Top: Beckhap's law from the 2003, 25th anniversary edition of the book Murphy's law by Arthur Bloch. [4] Bottom: Visual and verbal depiction of Beckhap's law: screen shot (and thinking-to-herself words) of Cynthia, described as the "Superbowl of women", in the 2000 film 100 Girls. [4]
In laws, Beckhap’s law is a colloquial premise on the relation between beauty and intelligence, which states that: [1]

Beauty times brains equals a constant.”

or in mathematical form:

$\text{beauty} \times \text{brains} = k$

or, stated another way, beauty is inversely proportional to brains:

$\text{beauty} \propto \frac{1}{\text{brains}} \,$

Beckhap’s law is sometimes referred to as the "beauty-brains paradox", the rule that intelligence is inversely proportional to beauty, that beauty varies inversely with intelligence, that it is very rare to find beauty and brains in the same package, or that people aren’t typically endowed with both beauty and brains, etc.

In simple terms, Beckhap's law captures the general understanding that beauty and brains tend not to be found in the same package; that best-looking or hottest people won’t typically be found in math class; that models tend to be dumb; that physicians tend not to be beauty queens; and so on, in the sense that if one is heightened in the looks department, he or she will likely be deficient in the brains department, and vice versa.

Politically correct | General statement
In more intuitive terms, using more politically correct language or rather "scientifically correct" terminology, this can be restated to the effect that Beckhap's law declares that, on average, one's level of physical attractiveness (beauty) A will be inversely proportional to one's intelligence (brains) I, a statement defined mathematically as :
 The 2001 film Shallow Hall is a classic parody of the "inner beauty" vs "outer beauty" paradox, aka Bekhap's law.

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

In more general terms, as captured well in the 2001 film Shallow Hal, Bekhap's law can be restated to equivalent conclusion that, in general:

“Inner beauty times outer beauty equals a constant.”

The term "beauty", however, in thermodynamic terms, is one of the more difficult conceptions to quantify (see below), in the sense that, knowing a person is powered molecule, it becomes difficult to distinguish the relative beauty of comparative smaller molecules, such as water H2O as compared to hydronium H3O+

The 1975 personal space studies of James Dabbs and Neil Stokes is one exception to the above, being that here average physical outer beauty was quantified in inches of personal space increase, in moving scenarios. [13]

Etymology | Peckham's or Beckhap's law
There is some inconsistency as to the eponym of this law, namely whether it is attributed to a person named Peckham (used commonly in the 1970s) or Beckhap (used commonly in recently decades). In a 1975 listing of eponym laws, the statement ‘beauty times brains equals a constant’ was listed as Peckham’s law, not Beckhap’s law? [6] Likewise, in the 1979 book 1,001 Logical Laws, the same saying was also listed as “Peckham’s law”. [8] If it is indeed Peckham, the saying could be attributed to American Supreme Court justice Rufas Peckham (1838-1909). [11] This, however, is only logical guess.

The name Beckhap's law (possibly a verbal to written typo) was cemented into fame, as seems to be the case, with its inclusion of the very-popular 1977 book Murphy’s Law and Other Reasons Why Things Go Wrong by American writer Arthur Bloch, which reprinted into the 25th edition. [2] The origin of who is Beckhap (or Peckham), however, may remain as elusive as is the origin of Murphy of Murphy's law. [2]

Usage
In 1983, without specifically naming the saying, sociologist David Myer stated that:

“On the idiom that it is ‘a strange illusion to suppose that beauty is goodness’, there might be a trace of truth to the stereotype. Children and young adults who are attractive tend to have higher self-esteem. They are more assertive, although they are also believed to be more egotistical. There are neither more nor less academically capable, contrary to the negative stereotype that ‘beauty times brains equals a constant’.”

 A 2005 depiction of Beckhap’s law on the home page of HumanThermodynamics.com. Caption: “Why is attractiveness A inversely proportional to intelligence I? Why in any random high school, for instance, are the least physically attractive students typically found in math class, and the most physically attractive students typically found in English class?” (based on poll, conducted by Libb Thims, of twenty different Chicagoans, from twenty different high schools, concerning which classroom on average will the most attractive, and least attractive, students typically tend to be found in any given high school).
Myer cites the 1979 paper of Jack Sparacino and Stephen Hansell, who in their various studies of high school and college students, founding that for high school students "attractiveness was not associated with achievement for boys [N=84] but was negatively associated for girls [N=83]" and also “found no relationship between attractiveness and grades for male college students”, as reference to back his assertion regarding the beauty-brains quote, being what he considers to be a negative stereotype. [7]

In 2005, and anon blogger posted a short blog entitled “What is Beckhap constant?”, stating that: “there has been a constant debate (pardon the pun) on what exactly the Beckhap constant is. Did this crazy guy simply mean a fixed constant or an arbitrary one?” [9] He summarized Beckhap’s law as such: [10]

“Using Beckhap's law, we are able to prove that the world is made up of two sorts of people, assuming that most people tend towards extremes of the two characteristics: (a) that if you are a genius, you will never be so good-looking; (b) that if you are a beauty, the magnitude of your good looks will exceed the magnitude of your intelligence. (and conversely).”

The following is a 2019 statement (ΡΊ) from just-one-liners.com:

Other usages can be found beyond this.
 A general visual of Beckhap's law: namely the premise of finding burning (hot) ice (ordered), i.e. a hot and ordered thing, generally speaking.

Thermodynamics
See main: Beckhap's law (proof)
A very simplified explanation of Beckhap's law, in physical science terms, is beauty governed within the confines of Boerhaave's law, namely that heat added to a system tends to increase the volume (or social volume), and hence the molecular motion, or "disorder" as is often implied, of the given body or system; and conversely, heat removed from a system tends to decrease the volume and hence the "order" of the system or body, and that at absolute zero of temperature perfect crystalline order is said to exist.

Some aspects of this phenomenon have been quantified volumetrically, as discussed in the 1975 study “Beauty is Power: the Use of Space on the Sidewalk” by James Dabbs and Neil Stokes. [13]

Hence, in conceptual speak, it is difficult in nature to find a "hot ice cube", such as depicted adjacent (burning ice cube), or in turn a "beautiful smart person" (e.g. a genius supermodel).

In 2002, American electrochemical engineer Libb Thims, in human chemical thermodynamics terms, attempted to correlate 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:

$H = k A \,$

second that entropy is inversely proportional to intelligence:

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

values which can used to represent the instantaneous 'state' of the reactive system at any given second on going from reactants to products. These 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: [5]

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

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.

 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.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.
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.

Corroboration

In a 2002, in corroboration of Beckhap's law, American chemical engineer Libb Thims (being unaware of either Beckhap's law or the saying) undertook an "Attractiveness vs. Intelligence" study of photos and corresponding graduation degrees of the 2,018 students of the graduating classes at the University of Illinois at Chicago (UIC) in the years 1969 and 1972, finding that when like groups of students are grouped (e.g. male engineers, female science majors, education degrees students, etc.) that within those groups physical attractiveness does indeed have an inverse relationship to intelligence. [3]

In more detail, in the course of the study, to determine if the inverse pattern, called by Thims the "beauty-brains paradox", exists statistically, in 2002 Thims conducted an ‘attractiveness A vs. intelligence I’ study, i.e. to determine if attractiveness is, in actuality, inversely proportional to intelligence.

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, on a 7-point Likert type scale (7 being most physically attractive), and had a second group of people rate the intellectual difficulty, on a 100 point scale (100 being most intellectually difficult), of each degree obtained, for the people in those photos, albeit only being shown the name of the degree.

These two 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, we obtain the plot shown above. [3]

 The 2001 film Legally Blond a parody of Beckhap's law, mixed in with a bit of the anomaly effect.
Anomalies
Some will be quick to point out so-called "anomalies" to Beckhap's law, and cite individuals who are considered "brainy" and "beautiful", such as supermodel Cindy Crawford who was a high school valedictorian, who was a first year chemical engineering student before being "discovered" for her beauty potential, and thereafter entering into the supermodel hall of fame.

German mathematics student Julia Ruscher, adjacent, and the 2001 comedy film Legally Blond are two other anomaly examples.

Note
A controversial, albeit well-discussed, 2004 paper “Why Beautify People are More Intelligent” by English economist Satoshi Kanazawa and American sociologist Jody Kovar attempts to present what seems to be a contrived “empirical evidence” based polemic to "prove" the stereotype that individuals perceive physically attractive others to be more intelligent than physically unattractive others. [12]

Trivia
The 1976 film Rocky, gives the following take on Beckhap’s law:

 A photoshop image of hot guy with a genius brain (Albert Einstein), with a tattoo of the mass-energy equivalency on his arm E=mc², an example of Beckhap’s law.

References
1. Beckhap’s law (list of eponymous laws) – RationalWiki.org.
2. (a) Bloch, Arthur. (1991). The Complete Murphy’s Law: a Definitive Collection (Quote: Beckhap's law: Beauty times brains equals a constant.). Price Stern Sloan.
(b) Bloch, Arthur. (1977). Murphy’s Law and Other Reasons Why Things Go Wrong. Price.
3. (a) Thims, Libb. (2002). “UIC: Attractiveness vs. Intelligence Date: 2,000 graduation photos rated for attractiveness and undergraduate degrees per each photo rated for intellectual difficulty”, IoHT Research Project.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two) (UIC: Attractiveness vs. Intelligence Study, pgs. 671-72). Morrisville, NC: LuLu.
4. (a) Bloch, Arthur. (2003). Murphy’s Law: 25th Anniversary Edition (Beckhap’s law, pg. 136). Perigee.
(b) Arthur Bloch - Wikipedia.
5. Thims, Libb. (2002). Human Thermodynamics (Volume One) (ch. 8, pg. 199). 330-pgs. (un-finished) Chicago: IoHT.
6. Anon. (1975). “Article” (section: Peckham's law: “beauty times brains equals a constant” The Coevolution Quarterly (pg. 139), Issues 8-12. Whole Earth Catalog.
7. (a) Myers, David. (1983). Social Psychology (pg. 479). McGraw-Hill.
(b) Sparacino, Jack. and Hansell, Stephen. (1979). “Physical Attractiveness and Academic Performance: Beauty is not always Talent” (abstract), Journal of Personality, 47: 449-69.
(c) Hansell, S., Sparacino, J., and Ronchi, D. (1982). “Physical attractiveness and Blood Pressure: Sex and Age Differences.” Personality and Social Psychology Bulletin, 8(1): 113-21; study found that unattractive females have significantly higher blood pressure than attractive women, a relationship not found among unattractive and attractive men.
8. Peers, John, Bennett, Gordon. (1979). 1,001 Logical Laws, Accurate Axioms, and Profound Principles, Trusty Truisms, Homey Homilies, Colorful Corollaries, Quotable Quotes, and Rambunctious Ruminations for all Walks of Life (Peckham’s Law, pg. 131). Doubleday.
9. Decipher. (2005). “What is Beckhap constant?”, Happenchance.Blogspot.com, Feb 11.
10. Decipher. (2005). “Special Report: Beckhap’s Law”, Happenchance.Blogspot.com, Feb 10.
11. Rufas Peckham – Wikipedia.
12. Kanazawa, Satoshi and Kovar, Jody. (2004). “Why Beautify People are More Intelligent” (abs), Intelligence, 32(3): 227-43.
13. Dabbs, M. and Stokes, N.A. (1975). “Beauty is Power: the Use of Space on the Sidewalk” (abs), Sociometry, 38: 551-57.