Euler's Formula and Existence of Solids

Question

Euler's formula tells us that the number of vertices, edges and faces of a 3D solid have to satisfy the relationship $V+F=E+2$. How about the converse, if I have a triple of numbers that fulfill this identity, how can I check if such solid (polyhedron) exists?

Answer 1

I followed the relevant (linked) resources and found the following.

---

Every face has at least three sides $2E \ge 3F$, and at least three faces meet at a vertex $2E \ge 3V$. Combining this with the Euler's formula $V+F=E+2$ for convex polyhedra, one has:

$$2F - 4 \ge V \ge F/2 +2$$

And every such pair $(V,F)$ should correspond to at least one convex polyhedron.

That is, it appears these necessary conditions are also sufficient: You only need to check these inequalities on top of the Euler's Formula to see if such polyhedron can exists or not.

I believe this answers your question.

---

If I'm not mistaken, this result is also stated as a theorem in Eulers Polyederformel, und die Arithmetisierung der Gestalt on page 9., which is a resource linked by Christian Blatter's answer on the related question which was linked in the comments by Shahab.

To see the intuition why it is sufficient, we observe a way to construct examples via truncation and expansion of pyramids and subsequently obtained polygons: (taken and shortened from this article):

Every $n$-sided polygon based pyramid has $V=F=n+1$, $n\ge 3$.

• Truncating (a pyramid) at a vertex where three faces meet: $V\to V+2,F\to F+1$.

• Expanding on one of the triangular faces (of a pyramid): $V\to V+1,F\to F+2$.

Both operations may be applied infinitely, provided that the polyhedron remains convex. (As for non-convex polyhedra, the Eulers formula might fail $-$ see Euler characteristic.)

---

---

Remark; Alternatively, You might be interested in the Steinitz's theorem, cited from Wikipedia:

"Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs."

Which is not needed for this particular question, but should be useful for similar problems.