Toroflux paradox: making things (dis)appear with math

TL;DR

Explore mathematical paradoxes involving vanishing and appearing tricks, including the Toroflux, Fibonacci sequence, torus knots, and more.

Transcript

Welcome to another Mathologer video. After the last insane Mathologer marathon in which Marty and I proved that e and pi are transcendental numbers I needed a bit of light relief. And maybe you do, too, right? So how to relax? Well we'll have fun with some of the most spectacular mathematical vanishing and materializing paradoxes, tricks and illusi... Read More

Key Insights

• ❓ The Toroflux toy demonstrates a vanishing paradox when collapsed due to the redistribution of length among the coils.
• 🌥️ The Fibonacci sequence can be used to create mathematical tricks involving the rearrangement of shapes and the creation of larger areas.
• #️⃣ Torus knots, represented by the Toroflux, have two numbers that count the looping around the torus and the central point, and these numbers are always relatively prime.
• 🔇 Mathematical vanishing and appearing tricks often involve the clever redistribution of length, area, or volume.
• #️⃣ The Toroflux paradox is unique because it involves a continuous transition between the number of coils, unlike other paradoxes that involve a discontinuous jump.

Questions & Answers

Q: How does the Toroflux demonstrate a vanishing paradox?

The Toroflux toy, when collapsed, appears to have one less coil than when it is expanded. This is due to the redistribution of length, where the excess length from each coil adds up to the extra coil counted.

Q: How does the Fibonacci sequence create a mathematical trick?

By choosing a Fibonacci number, a square can be cut and rearranged into a rectangle with a larger area. The sum of the two neighboring Fibonacci numbers determines the dimension of the cut segments.

Q: What are torus knots and how are they represented by the Toroflux?

Torus knots are closed loops that lie on the surface of a torus. The Toroflux is a torus knot toy that represents these knots. Each knot is characterized by two numbers that count the number of times the loop winds around the torus and the central point.

Q: How do torus knots relate to the sun-planet-moon model?

In the sun-planet-moon model, if the moon's orbit is at right angles to the planet's orbit, the overall orbit of the moon around the sun forms a torus knot. The numbers associated with the torus knot represent the number of orbits of the moon around the planet and the planet around the sun.

Summary & Key Takeaways

• The Toroflux is a toy that can be seen as a torus in shape and demonstrates a vanishing paradox when collapsed, with the number of coils decreasing by one.

• The Fibonacci sequence can be used to create a mathematical trick where a square can be rearranged into a rectangle with a larger area.

• Torus knots, represented by the Toroflux, have two associated numbers that count the number of times they loop around the torus and the number of times they loop around a central point.