Ramsey numbers, Einstein tile, and arithmetic progressions in 2023

[jedishrfu]

TL;DR

Ramsey numbers, Einstein Tiles and 3 Arithmetic Progressions

Covering:
- New Ramsey number bounds
- Aperiodic tiling discovery of an Einstein tile
- Three Arithmetic Progressions

 

[helenbooth]

geometry dash subzero[/color]​

TL;DR Summary: Ramsey numbers, Einstein Tiles and 3 Arithmetic Progressions

Covering:
- New Ramsey number bounds
- Aperiodic tiling discovery of an Einstein tile
- Three Arithmetic Progressions

First time I watch this video. It is incredible that Paul Erdos had a hand in all these initial discoveries. What an incredible Mathematician and human.

 

[pinball1970]

geometry dash subzero​

First time I watch this video. It is incredible that Paul Erdos had a hand in all these initial discoveries. What an incredible Mathematician and human.

And a bit (a lot) of an eccentric!

 

[jedishrfu]

Erdos was a wandering mathematician/ He never got tenure and instead would travel around crashing at friends places and coming up with novel math ideas. Some of his famous sayings were:
- my brain is open
- a mathematician is a device for converting coffee into theorems
- another roof, another proof
- if numbers aren't beautiful, I don't know what is

https://www.azquotes.com/author/4538-Paul_Erdos

https://en.wikiquote.org/wiki/Paul_Erdős

He believed that God has a book of the most beautiful proofs. Some other mathematicians created a book of the best proofs in honor of Erdos:

https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

There are a couple of excellent biographies of Erdos:

- My Brain is Open by Bruce Schechter

https://www.amazon.com/MY-BRAIN-OPEN-Mathematical-Journeys/dp/0684859807?tag=pfamazon01-20

- The Man who Loved only Numbers by Paul Hoffman
https://www.amazon.com/Man-Who-Loved-Only-Numbers/dp/0786884061?tag=pfamazon01-20

and on NOVA:

 

[jedishrfu]

I just found a Scientific American article on Paul Erdos:

https://www.scientificamerican.com/...s-the-most-prolific-mathematician-in-history/

 

[Greg Bernhardt]

I can't explain any of the words in the OP

 

[jedishrfu]

Ramsey Numbers:

Ramsey numbers are a way of understanding how quickly disorder or randomness can appear in a system.

Imagine you have a group of folks at a party; some are friends, and some are strangers. Ramsey numbers help you figure out how many people need to be at the party to guarantee that either a certain number of them are all friends or a certain number are all strangers.

Aperiodic Tiling:

An amateur tileist, a computer scientist, and a mathematician go to a bar to show that the tile pattern the tileist discovered is truly aperiodic. His first attempt worked but used mirror images of the tile in the patterns. But some folks balked saying its two tile types and therefore can't be an Einstein tile. He reworked it and found a genuine Einstein tile ie a single tile that can be utilized to make tile patterns that have no repeating patterns.

Arithmetic Sequences:

Mathematicians wondered if you can describe a set of numbers without a three-term sequence of numbers. After finally finding one, they found many more. 1,2,3 or 1,3,5 or 1,2,3,5,8 (fibonacci) all have at least one sequence of three numbers.

QuantaMagazine has an article to go along with the video:

https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/

 

[Hornbein]

I like this.



One of the three breakthroughs was discovered by an amateur.