Understanding Heesch Numbers and Tiling Problems

The video discusses tiling concepts, focusing on Heesch numbers, which indicate how many times a shape can be surrounded by copies of itself before it can't continue. Examples include shapes with Heesch numbers from zero to five, with the latter being the current record. The video also touches on the Einstein problem, which seeks a single tile that can tile a plane non-periodically. Joan Taylor's work on this problem is highlighted. The video concludes with a sponsorship message from Audible, recommending the book 'Dead Mountain'.

What is an unbalanced tile?

• A tile that can only be used once

• A tile that has equal positive and negative arcs

• A tile that cannot be used to tile a plane indefinitely

• A tile that can be flipped over

What is the Heesch number of a circle?

• Three

• One

• Zero

• Infinity

Which shape has a Heesch number of infinity?

• Hexagon

• Triangle

• Square

• Circle

What is the highest known Heesch number for a tile?

• Ten

• Five

• Three

• Two

Who discovered the tile with the highest known Heesch number?

• Joan Taylor

• Casey Mann

• Donnie Eichar

• Brady Haran

What is the Einstein problem in tiling?

• Finding a tile that can tile a plane periodically

• Finding a single tile that can tile a plane non-periodically

• Finding a tile with a Heesch number of zero

• Finding a tile that can tile a plane with two shapes

What was unique about Joan Taylor's solution to the Einstein problem?

• It was a periodic tiling

• It was a connected tile

• It was a single shape with two pieces

• It used two different shapes

What is the main topic of the book 'Dead Mountain'?

• A biography of a mathematician

• A guide to mountaineering

• A historical mystery about hikers

• A mathematical tiling problem

What benefit does Audible offer if you dislike a book?

• You can exchange it for another book

• You get a refund

• You receive a discount on your next purchase

• You can keep it for free

How many Audible originals can you choose each month with a membership?

• Four

• Three

• Two

• One