2205: Types of Approximation

|< < Prev Comic #2205 (September 20, 2019) Next > >|

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

This explanation may be incomplete or incorrect: Created by an APPROXIMATOR. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon. If you can address this issue, please edit the page! Thanks.

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 tolerance. 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 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 lower-left portion of each panel is a diagram of the wheel and hub type, showing two spokes going out to a curved rail. The two spokes connect to the rail with a small raised potiopn 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.

 add a comment! ⋅  add a topic (use sparingly)! ⋅  refresh comments!

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)