# 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?

• If you haven't already done so, google "Euler Charateristic"+ "Gibbs Phase Rule" you'll see that many authors discuss the likeness. Nothing jumps out, on a cursory glossing, as a definitive "yes, they have a connexion because blah", though (so I'm guessing you may already have done this). An interesting generalisation of the question would be "is there any meaning for a thermodynamic "genus"", i.e. are there other cases where the Gibbs phase rule gives the Euler charaterisatic of surfaces other than that of the sphere: $F+P-C=\chi(\Gamma)$ and what would the "surface" $\Gamma$ be? – Selene Routley Apr 17 '14 at 23:34
• Other interesting question is next. It is easy to deliver from Euler's formula all types of regular polyhedron (known as 5 platonic solids). Is there any analogy in thermodynamics? – aeiklmkv Apr 18 '14 at 0:01
• Just a few thoughts, albeit fairly obvious ones that you may already have thought of but which are a slightly different take on the question: to bear a relationship with the Euler formula means that there is some set $\mathbb{X}$, perhaps some space derived somehow from the total system phase space, kitted with the appropriate topology $\mathscr{T}$ such that the transformation wrought by phase changes of the matter corresponds to a homeomorphism of $(\mathbb{X},\,\mathscr{T})$ so that you can talk about topological invariants, simplicial complexes forming meshes for $\mathbb{X}$ and so .... – Selene Routley Apr 19 '14 at 1:59
• ...forth. So your question will be answered in the positive by defining $(\mathbb{X},\,\mathscr{T})$ and the appropriate complex. – Selene Routley Apr 19 '14 at 2:00
• polyhedron formula fails for some polyhedrons such as the great dodecahedron , If Gibbs phase rule also fails at there then there is a good correlation. – user28737 Aug 9 '14 at 12:43

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...)

• Have you read the comments above? In fact, the "$2$" in Euler's formula $V+F-E=2$ is also a special case. More general form is $V+F-E=\chi$, where $\chi$ is Euler characteristic. So the there is still possible connection between Gibbs' phase rule and Euler's polyhedral formula. Google gives some articles about special kind of graphs for this view. But I can't understand how far the parralel is true. – aeiklmkv Apr 18 '14 at 19:18

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.