Why don't they teach simple visual logarithms (and hyperbolic trig)?

TL;DR

By exploring the concept of anti-shapeshifters, the video reveals the natural connection between logarithms, hyperbolic functions, and other mathematical concepts.

Transcript

Welcome to another Mathologer   video. Have a look at this. First squish this square down by a factor of 2 and then stretch the resulting rectangle horizontally, also by a factor of 2. Is this rectangle larger or smaller than the square we started with? Easy, right? The answer is “neither”. They have the same area. First, squishing by a factor of 2... Read More

Key Insights

• 💠 Anti-shapeshifters are shapes that maintain their area under squish-and-stretch transformations.
• 🥺 The area function of an anti-shapeshifter behaves like a logarithm and leads to the discovery of the special number e.
• 👻 The logarithmic properties of the area function allow for the introduction of hyperbolic counterparts of trigonometric functions.

Questions & Answers

Q: What is an anti-shapeshifter?

An anti-shapeshifter is a shape that remains the same before and after squish-and-stretch transformations. It retains its area regardless of the transformation applied.

Q: How does the area function of an anti-shapeshifter relate to logarithms?

The area function behaves like a logarithm, with the special number e being the value for which the function equals 1. This connection allows for the derivation of logarithmic properties and the introduction of other logarithmic functions.

Q: What are the hyperbolic counterparts of trigonometric functions?

The hyperbolic counterparts are cosh and shine (or sin-huh), which can be defined using a hyperbolic sector analogy. These functions have similar properties to sine and cosine but with hyperbolic interpretations.

Q: What are the applications of hyperbolic functions?

Hyperbolic functions have widespread applications in physics and engineering. They can describe the shape of hanging chains and cables (cosh) and are used in Lorentz transformations in special relativity.

Summary & Key Takeaways

• The video introduces the concept of anti-shapeshifters, which are shapes that remain unchanged under squish-and-stretch transformations.

• The video demonstrates how the area function of an anti-shapeshifter behaves like a logarithm, leading to the discovery of the special number e.

• The video explores the relationship between the area function and logarithms, including the product, quotient, and power rules.

• The video delves into hyperbolic counterparts of trigonometric functions and their applications in physics and engineering.