“ These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?”
What makes you think lines and squares don’t exist in nature? And what on earth does that have to do with how long life took to emerge?!
Perfect lines and squares don’t exist as physical objects, sure, but geometry is less about material perfection than it’s about relationships. Nature constantly approximates geometric regularities because physics imposes them: energy minimization gives spheres, space-filling gives hexagons, angular momentum gives spirals.
Life didn’t need 4.5 billion years to “invent” geometry; geometry constrained life from the beginning. We only invented the formal language to describe it.
>> These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?
> And what on earth does that have to do with how long life took to emerge?!
I think you misunderstood that part you quoted. He's not claiming that it had a causative effect on how long life took to develop, he's claiming that it took 4.5 billion years after life first appeared for those geometrical concepts to emerge.
It's the usual "until we prove animals do _X_ we can safely assert only humans do _X_" trope of biology.
As we learn that animals do things like have homosexual relationships, giggle when tickled, and understand basic rules of economics... biologists are learning to phrase it as "until we prove animals do _X_ we cannot be sure if animals do _X_", which is much safer.
(Also, there are trillions of lines in nature - WTF? Squares are somewhat rarer, except on the ground in wombat territory...)
Rooks don't have taxicab geometry. Their metric space is compact even on an infinite board. I think you're thinking of the wazir: https://en.wikipedia.org/wiki/Wazir_(chess)
I would've also appreciated a discussion of how intuition of geometry might apply to chess playing abilities and how it might not be sufficient for playing chess well.
As a side note, I appreciated the small typos as a further signal that this was written by a human.
For anyone actually interested in the question of measuring animal intelligence, I recommend the book 'Are we smart enough to know how smart animals are?' by frans de waal.
(And if you care about measuring artificial intelligence, you should definitely care about what we've learned from trying to measure animal intelligence...)
I thought this article was going to be about how chess at its core is a game about intersecting lines or crosses (+ or x). Also, there are really interesting ideas that could be explored around why a rook on a bare board always controls the same number of squares no matter where it is placed, but for other pieces like the bishop or the knight, they control more squares the closer they are to the center of the board.
If you watch any Hikaru Nakamura content, you will see him draw "classic right angle triangle"s with three pieces, "classic wooden shield"s (a cross showing the scope of a centralized bishop), so he definitely uses some kind of geometry while playing.
Not sure if he just recognizes the shapes as they appear or tries to make them appear, would be nice if he came here to answer.
“ Humans are the only animals that we know that understand geometrical concepts. Things like lines and shapes (triangles, squares, circles etc.).”
False.
Crows for example understand geometry. I’d wager there are plenty more.
https://www.science.org/doi/10.1126/sciadv.adt3718
“ These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?”
What makes you think lines and squares don’t exist in nature? And what on earth does that have to do with how long life took to emerge?!
Perfect lines and squares don’t exist as physical objects, sure, but geometry is less about material perfection than it’s about relationships. Nature constantly approximates geometric regularities because physics imposes them: energy minimization gives spheres, space-filling gives hexagons, angular momentum gives spirals.
Life didn’t need 4.5 billion years to “invent” geometry; geometry constrained life from the beginning. We only invented the formal language to describe it.
>> These geometrical concepts do not exist in nature. There are no lines and squares. If it's obvious then why did it take 4.5 billion years since the development of life to emerge?
> And what on earth does that have to do with how long life took to emerge?!
I think you misunderstood that part you quoted. He's not claiming that it had a causative effect on how long life took to develop, he's claiming that it took 4.5 billion years after life first appeared for those geometrical concepts to emerge.
Ah, thanks. That makes sense. I still disagree that it took that long for these concepts to emerge. Perhaps words for these concepts.
https://youtu.be/EbzESiemPHs?si=4UNA7JGPt7OmfnOi&t=206
Here's Gromov, one of the greatest geometers of the last 50 years, discussing his viewpoint on this.
He always has these brilliant ond original perspectives on even the simplest concepts.
He also has this series of talks beginning with the question "What is probability, what is randomness?"
https://www.youtube.com/watch?v=aJAQVletzdY&list=PLx5f8IelFR...
Merely being able to differentiate a door from a wall, as dog does, takes an understanding of geometry.
I'd go even further and postulate that all intelligence is an understanding of geometry.
It doesn't. It takes an application of geometry. It does not require understanding.
You can use things without understanding them. See people asking chatgpt to do sth for them.
It's the usual "until we prove animals do _X_ we can safely assert only humans do _X_" trope of biology.
As we learn that animals do things like have homosexual relationships, giggle when tickled, and understand basic rules of economics... biologists are learning to phrase it as "until we prove animals do _X_ we cannot be sure if animals do _X_", which is much safer.
(Also, there are trillions of lines in nature - WTF? Squares are somewhat rarer, except on the ground in wombat territory...)
The title is "How geometry is fundamental for chess." but 60% of this article is about how animals don't have a sense for numbers or bad at geometry.
Only a couple brief mentions about how chess piece moves are lines and transforms of lines. Other than that the author never establishes the title.
I was actually looking for some insight about chess and did not get any.
I think the insight is that geometry on advanced level is fundamental for living creatures to create game of chess.
Chess geometry is not the same as physical geometry. See, e.g., https://en.wikipedia.org/wiki/R%C3%A9ti_endgame_study
Indeed, it's not even the same between pieces!
Kings have Chebysev geometry while Rooks have taxicab geometry: https://en.wikipedia.org/wiki/Taxicab_geometry#See_also
It's left as an exercise for the reader to figure out the geometry of the remaining pieces.
Rooks don't have taxicab geometry. Their metric space is compact even on an infinite board. I think you're thinking of the wazir: https://en.wikipedia.org/wiki/Wazir_(chess)
https://en.wikipedia.org/wiki/Chebyshev_distance for kings, but on a clear board the rook distance between any two squares is 1 or 2, whereas with Taxicab it could be as much as 14.
I found this article very interesting.
I would've also appreciated a discussion of how intuition of geometry might apply to chess playing abilities and how it might not be sufficient for playing chess well.
As a side note, I appreciated the small typos as a further signal that this was written by a human.
For anyone actually interested in the question of measuring animal intelligence, I recommend the book 'Are we smart enough to know how smart animals are?' by frans de waal.
https://www.goodreads.com/book/show/30231743-are-we-smart-en...
(And if you care about measuring artificial intelligence, you should definitely care about what we've learned from trying to measure animal intelligence...)
I thought this article was going to be about how chess at its core is a game about intersecting lines or crosses (+ or x). Also, there are really interesting ideas that could be explored around why a rook on a bare board always controls the same number of squares no matter where it is placed, but for other pieces like the bishop or the knight, they control more squares the closer they are to the center of the board.
Check out: https://en.wikipedia.org/wiki/Corresponding_squares
And the wiki page is a little limited but this subject can get complex but still very cool.
> Chimpanzees, instead of seeing 6 and 7, they feel 6ish-7ish.
I see what the author did there.
I've got a kid so "what the sigma" and "six seven" are a thing.
Type "six seven" in Google search and you should get the screen wobbling ; )
If you watch any Hikaru Nakamura content, you will see him draw "classic right angle triangle"s with three pieces, "classic wooden shield"s (a cross showing the scope of a centralized bishop), so he definitely uses some kind of geometry while playing.
Not sure if he just recognizes the shapes as they appear or tries to make them appear, would be nice if he came here to answer.
This is called “chunking” [0] — identifying grouped assortments of pieces as a single semantic unit - and has extensive research [1] behind it.
[0] https://www.chessprogramming.org/Chunking
[1] https://pmc.ncbi.nlm.nih.gov/articles/PMC4361603/
I was never particularly good at geometry.
I've beaten over 2500 ELO in Crazyhouse on Lichess (2518 to be exact). Currently rated around 1900.
Am I missing something?
Kinda disappointing article. Not much substance regarding the link between geometry and chess, as suggested by the title.
> Shapes are hypothesized to be formed by a programming language in the brain.
And what does this even mean? What does it mean for there to be a "programming language" in the brain?
Geometry is fundamental, period.
Someone call Bernard Parharm lmao.
[dead]