A donut is not a sphere | Things you can do on one surface but not the other

The video explores the topological differences between a torus and a sphere, using the game Asteroids as an example. It explains why the universe in Asteroids is a torus, not a sphere, by examining movement patterns. The video also highlights unique properties of a torus, such as non-contractible loops and vector fields, and discusses graph theory problems that can be solved on a torus but not on a sphere.

What topological shape does the universe in the Asteroids game resemble?

• A sphere

• A torus

• A cube

• A cylinder

Why can't the Asteroids game universe be topologically a sphere?

• Because it is flat

• Because it has no edges

• Because it is a 3D shape

• Because paths don't always return to the starting point

What is a unique property of loops on a sphere?

• They can be infinitely long

• They can form knots

• They can intersect at multiple points

• They can be contracted to a point

What does the hairy ball theorem state about vector fields on a sphere?

• Vectors can only exist at the poles

• All vectors must point in the same direction

• There must be at least one zero vector

• There can be no zero vectors

How does the three utilities problem demonstrate the difference between a torus and a sphere?

• It shows that a sphere can hold more nodes

• It shows that a sphere has more surface area

• It shows that a torus can have more edges

• It shows that a torus allows non-intersecting connections

What is the topological equivalence of a torus?

• A sphere

• A cube

• A pyramid

• A coffee mug

Which graph is impossible to draw on a plane or sphere without intersection but possible on a torus?

• K6

• K3

• K4

• K5