The error in approximating the perimeter of an ellipse increases with the eccentricity of the ellipse for every practical approximation as far as I know.

🧵 1/n I say practical because here's a useless approximation that is exact in the limit as eccentricity goes to infinity:

perimeter ≈ 4 * semi-major axis

You wouldn't want to use this approximation on, say, the orbit of a planet.

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A polyhedron is called harmonic if the number of vertices is the harmonic mean of the number of edges and faces.

🧵 1/n The notion of a harmonic polyhedron goes back to Philolaus (c. 470 – c. 385 BC). Philolaus was a Pythagorean, as depicted in the medieval woodcut below.

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What is the significance of the number 10^10^10^34?

🧵 1/n The number 10^10^10^34 is known as Skewes' number. When Skewes defined this number in 1933 it was the largest "useful" number that had been defined. What was it useful for?

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You can use the quadratic formula without understanding the problem that lead to using it.

But the formula gives two roots, and you have to know which one makes sense in context. Now you do need to understand the application.

This is a simple example of a common pattern. A sighting of the sun or of a planet at a particular time determines a circle of possible locations.

A second sighting at a different time gives a second circle.

These two circles generally intersect in two points, one of which hopefully you know cannot be your location.

Necessary conditions for a number ending in yz to be a square.

z must be 0, 1, 4, 5, 6, or 9.

If z = 1, 4, or 9, y must be even.

If z = 6, y must be odd.

If z = 0, y must be 0.

If z = 5, y must be 2. For example, can 2194 be a square? No, because 9 is odd.

Can 2184 be a square? Possibly, because 8 is even.

(In fact it's not a square. These are necessary conditions, but not sufficient.)

The observation below led Marcel Golay to discover what’s now called the Golay code. The code uses blocks of 23 bits, 12 for data and 11 added for error detection.

It can correct up to 3 erroneous bits per block.

How to mentally compute the cube root of the cube of a two digit number.

Thread (1/10) In base 10, the last digits of the cubes of digits are distinct.

If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.

If d = 2, 3, 7, or 8, then d^3 ends in 10-d.

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