How exactly is the formalism of thermodynamics based on contact geometry? There is a famous quote by the mathematician V. I. Arnold that goes like this:

Every mathematician knows it is impossible to understand an elementary
  course in thermodynamics. 

The source is Contact Geometry: the Geometrical Method to Gibbs' Thermodynamics, and It goes on like this (bold is mine):

The reason is that thermodynamics is based —as Gibbs has explicitly proclaimed— on a rather complicated mathematical theory, on contact geometry.

Then the author starts to explain, I imagine, how thermodynamics can be rigorously formulated using the formalism of contact geometry. I say "I imagine" because I have to admit that such a formalism is a bit too obscure for me, and I have no familiarity at all with the concept of "contact geometry". As a matter of fact, it is the first time that I hear about it, and the definition Wikipedia gives of it is completely unintelligible to me...
What I would like to know is, in terms accessible to someone with a "basic" mathematical background like me (mostly calculus): how exactly is the formalism of thermodynamics based on contact geometry?
 A: Here is the upshot:


*

*On one hand, a strict contact manifold $(M,\alpha)$ is a $(2n+1)$-dimensional manifold $M$ equipped with a globally defined one-form $\alpha\in \Gamma(T^{\ast}M)$ that is maximally non-integrable 
$$\alpha \wedge (\mathrm{d}\alpha)^n~\neq~ 0.\tag{1}$$ 
It is of interest to find submanifolds $N\subseteq M$ such that $TN\subseteq {\rm ker}(\alpha)\subseteq TM$. Such submanifolds of maximal dimension [which turns out to be $n$-dimensional] are called Legendrian submanifolds.

*On the other hand, the first law of thermodynamics 
$$\mathrm{d}U~=~ \sum_{i=1}^np_i\mathrm{d}q^i  \tag{2}$$
[where $U$ is internal energy and $(q^i, p_i)$ are thermodynamical conjugate variables] yields a contact form 
$$\alpha~:=~\mathrm{d}U- \sum_{i=1}^np_i\mathrm{d}q^i.\tag{3} $$
A concrete thermodynamical system [with an equation of state] becomes realized as a Legendrian submanifold.    
References:


*

*S. G. Rajeev, A Hamilton-Jacobi Formalism for Thermodynamics, Annals. Phys. 323 (2008) 2265, arXiv:0711.4319.

*J. C. Baez, Classical Mechanics versus Thermodynamics, part 1 & part 2, Azimuth blog posts, 2012.
A: 
in terms accessible to someone with a "basic" mathematical background like me (mostly calculus): how exactly is the formalism of thermodynamics based on contact geometry?

From what I understand (little), especially from Baez and Grmela, a logical sequence from thermodynamics to contact geometry is:


*

*classical thermodynamics
-> variational formulation (maximization of entropy)
-> differential geometry (one-forms)
-> contact geometry.


The differential geometry bit includes a Riemannian metric, and it may take the form of symplectic geometry (for even-dimensional manifolds) or contact geometry (for odd-dimensional manifold).
To learn more:


*

*Starting from calculus-level math, John Denker proposes to introduce differential forms and their application to thermodynamics.

*Salamon et al., in The mathematical structure of thermodynamics, offer what should be a quite smooth introduction to contact manifolds in thermodynamics.

*Starting from differential forms, Mrugala provides another introduction to the subject in On contact and metric structures on thermodynamic spaces (e-print).
And there are very relevant answers, discussions, and references in the older questions:
Introduction to differential forms in thermodynamics
Symplectic geometry in thermodynamics
Conjugate variables in thermodynamics vs. Hamiltonian mechanics
A: I suspect you are looking for a rather more low-fi answer. Perhaps you could clarify why this interests you, for context. 
My sense (not my physics forte tbh) is that this is about how to consider large sets of interacting things, with certain degrees of freedom. Say helium atoms can be treated as having 3 degrees (xyz axis), hydrogen atoms as H2 molecules have an extra way to spin and bump (xyz + spin around central axis), and so on eg. for tension & compression. 
Because thermodynamics is all about keeping one thing constant while changing other things, in a mathematical space that embodies interactions with certain degrees of freedom, you are going to generate surfaces where the demand for those variables to be constant are met, or manifolds. 
The trick of creating a mathematical space where you can identify patterns more easily is used a lot in physics. Like in creating the Lagrangian or Hamiltonian of system, which sort of boil things down to the core dynamic. Or, using complex variables (a number + an imaginary number counterpart) to keep track of different types of thing, while doing maths on the combination of them. I remember enjoying the moment I understood you can take the equations of two planetary orbits, and make a new equation which is like a slice across the orbits with dots on; then you just crank the handle to move the slice around and see if any of the dots ever overlap, and if they do the planets will eventually collide and the orbits aren't stable. 
Mathematical spaces are useful. Large sets of things interacting comes down to geometry. Hence contact geometry, with manifolds in 'phase' or mathematical space. 
