6.055J | Spring 2008 | Undergraduate

The Art of Approximation in Science and Engineering


Course Meeting Times

3 sessions / week, 1 hour / session


An approximate model can be better than an exact model!

This counterintuitive statement suggests a few questions. First, how can approximate models be at all useful? Should we not strive for exactness? Second, what makes some models more useful than others?

On the first question: An approximate answer is all that we can understand because our minds are a small part of the world itself. So when we represent or model the world, we have to throw away aspects of the world in order for our minds to contain the model.

On the second question: Making useful models means discarding less important information so that our minds may grasp the important features that remain.

I will show you how to discard the less important information and thereby to make the most useful approximate answers. The most useful techniques fall into three groups:

  1. Divide and conquer (managing complexity)
    • Heterogeneous hierarchies
    • Homogeneous hierarchies
  2. Symmetry and invariance (removing spurious complexity)
    • Proportional reasoning
    • Conservation/box models
    • Dimensionless groups
  3. Lying (discarding complexity)
    • Special cases
    • Spring models
    • Discretization

The two divide-and-conquer techniques help you manage complexity. The three symmetry techniques help you remove superfluous complexity. These first two groups do lossless compression. The three lying techniques help you discard complexity. This third group does lossy compression.

Using these techniques, we will explore the natural and manmade worlds. Applications include:

  • turbulent drag: or how falling coffee filters tell us the fuel efficiency of airplanes.
  • xylophone acoustics: or why pianos are tuned with the lower notes below the ideal, equal-tempered frequency and with the higher notes above the ideal, equal-tempered frequency.
  • the design of compact discs: or how Beethoven’s ninth symphony helps you find the spacing between the pits.
  • period of a pendulum as a function of amplitude: or how hard it was to navigate 300 years ago.
  • the size distribution of eddies in turbulent flow: or how stars twinkle.
  • the bending of starlight by the sun: or the size of a black hole.
  • biomechanics: how high an animal jumps as a function of its size.

None of these problems has a simple analytic solution. The world — whether manmade or natural — rarely offers problems limited to one field of study, let alone problems whose equations have an analytic solution. To understand aspects of the world even partially, we need to use the preceding techniques, to make models that keep only the important features of a problem.

By making such models, we make understanding and designing more enjoyable. So the hidden although less tangible purpose of this course is to amplify your curiosity about the world.


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


The prerequisites for this course are Physics I (8.01 Mechanics, GIR) and Calculus I (18.01 Single Variable Calculus, GIR). A dose of curiosity and open-mindedness to attack problems about which you know little are recommended.


There will be six assigned problem sets during the semester and one optional problem set for you to chew on over the summer months. For each problem set there are a series of warmups and problems. Write solutions to all of them based on the open universe policy:

Collaboration, notes, and other sources of information are encouraged. However, avoid looking up answers until you solve the problem (or have tried hard). That policy helps you learn the most from the problems.

Bring a photocopy of your solution to class on the due date, trade it for a solution set, and figure out or ask me about any confusing points.


Homeworks (6) 100%

The purpose of this course is to learn, not to grade. If you come to class, do your homework and hand it in, you can succeed in this course.

Your work will be graded lightly: P (made a reasonable effort), D (did not make a reasonable effort), or F (did not turn in).

Course Info

Learning Resource Types
Problem Sets with Solutions
Online Textbook