2205: Types of Approximation

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Types of Approximation

Title text: It's not my fault I haven't had a chance to measure the curvature of this particular universe.

Explanation

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 that can use relatively simple 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, which deviate from ideal shapes. Dimensions may be known to a certain tolerance. Megan, the engineer, also assumes that the curve is similar to a circle, with a deviation factor of 1/1000 or less.

The joke arises when Ponytail, the cosmologist, uses the much less precise^{[citation needed]} approximation of pi (π) equal to 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. Pi is an irrational number that can be approximated by 3.14, so it is very close to the 3.16 cutoff point. The closest order of magnitude to pi is 10⁰, or 1. Furthering the joke, Ponytail's calculations are so "coarse" she doesn't even particularly mind whether pi is approximated to 1 or the other reasonable Fermi approximation, 10¹, 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 3.1415926535897932384626, the value for this ratio in a flat geometry. But in a curved space, the ratio might be different. The title text makes use of the fact that almost every number can be this ratio depending on the curvature of the space the circle is in. The cosmologist doesn't know the curvature of "this particular universe" (a funny way to state the universe the cosmologist lives in, which is not perfectly flat), and so pi may not be the best value to use for the ratio between a circle's circumference and diameter.

This comic is a parody of the tendency of cosmology to use much rougher approximations in their work that would horrify engineers, other physicists, mathematicians, etc. In general, cosmologists deal with distances, time spans, masses, etc. that are so vast, with such large estimated errors, that approximations that would be ridiculous elsewhere still yield useful answers in cosmology. When dealing with the large numbers in cosmology, small multiplicative factors like 3 vanish into the rounding error: there probably isn't a useful difference between 10¹⁰⁰ and 10^100.497, even though these numbers differ by a factor very close to pi -- an error that would greatly disturb most physicists and engineers.

Approximating pi as 1 may also refer to the habit astronomers have of changing the units of measure such that important constants of the universe (such as the speed of light or the gravitational constant) are equal to 1, which highly simplifies the formulas without compromising the math. The number pi, however, is a dimensionless ratio, which doesn't depend on the unit of measure.

Transcript

[Three panels show the same setup with three different characters. In the upper-right corner of each panel is the lower-left portion of a wheel and hub diagram, showing two spokes going out to a curved rail. The two spokes connect to the rail with a small raised portion on the inside of the rail. There are both readable and unreadable text/symbols both outside and inside the curve and an equation below the curved rail. There are two small squares with readable labels. The three different characters are all holding a pointer up to the diagram while explaining an assumption. In the last panel an off-panel voice interrupts the speaker. This means the text from the reply to this comment goes further down over the diagram, so the top is hidden by text, compared to the first two. Above each panel is a label with the character's profession. As the text on the diagram is the same on all three panel, this text is shown here:]

r₁

r₂

d=2π(r₁+r₂)/2

[Panel 1 - Cueball. Caption above:]

Physicist Approximations

Cueball: We'll assume the curve of this rail is a circular arc with radius R.

[Panel 2 - Megan. Caption above:]

Engineer Approximations

Megan: Let's assume this curve deviates from a circle by no more than 1 part in 1,000.

[Panel 3 - Ponytail. Caption above:]

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)