2205: Types of Approximation

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Revision as of 02:02, 21 September 2019 by 172.69.22.134 (talk) (Explanation: different sorts of approximation - tie together, reduce duplication on cosmology)
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Types of Approximation
It's not my fault I haven't had a chance to measure the curvature of this particular universe.
Title text: It's not my fault I haven't had a chance to measure the curvature of this particular universe.

Explanation

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In physics and engineering, problem solving typically requires approximations, as physical properties of the universe can be difficult to model. For example, in introductory physics classes, theories are introduced in frictionless environments. The level of precision required in a calculation or approximation varies depending on the context.

In the comic, Cueball, the physicist, generally dealing with theoretical constructs can use straight math, is introducing a problem with the assumption that the particular curve is a (perfectly) circular arc with a radius represented by R. Engineers have to deal with real things, whose dimensions may be known to a certain tollerance. Megan, the engineer, also assumes that the curve is similar to a circle, with a deviation factor of 1/1000.

The joke arises when Ponytail, the cosmologist, uses the much less precise approximation of pi (π) equal to 1. Pi is an irrational number, usually truncated to 3.14. The closest order of magnitude to that is 10 to the 0 power, or 1.

Ponytail offering to use 10 instead of 1 alludes to Fermi approximations, as shown in Paint the Earth. Numbers are rounded to the nearest order of magnitude (1, 10, 100, etc.) using a base 10 logarithmic scale. On this scale, "halfway" between 1 and 10 would be √10 ≈ 3.16. Thus, numbers between about 0.316 and 3.16 are rounded to 1, between 3.16 and 31.6 are rounded to 10, and so on. At about 3.14, pi falls close to this cutoff point, and so by using this form of estimation it doesn't really matter to Ponytail whether pi is approximated to 1 or 10.

Pi is defined as the ratio of the circumference of a circle divided by its diameter. This number is an irrational starting with 3.14 when the geometry is flat. But in curved spaces, the ratios are different. Almost every number can be pi depending on the curvature of the place the circle is residing. The cosmologist doesn't know the curvature of the universe, and so traditional values of Pi may not be more accurate.

This is a parody of the tendency of cosmology to use much rougher approximations in their work. In general, astronomers deal with masses and distances that are so vast that approximations that would be ridiculous elsewhere still yield reasonable answers in astronomy.

Approximating Pi as 1 may also refer to the habit astronomers have of changing the units of measure such that important constants (such as the speed of light or the gravitational constant) are equal to 1, which highly simplifies the formulas without compromising the math. In this case, the number pi is a dimensionless factor, not a directly measured quantity, which means the math will not work.

Transcript

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[Three nearly identical panels showing the lower-left portion of a wheel and hub diagram with symbols and an equation, each with a different character holding a pointer up to the diagram and a label above the panel with the character's profession:]
[Physicist Approximations]
Cueball: We'll assume the curve of this rail is a circular arc with radius R.
[Engineer Approximations]
Megan: Let's assume this curve deviates from a circle by no more than 1 part in 1,000.
[Cosmologist Approximations]
Ponytail: Assume pi is one.
Off-panel voice: Pretty sure it's bigger than that.
Ponytail: OK, we can make it ten. Whatever.


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Discussion

The cosmologist is probably using Fermi's a la What-If 84: Paint the EarthOhFFS (talk) 20:34, 20 September 2019 (UTC)

In that What-If, the rounding formula for Fermi problem estimation is given as "Fermi(x) = round10(log10(x))". log10(pi) (Google search, shows calculator) is roughly .4971... so close enough that someone could do a "Fermi rounding" to either 1 or 10 and not really care one way or another. 162.158.142.118 21:19, 20 September 2019 (UTC)

As a physics Phd (though not working in astrophysics), approximating pi to 1 is not all that bad. Especially when the measurable quantities that go into the calculation usually have huge error bars.--172.68.59.120 21:03, 20 September 2019 (UTC)

Using natural units (setting c=hbar=1) is different from setting pi to 1. Using different units is always allowed and not an approximation. Setting pi to 1 on the other hand, is an approximation and is only justifiable if the other quantities in the calculation have huge uncertainty. --172.68.59.120 21:07, 20 September 2019 (UTC)