The Art of Approximation

Question: What is the spacing between the pits on a CDROM? The pits (indentations) are the
memory elements, each pit storing one bit of information. Answer(s) below…

The Art of Approximation in Science and Engineering is a MIT course & book that I found really interesting and helpful for guiding thinking through complex problems.

CDROM Answers (Approximations!)

  • Turn the CD over: the brilliant colors imply diffraction in the visual light spectrum (you may recall those wavelengths are around 400~800μm), so the pits would be on the order of 1μm
  • Dimensional analysis: we’re looking for a characteristic Length per Bit, L/N
    • CD area: A ~ (10 cm)2
    • CD holds around 700MB: N ~7 gigabits, or 7 x 109
    • Pit size is of unit length so square root of Area / Bits
    • L~√(A/N) ~ 10cm/105 ~ 1μm
  • (You might know) CDROMs use a near-infrared laser to burn discs –> pit burned will be about the same size as the laser wavelength and a little longer than red light (~800μm) so 1μm is the right ballpark
  • (Allegedly) CDs were engineered to hold Beethoven’s Ninth Symphony, of about 74 minutes
    • Sampling rates are around 44~48kHz
    • 32 bits per sample is typical
  • (work the math with the powers of ten trick below) Capacity is about 5 x 109 which pretty close to our other approximation

The Powers of Ten

Love the framework of:

  • Scientific notation is your friend: separate into the powers of ten, and the mantissas (the parts in front of the power of ten)
  • Multiply the powers of ten together. Then, approximately multiply the mantissas together
  • There are only two numbers in the world: 1 and “few”
  • few2 ~ 10

Aside: I can’t say “Powers of Ten” without sharing this excellent classic

Further Readings

Apparently, there’s an updated course and book – it’s called The Art of Insight in Science and Engineering: Mastering Complexity

Truly, very cool that MIT has posted such great, free information through OpenCourseWare and MITx

Handy Cheat Sheet

Handy Cheat Sheet