A visual depiction of Beckhap's law, which says that beauty (physical attractiveness) is inversely proportional, on average, to brains (intellect). In 2002, American electrochemical engineer Libb Thims collected data on actual "beauty" and "brains" estimates of the 1969 to 1972 graduating classes of the University of Illinois, Chicago, giving quantitative proof to the colloquial folklore legend.
In proofs, Beckhap’s law proof was a quantitative study, conducted in 2002 by American electrochemical engineer Libb Thims, wherein the attractiveness of graduation photos were plotted against the difficulty of degrees obtained in the 1969 to 1972 students of the University of Illinois at Chicago, to determine if beauty and brains, on average, are inversely proportional, done to quantitatively prove Beckhap's law, which asserts that beauty is inversely proportional to brains.

Overview
In 2002, American electrochemical engineer Libb Thims had one group of people (N=14) estimated intellectual difficulty of college degrees, obtained by the University of Illinois, Chicago (UIC) graduating classes of 1969 and 1972, on a scale of 1 (easiest) to 100 (hardest), which he then plotted, on the ordinate, against the estimated, by another group of people (N=2) average physical attractiveness, on the abscissa, of the graduation photos of those who obtained those degrees, on a scale of 1 (least attractive) to 7 (most attractive).

Specifically, to determine if physical attractiveness, statistically, is inversely proportional to intelligence, on average, Thims had one group of people rate the physical attractiveness of 2,018 college graduation photos, graduating classes of 1969 and 1972 at the University of Illinois at Chicago, and had a second group of people rate the intellectual difficulty of each degree obtained, for the people in those photos, albeit only being shown the name of the degree.

The study found that attractiveness is inversely proportional to intelligence, on average, in accordance with Beckhap’s law. The second part of the study attempted to explain the results chemical thermodynamically, via showing how Gibbs free energy function can be used to explain the findings.

Physical attractiveness | Beauty
Thims had two people, a married couple in their late 20s, who each scored in the 8.5-9.0 range, on the 10-point hot-or-not.com photo ranking website (when it was active), sort through and rank by attractiveness some 2,000+ photos of the UIC graduating classes of 1969, 1970, 1972, 1973, and 1980, the male ranking the females, the female ranking the males, without knowing what their degree obtained was; example rankings shown below: [1]

 Female | Examples Male | Examples 7 (most attractive)↑_ ↓(least attractive)1 Top: more physically attractive females, attractiveness scores of 7, who obtained easier degrees in: elementary education and English.Bottom: lesser physically attractiveness females, attractiveness scores of 1, who obtained harder degrees in: German, anthropology, history, and sociology. Top: a more physically attractive male, attractiveness score of 7, who obtained an intellectually easier degree of communications.Bottom: lesser physically attractive males, attractiveness scores of 1, who obtained harder degrees in: psychology, history, computer science, and chemical engineering.

Intellectual difficulty | Brains
See main: College degrees by intellectual difficulty
The following shows the ranking of “intellectual difficulty” (I), on a scale of 1-100 (100=harder, 0=easier), of the 90 attained degrees for female graduating students of the University of Illinois, Chicago, for the graduating classes of 1969 and 1970 combined, according to the polled opinion of American college students (N=14). [2]

 Hardest Easiest

Data | Analysis
The data sets were sorted by sex and grouped into similar categories. The results confirmed the theory. In the graduating classes of 1969 and 1972, for example, 670 female students obtained 67 different degrees. By comparing females who obtained science-related degrees, among other related groups, the following plot is obtained: [3]

 Description: A plot of the ranked data results, of the group "female science majors", from the 2002 study of 2,018 University of Illinois at Chicago (UIC) college graduation photos, graduating classes of 1969 and 1972, showing that attractiveness is inversely proportion, on average, to intelligence, a finding which corroborates Beckhap's law.Key: P = psychology, B = biology, C = chemistry, and M = mathematics, each with 41, 20, 13, and 21 students, respectively. Similarly, A = physical attractiveness (of group); on a scale of 7.0 = most physically attractive to 1.0 = least physically attractive; and I = intellectual difficulty (of degree); on a scale of 100 = most intellectually difficult to 10 = least intellectually difficult.

A similar inverse trend was found with male engineering students, namely that the physical attractiveness of students, on average, was found to be inversely proportional to intellectual difficulty of degree obtained. Other gradating classes were analyzed as well, including: 1973, 1980.

Thermodynamics | Human chemistry
See main: Human chemical thermodynamics
Thims then attempted to explain this finding by correlating the initial state Gi and final state Gf of the free energy change for a typical mating reaction to bulk values of attractiveness and intelligence involved in mate selection. A solution was found using the following two assumptions, first that enthalpy is proportional to physical attractiveness:

$H = k A \,$

This would concur, in some sense, with Frederick Rossini's 1971 "Chemical Thermodynamics in the Real World" argument that enthalpy is a measure of "security" in social reaction existence, meaning that people will tend to want to bond with physically attractive individuals in relationships, and hence be seemingly more "secure" in their social existence or in the social structure, whereas less physically attractive individuals will tend to remain single, e.g. homebodies, cat ladies, and or outcasts, e.g. hobos, bag ladies, etc., give or take, baring more detailed discussion.

The second assumption made was that entropy is inversely proportional to intelligence:

$S = \frac{k}{I} \,$

This would concur, in some sense, with Stephen Hawking's 1996 argument that reading decreases the neurological entropy of a person by so many units, meaning that intellectual mastery would be inversely proportional to entropy of a person, in a roundabout sense, using a combination of the 1862 entropy as a measure of disgregation model of Rudolf Clausius and the 1882 characterization by Hermann Helmholtz of the magnitude of entropy |S| as the measure of disorder of the particles of the system with respect to each other.

With these approximations in place, one can employ intelligence and physical beauty as correlative measure of entropy and enthalpy, respectively, which can thus be used to represent the instantaneous 'state' of the reactive system at any given second on going from reactants to products. These, in turn, can then be substituted into the Gibbs equation:

$\Delta G = H_f - H_i - T (S_f - S_i) \,$

to yield for an inverse relationship plot. Skipping over much of the derivation and discussion, using the two above approximations, and assuming that initial state of the reaction, in which two individuals, one male molecule Mx and one female molecule Fy, of varying levels of intelligence and beauty, is the day the pair fall in love at first sight, that they pair conceives one child, Bc, three years later, and that the end state of the reaction, coincides with the point of the fifteenth year of the growth of the child, after which the precipitate child molecule begins to detach from the parental structure. This gives the following simplified overall reaction mechanism:

$M_X + F_Y \rightarrow B_C \,$

On this model, the following variables can be be defined at day one (-3 years before conception) and the final day (+15 after conception):

 $G_f = G_C^{15} \,$ Gibbs free energy of the state of the child, Bc, detached at age 15. $G_i = G_X^{-3} + G_Y^{-3} \,$ Gibbs free energy of the state of two reactants, the male Mx and female molecule Fy, at the point of love at first sight. $H_f = H_C^{15} \,$ Enthalpy of the state of the child, Bc, detached at age 15. $H_i = H_X^{-3} + H_Y^{-3} \,$ Enthalpy of the state of the two reactants, the male Mx and female molecule Fy, at the point of love at first sight. $S_f = S_C^{15} \,$ Entropy of the state of the child, Bc, detached at age 15. $S_i = S_X^{-3} + S_Y^{-3} \,$ Entropy of of the state of the two reactants, the male Mx and female molecule Fy, at the point of love at first sight.

Using these time-specific variables, through a bit of substitution, one can derive the following result: [9]

$A_X^{-3} = \frac{C_1}{I_X^{-3}} + C_2 \,$

which says that, owing to the constraints of the Gibbs equation, otherwise known as the combined law of thermodynamics, the physical attractiveness of the individual, in this case the male, will vary inversely with the intellect of the individual, on average, at the initial start to a typical romantic male-female reaction.

There are many issues, to note, with this proof, one being that the second assumption, that of entropy, using the disorder model of entropy, in human reactions, being inversely proportional to intelligence (mental order), is derived from gas theory, particularly the Boltzmann chaos assumption, in which particles are assumed to have non-correlative velocities, which is not the case with human molecules.

Discussion