The cube shadow theorem (pt.1): Prince Rupert's paradox

TL;DR

Discover recently discovered mind-boggling facts about cubes, including the shadow theorem and the possibility of passing a cube through itself.

Transcript

Welcome to a different kind of Mathologer video. All my students at uni know that I have a bit of a cube fetish-- mathematically cubes, Rubik's cubes, anything goes. What I want to do today is introduce you to some very recently discovered mind-boggling facts about cubes that hardly anybody knows about, not even mathematicians. Alright, so I've got... Read More

Key Insights

• 🧊 Cubes can have shapes other than a traditional six-sided cube, such as the Skewb's rhombic dodecahedron shape.
• 🧊 The area of a cube's shadow and the height difference of the cube are equal when dealing with a unit cube.
• 🧊 The shadow theorem simplifies calculations for determining the smallest and largest possible shadows of a cube.
• ❎ It is possible to pass a cube through itself by drilling a hole with a square cross-section.
• 🧊 The concept of passing a cube through itself was discovered by Prince Rupert in the 17th century.
• 😎 The Prince Rupert's drop, a glass tadpole with unique properties, is unrelated to the concept of passing a cube through itself.
• 🥳 The video is divided into two parts, with part 1 introducing basic facts and part 2 delving deeper into the mathematics.

Questions & Answers

Q: What is the shape of a Skewb?

A Skewb is a twisty puzzle that is cube-shaped but not a cube itself. It has a shape known as a rhombic dodecahedron, consisting of twelve rhombuses.

Q: How does the shadow theorem work?

When dealing with a unit cube, the area of the cube's shadow and the height difference of the cube will always be equal, regardless of its orientation or the shape of the shadow.

Q: What practical applications does the shadow theorem have?

The shadow theorem allows us to determine the smallest and largest possible shadows of a cube by finding the smallest and largest possible height differences.

Q: How is it possible to pass a cube through itself?

By drilling a square hole through the cube, it becomes possible to pass another cube of the same size through the hole. This can be repeated with progressively larger holes and cubes.

Summary & Key Takeaways

• The video introduces some lesser-known facts about cubes, such as the shape of a Skewb and the 2D shadow of a 666-dimensional cube.

• The shadow theorem states that the area of a shadow and the height difference of a cube will be equal when the cube is a unit cube.

• These facts have practical applications, such as determining the smallest and largest possible shadows of a cube.