Is there a true parallel between Gibbs' phase rule and Euler's polyhedral formula? Gibbs' phase rule states: $$F=C-P+2$$ where $F$ is number of degrees of freedom, $C$ is number of components, $P$ is number of phases.
Euler's polyhedral formula states: $$V+F-E=2$$  where $V$ is number of vertices, $F$ is number of faces, $E$ is number of edges.
It is easy to see that these formulas are similar. Is there a true parallel between them? Otherwise, what is the mathematical meaning of Gibbs' phase rule?
 A: The "$2$" in the "phase rule" conventionally written as $F=C-P+2$, where $F$ is the degrees of freedom, $C$ is the number of components and $P$ is the number of phases, refers to the temperature and pressure that are the usual intensive parameters in chemical equilibrium of several phases and components.  But this equation is only a special case of a more general one where there are other intensive parameters representing other than thermal and mechanical interactions, such as magnetic or electric fields. In fact, if the number of possible non-chemical interactions is denoted by $N$ then the phase rule is $F=C-P+N$ where now we can count the temperature for thermal and pressure for volumetric mechanical among the possible non-chemical interactions, if any. $C$ can be viewed as representing the number of chemical interactions, if you wish. (I think the connection between the formulas of Euler and Gibbs are as deep or shallow as that two assassinated Presidents Lincoln and Kennedy both had VPs named Johnson...)
A: Gibb's phase rule is much simpler than Euler's formula.
First of all, we're talking about a phase diagram which has $C + 2$ dimensions, where $C$ of the coordinates are chemical potentials for the $C$ many components, and the extra two are pressure and temperature, as hyportnex says.
Suppose there is a special submanifold of the phase diagram which lies at a co-existence junction between $P$ different phases. Then Gibb's phase rule just says that since $P$ is the codimension, the dimension of this submanifold, namely the number of degrees of freedom $F$, is the dimension minus the codimension, or
$$F = C + 2 -P.$$
It's equivalent to the rank-nullity theorem in linear algebra. This is a far simpler result compared to Euler's formula, which is a global property of graphs on the 2d sphere. Gibb's phase rule meanwhile is simply a local property in the phase diagram, which has different meaning at different points. It tells us nothing of the global structure.
If one were to invent a correspondence between them, it would be quite ad-hoc, since such a correspondence would have to say that there is something very special about temperature and pressure, ie. about the number 2. In topology 2 is very special, since it is the largest Euler characteristic among connected surfaces. If we included another tuning parameter, then the phase rule would read
$$F = C+3 - P,$$
and there is no connected surface of Euler characteristic 3, so nowhere to draw the graph that's supposed to be associated to each point of the phase diagram.
