Combinatorics and Higher Dimensions - Numberphile
video description
Date: 2022-04-08
Related videos
Comments and reviews: 9
education
This uses some lingo, but the connection is really nice. Ardila in the video is showing a connection between power sets and an abstract vector space.
Abstract vector spaces are essentially sets of labels for each axis/dimension and an -abstract number line- attached to each label. Usually we think of the axes as being indexed over the real line, but we can replace the reals with some other field (in the abstract algebra sense. A field is just some set of things where we can do normal arithmetic, i. e. addition, subtraction, multiplication and division. For example, if we take the 2D plane and replace R with modular arithmetic, we get a torus!
The cool thing is that linear algebra Just Works, even in these more abstract spaces. In particular, this video uses the field Z/2Z which is just the 1-bit numbers. There are technical caveats to be aware of, but this shows a connection between binary numbers and high-dimensional linear algebra! It turns out that cryptography, compression algorithms, etc. often make use of this fact.
reply
This uses some lingo, but the connection is really nice. Ardila in the video is showing a connection between power sets and an abstract vector space.
Abstract vector spaces are essentially sets of labels for each axis/dimension and an -abstract number line- attached to each label. Usually we think of the axes as being indexed over the real line, but we can replace the reals with some other field (in the abstract algebra sense. A field is just some set of things where we can do normal arithmetic, i. e. addition, subtraction, multiplication and division. For example, if we take the 2D plane and replace R with modular arithmetic, we get a torus!
The cool thing is that linear algebra Just Works, even in these more abstract spaces. In particular, this video uses the field Z/2Z which is just the 1-bit numbers. There are technical caveats to be aware of, but this shows a connection between binary numbers and high-dimensional linear algebra! It turns out that cryptography, compression algorithms, etc. often make use of this fact.
reply
John
It's easier to think about higher dimensions when you aren't trying to picture geometric shapes. For example, most people can easily imagine a person as multi-dimensional in the sense that a person has the dimensions of intelligence, athletics, empathy, sympathy, social interaction (one on one, social interaction (one against many, social interaction (several vs several. etc. Since it's impossible to see the complete structure of a 4+ dimensional object geometrically in our 3D world, this way of thinking about higher dimensions can be useful.
Think of a dimension as another way that something can be 'moved', such that it doesn't affect the other things that make it up. Just like with a person, they can have a dimension of intelligence that isn't necessarily affected by their dimension of social skill. So when you add dimensions past 3 geometrically, just know that what you are adding is not taking away from the previous things, but rather allowing more (unique) combinations.
reply
It's easier to think about higher dimensions when you aren't trying to picture geometric shapes. For example, most people can easily imagine a person as multi-dimensional in the sense that a person has the dimensions of intelligence, athletics, empathy, sympathy, social interaction (one on one, social interaction (one against many, social interaction (several vs several. etc. Since it's impossible to see the complete structure of a 4+ dimensional object geometrically in our 3D world, this way of thinking about higher dimensions can be useful.
Think of a dimension as another way that something can be 'moved', such that it doesn't affect the other things that make it up. Just like with a person, they can have a dimension of intelligence that isn't necessarily affected by their dimension of social skill. So when you add dimensions past 3 geometrically, just know that what you are adding is not taking away from the previous things, but rather allowing more (unique) combinations.
reply
LotRFam
There is a much simpler way of illustrating higher dimensional cuboids. I discovered it by accident when connecting lessons from my grade 11 physics class to my computer science class (I'm a highschool student in Canada. My friend decided to name my method after myself: Nigelian Projection. It can help with understanding higher dimensions at earlier ages, such as 10 years or even younger, plus gives a whole new way of viewing higher dimensions. It's very similar to the method described in this video, but much more structured.
Basically, you take the vertices of a square (this is a sped up version) and copy and paste them to the side, creating a line that divides their two regions. That creates a cube. Take the cube and do the same, but in a perpendicular direction to the first doubling. That's a 4D cube. And you continue this doubling method.
reply
There is a much simpler way of illustrating higher dimensional cuboids. I discovered it by accident when connecting lessons from my grade 11 physics class to my computer science class (I'm a highschool student in Canada. My friend decided to name my method after myself: Nigelian Projection. It can help with understanding higher dimensions at earlier ages, such as 10 years or even younger, plus gives a whole new way of viewing higher dimensions. It's very similar to the method described in this video, but much more structured.
Basically, you take the vertices of a square (this is a sped up version) and copy and paste them to the side, creating a line that divides their two regions. That creates a cube. Take the cube and do the same, but in a perpendicular direction to the first doubling. That's a 4D cube. And you continue this doubling method.
reply
Avana
I like your videos but I sometimes cannot watch them because I can't stand the sound of the sharpie on paper. It's like fingernails on a blackboard. Can't you write on a whiteboard? Such a waste of paper anyway, and you can't erase. I don't see the point of the brown paper and it makes me cringe to hear it written on. Or don't record the sound so close to the paper.
Also captions would be useful alternative so one could listen on mute, and it would also be more accessible for the hearing impaired.
reply
I like your videos but I sometimes cannot watch them because I can't stand the sound of the sharpie on paper. It's like fingernails on a blackboard. Can't you write on a whiteboard? Such a waste of paper anyway, and you can't erase. I don't see the point of the brown paper and it makes me cringe to hear it written on. Or don't record the sound so close to the paper.
Also captions would be useful alternative so one could listen on mute, and it would also be more accessible for the hearing impaired.
reply
1smallstep
-And He Built a Crooked House- by Robert Heinlein dealt with the topic of the 4th dimensional cube; called a tesseract. His description of the main character drawing one and making one out of toothpicks was great; I was able to reproduce them and sit them next to me as I read the story; then I spent hours exploring the implications of the object - yeah, I was an odd middle schooler. Great story; one that stuck in my mind the past 40 plus years.
reply
-And He Built a Crooked House- by Robert Heinlein dealt with the topic of the 4th dimensional cube; called a tesseract. His description of the main character drawing one and making one out of toothpicks was great; I was able to reproduce them and sit them next to me as I read the story; then I spent hours exploring the implications of the object - yeah, I was an odd middle schooler. Great story; one that stuck in my mind the past 40 plus years.
reply
clickaccept
I think -what is the benefit of looking geometrically at combinatorics- is a fair question. Its actually more like a stipulation though. If automorphisms of your combinatorial structure form a Coxeter group, then you can think about it geometrically. Conversely, the reason why we can say so many things about Coxeter groups, is because they are -geometric-.
reply
I think -what is the benefit of looking geometrically at combinatorics- is a fair question. Its actually more like a stipulation though. If automorphisms of your combinatorial structure form a Coxeter group, then you can think about it geometrically. Conversely, the reason why we can say so many things about Coxeter groups, is because they are -geometric-.
reply
TheTruthSentMe
I think for many people (me included) the cost of thinking in higher dimensions (geometrically) doesn't really merit the cause here. He didn't have an answer to what additional insight this view of combinatorics and subsets brings. In the end we get the same weird looking 2D projection of a 4D cube drawn on paper many of us have seen before.
reply
I think for many people (me included) the cost of thinking in higher dimensions (geometrically) doesn't really merit the cause here. He didn't have an answer to what additional insight this view of combinatorics and subsets brings. In the end we get the same weird looking 2D projection of a 4D cube drawn on paper many of us have seen before.
reply
D.
This really helped me finally grasp going into higher dimensions. Too often I've had mathematicians just show me those damned 3-D -shadow- polytopes or whatever without explaining to me a simple shorthand like you've got here: every time you are looking at sets and have to go to another dimension, you double the number of points!
reply
This really helped me finally grasp going into higher dimensions. Too often I've had mathematicians just show me those damned 3-D -shadow- polytopes or whatever without explaining to me a simple shorthand like you've got here: every time you are looking at sets and have to go to another dimension, you double the number of points!
reply
GEM4sta
I'm confused with how this is better than a tree. Wouldn't a tree be strictly better at showing this? To create the next layer of the tree, you always use the answer from the previous node, and the answer from the previous node with the new element, so it's easy to see adding a new element doubles the width of the row.
reply
I'm confused with how this is better than a tree. Wouldn't a tree be strictly better at showing this? To create the next layer of the tree, you always use the answer from the previous node, and the answer from the previous node with the new element, so it's easy to see adding a new element doubles the width of the row.
reply
Add a review, comment
Other channel videos
