FELINE FIZZIX

Cats. They like to explore. So, if your cat ever finds a tunnel through the centre of the earth, you can bet she’ll jump in.

A tunnel is a hole without a bottom. Your cat will fall at increasing speed toward the centre. Then she will continue to fall… upward. Eventually she’ll reach the other side of the planet.

But when?

Next time this happens, be ready with a catnip mouse. As soon as the cat jumps in, fling the toy mouse into a low orbit about the earth. …

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A comment poured in…

A viewer on my YouTube channel requested that I cover this problem from the 1997 International Mathematical Olympiad (Problem #5):

Find all pairs (a, b) of integers a, b ≥ 1 that satisfy the equation
aᵇ² = bᵃ

We’ll look at this on a YouTube Livestream Sunday, June 27, 2021 at 9 a.m. Pacific Time. Be there or be a regular right quadrilateral!

(If you miss the livestream, this link will let you watch the video post-broadcast.)

Calculus Made Easy

Sylvanus Thompson published an accessible introduction to Calculus in 1914. [Purists…

I’ll present this diagram in the form of an algorithm.

FOR n = 0 TO INFINITY
    PRINT [(1/2)^n]sin[(2^n)x]

It’s an infinitude of sine waves. The first function, sin x, runs from 0 to π. We take a bite out of the area above the x-axis. We do this with a sine wave having half the amplitude and twice the frequency.

Create a third sine wave, half the amplitude and twice the frequency of the previous. Use it to take a bite out of the remaining area.

Repeat.

Forever.

The remaining shaded area looks like a batwing. What is the size of the batwing?

Have fun with this. We’ll have a look during the MathAdam YouTube Livestream: Sunday, July 25, 9 A.M. Pacific Time. If you miss the livestream, the link will take you to the replay.

Cheers!

Adam

This comes from Sylvanus Thompson’s Calculus Made Easy. As presented in the text (page 86):

Have you seen The Good Place? Hypatia of Alexandria points to the 5 on her shirt. “Is this an S? Or a math?” Centuries of unencumbered bliss in Heaven have rendered her and her companions intellectual zombies.

I was resting in peace myself until a couple of weeks ago. This email fluttered into my inbox.

You have 5 holes and 1 fox. The holes are in a line, left to right, A, B, C, D and E. The fox is in one of these holes.

Mathematicians have been slicing open cones since the 4th Century BC. Here, we study one of these famous cross sections — the ellipse. Along the way, we’ll discover a beautiful proof from Germinal Dandelin of the 19th Century.

The Ellipse — Two Seemingly Unrelated Descriptions

Slice a right circular cone at an angle to the horizontal. The cross-section forms an ellipse.