## 6.055J/2.038J (Spring 2008) Art of approximation in science and engineering

Simple reasoning techniques for complex phenomena: divide and conquer, dimensional analysis, extreme cases, continuity, scaling, successive approximation, balancing, cheap calculus, and symmetry. Applications from physical and biological sciences, mathematics, and engineering. Examples include bird and machine flight, neuron biophysics, weather, prime numbers, and animal locomotion. Emphasis on low-cost experiments to test ideas and on fostering curiosity about phenomena in the world.

Newest items are in yellow.

Your comments on the course: This document has all your comments from the HKN evaluation. Thank you for the comments and suggestions. The plan for next year: more demonstrations, more problems, and a textbook. Happy estimating!

 Time/place MWF 1-2, Room 4-145 Instructor Sanjoy Mahajan (primary), Rohan Abeyaratne Handouts Grading philosophy |  Logarithms using musical intervals Homework HW 1 |  HW 2 |  HW 3 |  HW 4 |  HW 5 |  HW 6 |  HW 7 Solutions Solution 1 |  Solution 2 |  Python code for Solution 2, Problem 9 |  Solution 3 |  Solution 4 |  Solution 5 |  Solution 6 Notes (All the notes in one PDF file) Useful constants Preview Organizing complexity 1. Heterogeneous hierarchies:  |  Pit spacing |  Tree representations |  Oil imports |  Gold or bills? |  Random walks for accuracy |  Unix 2. Homogeneous hierarchies: Squeezing out fake complexity: Triangle bisection |  Pentagonal heat flow |  Calculus (maximizing a quadratic) |  Calculus (maximizing gain of a second-order system) 3. Proportional reasoning:  |  Intro plus spring-mass period |  Mountain heights |  Jump heights |  Power limits to jumping |  Falling cones |  Drag and jumping |  Cycling, swimming, flying 4. Conservation/box models:  |  Conservation cube |  Flight (scaling) |  Flight (explicit range and speed) 5. Dimensions:  |  Algebraic symmetry |  Economics example; pyramid example |  Dimensionless groups |  Hydrogen size |  Densities |  Hydrogen size by physics |  Bending of light |  Buckingham Pi theorem Discarding actual complexity 6. Special cases:  |  Pyramid volume |  Pulley acceleration |  Drag! 7. Discretization:  |  Random walks and keeping warm 8. Spring models:  |  Waves Afterword Interesting papers G. B. West et. al., "A General Model for the Origin of Allometric Scaling Laws in BiologyG. B. West et. al., "The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms" Interesting books Arthur Engel, Problem-solving Strategies (Springer, 1998)Knut Schmid-Nielsen, Scaling: Why Animal Size is So Important (CUP, 1984)Steven Vogel, Life in Moving Fluids (Princeton, 1996)George Polya, Induction and Analogy in Mathematics (Princeton, 1954) Interesting links Dangers of the resistive grid (comic)