Dr. Who, Fibonacci’s Rabbits, and the Wasp Apocalypse (Updated)

Chapter XII from Fibonacci’s Liber abaci describes the following scenario:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Solving this riddle, of course, yields the famous Fibonacci sequence:

\text{F}_n = 1,1,2,3,5,8,13,21,34,55,89,144,223,377,610,987,1597,2584,4181,6765...

where the nth term in the sequence is the sum of the previous two terms (n-1 and n-2). That is, \text{F}_n = \text{F}_{n-1} + \text{F}_{n-2} where n > 2.

What is much less well known is that a renegade group of rabbits escaped this enclosure and were later captured after trying to overrun a garden owned by a resident in a nearby town. These rabbits, they discovered, have a unique physiology: they procreate at a much faster rate. We’ll call this the “renegade sequence”:

\text{R}_n = 1,2,4,6,15,18,46,50,115,120,....



\text{Figure 1:} g(x) = 0.7131e^{0.5477x}

In the original storyline for “The Ark in Space” (1975), Dr. Who battles the Wirrn, “a wasp creature [that lays] its eggs inside cryo-preserved humans”; it just so happens the Wirrn’s reproductive pattern follows \text{R}_n (but at a much faster rate—in thousands of births per hour), and after traveling back to the twelfth century to recover the rabbits in order to study their anomalous physiology, Dr. Who tries unsuccessfully to unravel the sequence, a necessary step toward predicting the exact date the Wirrns will take over the earth while also determining whether a proposed vaccine is guaranteed to work fast enough to avert human extinction. Unfortunately, everyone dies (including Dr. Who, whose regenerative powers are neutralized by a wasp-like venom), the invading creatures repopulate the earth, and the series ends. (Cooler heads prevailed, though, and the story was rewritten.)

Figure 1 shows \text{R}_n as a function of time (in n hours) plotted with the regression line (red) defined in the caption (R^2 = 0.9829). Before he received the fatal sting, however, Dr. Who realized the renegade sequence can be derived from an underlying sequence, call it \text{S}_n, defined by \text{R}_n = \sum_{i=1}^n \text{S}_{i}, but he was unable to define an equation that calculates \text{R}_n precisely, one that predicts the total growth of Wirrns for any hour n after the initial infection.

Challenge: Find an equation that calculates any n in \text{R}_n by defining a function f : \text{Z}^+_0\to \text{Z} that generates \text{S}_n.

(I developed this sequence from my work on stock-market trends. I’ll post the answer in an update TBD.)

Spoiler Alert

The answer, based on \text{S}_n, is

\text{R}_n = \sum_{i=1}^n \lceil{\frac{x_i}{2}}\rceil^{3-2\bar{x_i}} + 0^{\bar{x_i}}

where \bar{x} \equiv x\pmod{2}.


Toward a quantification of intellectual disciplines

As a mathematician, I often find myself taking the STEM side of the STEM-versus-liberal-arts-and-humanities debate—this should come as no surprise to readers of this blog—and my principal conceit, that of a general claim to marginal productivity, quite often (and surprisingly, to me) underwhelms my opponents. So, I’ve been thinking about how we might (objectively) quantify the value of a discipline. May we argue, if we can, that quantum mechanics is “more important” than, say, the study of Victorian-period literature? Is the philosophy of mind as essential as the macroeconomics of international trade? Are composers of dodecaphonic concert music as indispensable to the socioeconomic fabric as historians of WWII? Is it really possible to make such comparisons, and should we be making them at all? Are all intellectual pursuits equally justified? If so, why should that be the case, and if not, how can society differentiate among so many disparate modes of inquiry?

To that end, then, I’ve quickly drafted eleven basic categories I believe can aid us in the quantification of an intellectual pursuit:


I. Demand

This will perforce involve a few slippery statistical calculations: average annual salary (scaled to cost-of-living expenses) for similar degree holders (e.g., BSc, PhD, etc.), the size of associated university departments, job-placement rates among graduates with the same terminal degree, the number of relevant publications (both popular and academic), and anything that betrays a clear supply-and-demand approach to the activities of participants within a discipline and the output they generate.

II. Influence

How fertile is the (inter-field) progeny of research? How often are articles cited by other disciplines? Do the articles, conferences, and symposia affect a diverse collection of academic research in different fields with perhaps sweeping consequences, or does the intellectual offspring of an academic discipline rarely push beyond the confines of its participants?


III. Difficulty

What is the effort required for mastery and original contribution? In general, we place a greater value on things that take increased effort to attain. It’s easier, for example, to eat a pizza than to acquire rock-hard abs. (As an aside, and apart from coeval psychosexual aspects of attraction—obesity was considered a desirable trait during the twelfth to fifteenth centuries because it signified wealth and power—being fit holds greater societal value because it, among other things, represents the more difficult, ascetic path, which suggests something of an evolutionary advantage.) Average time to graduation, the number of prerequisite courses for degree candidacy, and the rigor of standardized tests might also play a useful role here.

IV. Applicability 

How practical is the discipline’s intellectual import? How much utility does it possess? Does it (at least, eventually) lead to a general increase in the quality of life for the general population (e.g., the creation of plastics), or is it limited in its scope and interest only to those persons with a direct relationship to its machinery (e.g., non-commutative transformational symmetries in the development of Mozart’s Piano Sonata no. 12 in F major K. 332)? A less diplomatic characterization might involve asking the simple question: Who cares?

V. Recognition

Disciplines and academic fields that enjoy major prizes (e.g., Nobel, Pulitzer, Fields, Abel, etc.) must often succumb to more rigorous scrutiny and peer-reviewed analysis than those disciplines whose metrics more heavily rely upon the opinion of a small cadre of informed peers and the publish-or-perish repositories of second-tier journals willing to print marginal material. This isn’t a rigid metric, of course: Many economists now reject the Nobel-winning efficient-market hypothesis, and the LTCM debacle of the late 90s revealed the hidden perniciousness crouching behind the Black-Scholes equation, which also earned its creators a Nobel prize. (Perhaps these examples suggest something deficient about economics.) In general, though, winning a major international prize is a highly valued accomplishment that validates one’s work as enduring and important.

VI. Objectivity

Can we prove the propositions of an academic discipline, or are its claims wholly unfalsifiable? Is the machinery of an intellectual discipline largely based upon subjective and intuitive interpretation or rigorously defined axioms? Can the value and importance of a conceit change if coeval opinion modulates its position? It seems desirable to prefer an objective and provable claim to one based on subjectivity and a mushy, ever-changing worldview. 

math is purity

VII. Future value

What is the potential influence surrounding the field’s unsolved problems? Do experts generally believe resolving those issues might eventually lead to significant breakthroughs (or possibly chaos!), or will the discipline’s elusive solutions effectuate only incremental and localized progress when viewed through the widest possible lens?

VIII. Connectivity

What might be the long-range repercussions of eliminating a discipline? Would anyone beyond its active members notice its absence? How essential is its intellectual currency to our current socioeconomic infrastructure? One or two generations removed from our own? There exists inherent value in the indispensable.

IX. Ubiquity

How many colleges and universities offer formal, on-campus degrees in the field? Is its study limited to regional or localized interests, or is it embraced by a truly international collective? Wider academic availability, regardless of where you live, suggests a greater general value.

median earnings

X. Labor mobility

Is employment contingent upon a specific geographic area or narrowly defined economies? Does an intellectual discipline provide global opportunity? Do gender gaps or racial-bias issues exist that might impede entry for qualified candidates? How flexible is the discipline’s intellectual infrastructure? Do the skills you acquire permit productivity within a range of disparate occupations and applications, or do they translate poorly to other sectors of the the labor market because graduates are pigeonholed into a singular intellectual activity?

Can you find meaningful employment without going to graduate school, or must you finish a PhD in order to be gainfully employed? There are certain exceptions, of course: brain surgeons, for example, enjoy a very limited employment landscape—and earning anything less than an M.D. degree means you can’t practice medicine—but this is an example of an outlier that offer counterbalancing compensation within the larger model.

XI. Automation

What is the probability a discipline will be automated in the future? Can your field easily be replaced by a robot or a sufficiently robust AI (or even new advances in classical computer algorithms) in the next 15 years? (Luddites beware.)


Not perfect, but it’s a pretty good start, I think. The list strikes a decent balance across disciplines and, taken as a whole, doesn’t necessarily privilege any particular field. A communications major, for example, might score near the top in labor mobility, automation, and ubiquity but very low in difficulty and prize recognition (and likely most other categories, too). I also eliminated certain obvious categories (like historical import) because the history of our intellectual landscape has often been marked by hysteria, inaccuracy, and misinformation. To privilege, say, music(ology) because of its membership to the quadrivium when most people believed part of its importance revolved around its ability to affect the four humors seems unhelpful. (It also seems unfair to penalize, say, risk analysts because the stock market didn’t exist in the sixth century.)

Where we go from here is anyone’s guess. Specific quantifying methods might only require the most obvious metric: a function f : \text{R}^n\to \text{R} with a series of weightings where n is the total number of individual categories, c_i, and the total value of a discipline, v_j, is calculated by a geometric mean, provided no category can have a value of zero: v_j = \left(\prod_{i=1}^n c_i\right)^{1/n}.

Comments and suggestions welcome.


On making “Chewbacca Mom” Disappear

I take great solace in the fact that we could make “Chewbacca Mom” (hereafter CM) vanish—without being hurt in any way—if we could create the required Lorentz contraction utilizing Einstein’s gamma function:

L = L_r(1 - (\frac{v}{c})^2)^{1/2}

where L_r is CM’s length at rest. As her velocity approaches the speed of light (i.e., as v/c \to 1), CM (essentially) disappears before our very eyes! (And, yes, if the vehicle were large enough, we could fit the Kardashians inside, too. Don’t you just love science?)

Until technology catches mathematics and physics, though, I guess I’ll just have to keep filtering my news feed.