The fix-the-wobbly-table theorem

TL;DR

The Wobbly Table Theorem states that a square table can be stabilized by rotating it, and this can be extended to rectangular tables as well.

Transcript

Welcome to another Mathologer video. Today's video is about the absolutely wonderful wobbly table theorem. Some of you may already be familiar with a very pretty special case of this theorem which became quite well known a few years ago when Numberphile dedicated a nice video to it. This special case of the theorem runs as follows: take a square ta... Read More

Key Insights

• 🦶 The Wobbly Table Theorem states that a square table can be stabilized by rotating it instead of using objects to stabilize the wobbling feet.
• 🪘 This theorem can be extended to rectangular tables, as long as the ground is not too uneven and the table's legs are sufficiently long.
• 🛝 The proof of the theorem involves the intermediate value theorem and requires the ground to be Lipschitz continuous with a maximum slope of 35.26 degrees.
• 🇰🇪 The Wobbly Table Theorem was popularized by Martin Gardner's Mathematical Games column and later expanded upon by mathematicians like Miodrag Novakovic, Ken Austin, and the Mathologer.
• 🤣 The theorem has practical applications in real-life situations where tables may wobble due to uneven floors or different leg lengths.
• 🚱 There is an ongoing open problem to determine if there are non-rectangular tables that can be stabilized by rotating.

Questions & Answers

Q: What is the Wobbly Table Theorem?

The Wobbly Table Theorem states that a square table can be stabilized by rotating it, without using objects to wedge under the wobbling feet. It can also apply to non-square tables if the ground is not too uneven.

Q: How does the proof of the theorem work?

The proof involves the intermediate value theorem, which states that a continuous function changing from positive to negative must have a zero value. By demonstrating that the difference between the heights of the table feet changes continuously during rotation, it can be shown that there must be a point where all four feet touch the ground.

Q: Are there any limitations or scenarios where the theorem fails?

The theorem holds true for most scenarios involving square or rectangular tables, as long as the ground is Lipschitz continuous (slope of at most 35.26 degrees) and the table legs are sufficiently long. However, there may be unusual setups or objects with non-rectangular shapes where the theorem does not apply.

Q: Who were the key contributors to the development of the Wobbly Table Theorem?

The idea of stabilizing a table by rotating it was popularized by Martin Gardner, who credited Miodrag Novakovic and Ken Austin for discovering the trick. The theorem was later rigorously proven and extended by a group of mathematicians, including Bill, Marty, Reiner, and the narrator (Mathologer).

Summary & Key Takeaways

• The Wobbly Table Theorem states that a square table can be stabilized by rotating it, rather than using objects to wedge under the wobbling feet.

• This theorem also applies to non-square tables, as long as the ground is not too uneven.

• The proof of the theorem involves the intermediate value theorem and requires the ground to be Lipschitz continuous and the table legs to be sufficiently long.