I develop tools to amplify our mental senses: our intuition and reasoning abilities. The first five chapters—based on the Order of Magnitude Physics class taught at Caltech by Peter Goldreich and Sterl Phinney—form part of a textbook on dimensional analysis, approximation, and physical reasoning. The text is a resource of intuitions, problem-solving methods, and physical interpretations. By avoiding mathematical complexity, order-of-magnitude techniques increase our physical understanding, and allow us to study otherwise difficult or intractable problems. The textbook covers: (1) simple estimations, (2) dimensional analysis, (3) mechanical properties of materials, (4) thermal properties of materials, and (5) water waves.
As an extended example of order-of-magnitude methods, I construct an analytic model for the flash sensitivity of a retinal rod. This model extends the flash-response model of Lamb and Pugh with an approximate model for steady-state response as a function of background light Ib. The combined model predicts that the flash sensitivity is proportional to Ib-1.3.This result roughly agrees with experimental data, which show that the flash sensitivity follows the Weber-Fechner behavior of Ib-1 over an intensity range of 100. Because the model is simple, it shows clearly how each biochemical pathway determines the rod's response.
The second example is an approximate model of primality, the square-root model. Its goal is to explain features of the density of primes. In this model, which is related to the Hawkins' random sieve, divisibility and primality are probabilistic. The model implies a recurrence for the probability that a number n is prime. The asymptotic solution to the recurrence is (log n)-1, in agreement with the prime-number theorem. The next term in the solution oscillates around (log n)-1 with a period that grows superexponentially. These oscillations are a model for oscillations in the density of actual primes first demonstrated by Littlewood, who showed that the number of primes ≤ n crosses its natural approximator, the logarithmic integral, infinitely often. No explicit crossing is known; the best theorem, due to to Riele, says that the first crossing happens below 7 x 10370. A consequence of the square-root model is the conjecture that the first crossing is near 1027.
