Infinite Circles - Numberphile

TL;DR

Can an infinite sheet of paper be covered with circles that don't touch each other? While it is impossible in 2-dimensional space, it becomes possible in 3-dimensional space by using spheres.

Transcript

Today I'm going to tell you something about  circles, okay. So you know some things in math we   do because like we're trying to solve a problem, right, like we want to get a rocket ship to the   moon or whatever; and some things in math we do  because we've just been doodling on a piece of   paper and wondering what's going on. So this is  more in... Read More

Key Insights

• 👾 Covering an infinite sheet of paper with circles that don't touch each other is not possible in 2-dimensional space.
• 👾 In 3-dimensional space, spheres can be used to cover the entire space by removing two points from a sphere and dividing it into circles.
• 👾 The principles used for covering 3-dimensional space with spheres do not apply to higher dimensions like 4-dimensional space.

Questions & Answers

Q: Can an infinite sheet of paper be completely covered with circles that do not overlap or share any points?

No, it is not possible to cover an infinite sheet of paper with circles that don't touch each other. When circles do not overlap or share points, there will always be at least one point left uncovered.

Q: How can spheres be used to cover 3-dimensional space?

Spheres in 3-dimensional space can be used to cover the entire space by removing two points from the surface of the sphere and dividing it into circles. These circles, when repeated in all directions, can cover the entire 3-dimensional space.

Q: Can the same principles be applied to cover 4-dimensional space with spheres?

The principles used to cover 3-dimensional space with spheres do not directly apply to covering 4-dimensional space. The construction for circles in 3-dimensional space does not provide a guide for covering higher dimensions.

Q: What are the prime divisors of each element in the sequence of covering dimensions?

The prime divisors refer to the prime numbers that divide each element in the sequence of covering dimensions. The specific prime divisors of this sequence are not discussed in the content provided.

Summary & Key Takeaways

• In 2-dimensional space, it is impossible to cover an infinite sheet of paper with circles that don't touch each other.

• However, in 3-dimensional space, spheres can be used to cover the entire space by removing two points from a sphere and partitioning it into circles.

• The same principles do not apply when attempting to cover 5-dimensional space with 2-dimensional spheres.