Cannons and Sparrows - Numberphile

TL;DR

This video discusses the problem of cutting a polygon into equal area and equal perimeter pieces and explores the solutions for different numbers of pieces.

Transcript

The video you're about to watch was described to me as firing cannons at sparrows. The sparrow being the pretty simple mathematical problem we're about to tell you about, and the cannon being the heavy artillery, the mathematical machinery, that has been used to try to conquer it. Now in the middle of the video we will encounter some of that heavy ... Read More

Key Insights

• 🕰️ The problem of cutting a polygon into equal area and equal perimeter pieces becomes more complex as the number of pieces increases.
• 🔨 The concept of optimal transport and equivariant obstruction theory are mathematical tools used to study the problem.
• ✊ The problem has been solved for prime power values, indicating that polygons with those specific numbers of pieces can be cut according to the specified conditions.

Questions & Answers

Q: What was the initial problem presented in the video?

The initial problem was to cut a polygon into a certain number of pieces with equal area and equal perimeter.

Q: How was the problem solved for the cases of two and three pieces?

For two pieces, the polygon was cut with a straight line that rotated continuously, ensuring equal area while the perimeter varied. For three pieces, a map from the space of configurations of three points to the space of three numbers was used to determine if equal perimeters were achieved.

Q: What is the significance of the values that have been solved for in the problem?

The values that have been solved for are the prime powers, indicating that the problem can be solved for polygons with those specific number of pieces.

Q: Are there still open questions in this mathematical problem?

Yes, the problem remains unsolved for certain numbers such as 6, 10, 12, 14, 15, and 21.

Summary & Key Takeaways

• The video introduces a mathematical problem where a polygon needs to be cut into a specific number of pieces with equal area and equal perimeter.

• The problem is initially solved for the cases of two and three pieces, but becomes more complex for larger numbers of pieces.

• The concept of optimal transport and equivariant obstruction theory is introduced as mathematical tools used to explore the problem.