial types of $P$.

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  • $\begingroup$ Thank your for your answer! I need to better understand your definitions: why do you explicitly speak of spherical polytopes, and why are they subsets of $\Bbb S^n$ and not $\Bbb R^n$? $\endgroup$ – M. Winter Jan 27 at 17:06
  • $\begingroup$ Yes, spherical polytopes are subsets in $\mathbb S^n$ bounded by geodesic $n-1$-spheres. To each vertex $p$ of an Euclidean polytope $P $ in $\mathbb R^n$ we associate a spherical polytope in $\mathbb S^{n-1}$: take a small radius $r$ sphere $S_r^{n-1}$ in $\mathbb R^n$ centred at $p$ and take the intersection $S_r^{n-1}\cap P$. This is the spherical polytope, just scale it by $1/r$ so that it lies in $\mathbb S^{n-1}$ (i.e. in a sphere of radius $1$). $\endgroup$ – Dmitri Panov Jan 27 at 17:30
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In the case $d=2$ it is not possible if you exclude coplanar faces.

Proof:

Consider the set $S$ of all angles of the faces in $\mathcal P$. Since $\mathcal P$ is finite, there's a minimal such angle, let's say it is $\varepsilon > 0$. Now consider the set $V$ of all angles below $2\pi$ you can produce by summing elements of $S$. $V$ is finite, since each element of $V$ is a sum of at most $2\pi/\varepsilon$ elements of $S$. So there is a maximal element of $V$, let's say it's $2\pi-\delta$ where $\delta > 0$.

Now let's choose a polyhedron made from polygons in $\mathcal P$. At each vertex, this polyhedron must have nonzero angular defect, since otherwise the polyhedron would not be convex, or it would have coplanar faces. So the angular defect is $2\pi-v$ for some $v \in V$, so it is at least $\delta$. Now by Descartes' theorem on total angular defect the total angular defect is equal to $4\pi$, so the number of vertices of the polyhedron is bounded by at most $4\pi/\delta$.

Because of this, the total sum of angles at vertices can be at most $(4\pi/\delta)(2\pi-\delta)$. Since each face has an angle sum of at least $\pi$, the amount of faces is bounded by $n := (4/\delta)(2\pi-\delta)$, and this value depends only on $\mathcal P$.

Now there are only finitely many ways to glue together at most $n$ shapes of $\mathcal P$, and each such way describes a unique polyhedron (if any) by Cauchy's theorem. Therefore the amount of polyhedra you can form using faces in $\mathcal P$ is finite.

I'm fairly certain it is possible to generalize this proof to other values of $d$.

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  • $\begingroup$ Thank you. My argument for $d=2$ was exactly along these lines and I hoped to generalize. But I do not know about a higher dimensional analogue of Descartes' theorem. $\endgroup$ – M. Winter Jan 27 at 13:34
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    $\begingroup$ There's Chern's theorem which could discretize to the desired statement, but it only applies in the case of even $d$. $\endgroup$ – Magma Jan 27 at 13:39
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Not an answer, but an "almost counterexample" for $d=2$ that came to my mind when I saw that question. Starting with a regular dodecahedron, we can successively insert circular "belts" of hexagons, in a way that the whole polyhedron is convex at each stage. The illustrations should give an idea. The shapes of the hexagonal facets can come very close to each other (in the appropriate sense), though it is clear that there can only be so many of a precise shape. enter image description here

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  • $\begingroup$ That's a great illustration of what I was afraid would happen (at least in higher dimensions). $\endgroup$ – M. Winter Jan 29 at 12:07
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Perhaps you should add a condition to exclude coplanar facets, i.e., to insist upon strict convexity. Otherwise the deltahedra lead to an infinite number of combinatorially (but not geometrically) distinct polyhedra:


         
          Wikipedia image.


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    $\begingroup$ For me, a face of a convex polytope is the intersection of it with a touching hyperplane, so your example would still have only four facets. I will edit this into the question. $\endgroup$ – M. Winter Jan 27 at 13:09

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