Cube $C$ and Octahedron $O$ are dual Platonic solids in the sense that the the faces and the vertices are interchanged.

Duality of Cube and Octahedron

Often this is expressed like this: There is an bijection $B\colon C\to O$ which reverses inclusion.

What exactly is meant by reverse inclusion?

For example I found the following:

Let $S(C)$ be the set of all faces of the Cube and $S(O)$ the set of all faces of the Octahedron. Then duality between C and O means that there exists a bijective map $$ f\colon S(C)\to S(O) $$ which reverse inclusion. What does this mean?

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    $\begingroup$ I don't have time to spell this out comprehensively, but it's probably things like "this face of a cube contains these four vertices" is reversed to say "these four faces of an octahedron contain this vertex". $\endgroup$ – Mark S. Jul 26 '17 at 13:05

The context you are citing seems to be using the phrase "reverses containment" in a problematic way.

The normal meaning of "$f$ reverses containment" is this: if $x\subseteq y$, then $f(x)\supseteq y$". Really we are just talking about a function reversing a partial order, of which containment is an example. So the first problem is that I do not see any sensible concept of "partial order" for the faces of a polygon.

Apparently, then, this interpretation is out the window.

Another candidate is that it rather refers to the placement of the polyhedron after applying the duality transformation. The idea is that if you let $f$ associate each face of a cube with its midpoint, you wind up with a set of points forming the vertices of a octahedron inside the cube. Applying an analogous mapping to the octahedron, you get a smaller cube inside the octahedron, not the same cube you started with. But we could identify these polytopes with each other.

Still, this interpretation is problematic with the rest of the context, because it does not furnish a map between $S(C)$ and $S(O)$.

Finally, the other likely candidate is the "containment" of the central point of a face inside the face itself. A better word would be "membership" rather than containment. This is also reasonable, but it still suffers the conflict with the context that it is not a map from $S(C)\to S(O)$. Still, this interpretation is what I believe is the normal concept of the duality of these two polyhedra (and of duality between other pairs.) As far as I can see, the author just misexpressed the description of $f$.

If I were attempting to reconstruct a plausible version of the exposition you are describing, I would guess it would be this:

Let $C$ be a cube and $O$ be an octahedron, and $V(P)$ be the set of vertices of polytope $P$ and $F(P)$ be the set of faces of the polytope $P$.

Then $C$ and $O$ are dual in the sense that there is a special bijection $f:F(C)\to V(O)$, mapping each face to its center with the property that $f(F_1)\in F_1$ for any face $F_1\in F(C)$, and there is, likewise, a bijection $g:F(O)\to V(C)$ with analogous properties.

Alternatively, if the "polytope containment" interpretation seems more likely, you could say that $f$ is a function from the set of cubes in $\mathbb R^3$ to the set of octahedrons in $\mathbb R^3$, where $f$ sends a cube to the octahedron it contains.


A face includes its edges. Edges include their end points. Thus, "reverse inclusion" means, for instance, that the vertices of the octahedron, which are included in the edges of the octahedron, becomes faces of the cube, which include the edges of the cube. (I assume this is what they mean. Also, perhaps I should have used "contain" rather than "include"? "The bijection reverses containment"?)


First thing is to remember is that the Platonic Solids are not solids but geometrical structures created by connecting imaginary lines based on the centres of naturally attaching spheres.

We find within the Platonic Solids that there are two geometrical series which reflect the two ways at looking at these spherical arrangements. One is created by connecting the centres of adjacent spheres which forms a tetrahedron, octahedron and icosahedron, while the inverted tetrahedron, cube and dodecahedron are created by placing a face at the central point of each sphere perpendicular to its central axis

So, the octahedron within a cube is correct, while placing a cube within an octahedron is only a geometrical doodle with no meaning.

Defining the Platonic Solids this way helps to understand them and realize they are only part of a greater series called the Ffellonic Forms.


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