Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Counterintuitive Properties of High Dimensional Space

marckhoury.github.io

Post reply on HN
Print view View original

51–60 of 68 posts

Re: Counterintuitive Properties of High Dimensional Space

#51

Post by arketyp »

"Sparse Distributed Memory" is a lovely book by Pentti Kanerva in which he hypothesizes that high dimensional binary vectors (hypervectors) is key to understanding human memory and cognition. To a large part Kanerva is concerned with constructing the theory for a feasible implementation of hypervector storage and arithmetics in modern hardware. (While the book is some decades old the main strokes and ideas still hold relevant.) As a fun fact, and quite telling of what kind of book it is, Douglas Hofstadter has written the foreword.

The book is quite hard to get hold of, but here is a recent talk: https://www.youtube.com/watch?v=zUCoxhExe0o

Re: Counterintuitive Properties of High Dimensional Space

#52

Post by ilitirit »

dvh wrote: post #4

Just today on numberphile channel they showed how to pass circle through smaller square hole by bending it in higher (3rd) dimension: https://m.youtube.com/watch?v=AvFNCNOyZeE

There's a (sort of) intuitive way of understanding why things like these are possible without even seeing it visually.

How can you make two solid 2D squares occupy the space? They can't travel through each other, so...? Easy. Lift one of the squares into the 3rd Dimension, move it to the same X,Y coords, then drop it on the other (think of putting stacking 2 pieces of paper).

Similarly, you can have two 3D objects occupy the same space by "lifting" one into the 4th dimension, moving it to the same X,Y,Z coords as the other, and then "dropping it". You can "cheat" by thinking of the 4th dimension as time - one object travels through time, places itself at the same position of where the other object is, then returns. But as I said, that's "cheating".

Re: Counterintuitive Properties of High Dimensional Space

#53

Post by coinerone »

"You may be used to using the word “circle” in two dimensions and “sphere” in three dimensions. However, in higher dimensions we generally just use the word sphere, or d-sphere when the dimension of the sphere is not clear from context. With this terminology, a circle is also called a 1-sphere, for a 1-dimensional sphere. A standard sphere in three dimensions is called a 2-sphere, and so on. "

So, what is a 2-dimensional Sphere called if the 3-dimensional one is a 2-sphere?

Re: Counterintuitive Properties of High Dimensional Space

#54

Post by jgtrosh »

coinerone wrote: post #53

"You may be used to using the word “circle” in two dimensions and “sphere” in three dimensions. However, in higher dimensions we generally just use the word sphere, or d-sphere when the dimension of the sphere is not clear from context. With this terminology, a circle is also called a 1-sphere, for a 1-dimensional sphere. A standard sphere in three dimensions is called a 2-sphere, and so on. " So, what is a 2-dimensi…

I think it should be “in three-dimensional space”. The sphere is two dimensional.

Re: Counterintuitive Properties of High Dimensional Space

#55

Post by IngoBlechschmid »

I love Matt Parker's take on the fourth dimension: see for instance this one-hour long but very enjoyable talk for general audiences at the Royal Institution: https://www.youtube.com/watch?v=1wAaI_6b9JE (Things to See and Hear in the Fourth Dimension)

If you know German, then you might like this talk as well: https://www.youtube.com/watch?v=d19zmiVBLS8 (The curious world of four-dimensional geometry)

Re: Counterintuitive Properties of High Dimensional Space

#56

Post by OscarCunningham »

coinerone wrote: post #53

"You may be used to using the word “circle” in two dimensions and “sphere” in three dimensions. However, in higher dimensions we generally just use the word sphere, or d-sphere when the dimension of the sphere is not clear from context. With this terminology, a circle is also called a 1-sphere, for a 1-dimensional sphere. A standard sphere in three dimensions is called a 2-sphere, and so on. " So, what is a 2-dimensi…

The convention is to name things for their inherent dimensionalitly, not the space they live in.

A sphere in three dimensional space is a two dimensional object, in the sense that it's a surface on which the points can be described by two coordinates (say longitude and latitude).

Similarly a point on a circle can be described by a single coordinate (distance around the circle).

So the 1-sphere is a circle, and normally lives in 2 dimensional space, while the ordinary sphere is called a 2-sphere, and lives in three dimensional space.

Re: Counterintuitive Properties of High Dimensional Space

#57

Post by Sharlin »

resource0x wrote: post #25
ttoinou wrote: post #21

Earlier quoted context omitted.

That's what I thought reading that too. In general I don't think it's a good idea for maths people to just forget about units ("because it is for physics")

Speaking about math: I'm not even sure what the meaning of polynomial is. E.g. x^2+3x+1. Each component is measured in different units. Is it intuitive?

In abstract algebra, polynomials are just tuples (a_n, ... a_0) with addition (simple elementwise) and multiplication (more involved) defined. The +s and xs in the "x²+3x+1" notation are just syntax without semantic relevance; in particular x is not a variable and the substitution a ↦ P(a), a ∈ 𝗦 for any a and 𝗦 is not defined. This is because most interesting things about polynomials can be reasoned about without assuming anything about what x is.

Re: Counterintuitive Properties of High Dimensional Space

#58

Post by nwellnhof »

A simple observation that makes these properties a bit less counterintuitive is that the diameter of an n-dimensional unit cube (Euclidian distance between two opposite corners) is √n while the diameter of a unit sphere is always 1. So as the number of dimensions grows, the diameter of the unit cube can become arbitrarily large and the corners of the unit cube move further and further away from the unit sphere.

Re: Counterintuitive Properties of High Dimensional Space

#59

Post by algorias »

OscarCunningham wrote: post #56
coinerone wrote: post #53

"You may be used to using the word “circle” in two dimensions and “sphere” in three dimensions. However, in higher dimensions we generally just use the word sphere, or d-sphere when the dimension of the sphere is not clear from context. With this terminology, a circle is also called a 1-sphere, for a 1-dimensional sphere. A standard sphere in three dimensions is called a 2-sphere, and so on. " So, what is a 2-dimensi…

The convention is to name things for their inherent dimensionalitly, not the space they live in. A sphere in three dimensional space is a two dimensional object, in the sense that it's a surface on which the points can be described by two coordinates (say longitude and latitude). Similarly a point on a circle can be described by a single coordinate (distance around the circle). So the 1-sphere is a circle, and normal…

If you talk about the "volume of a sphere" (like the article does), then a sphere embedded in 3 dimensions should be 3 dimensional. This would usually be called a ball and not a sphere, though.

Note that analogously, a cube in 3 dimensional space is always considered to be 3 dimensional.

Re: Counterintuitive Properties of High Dimensional Space

#60

Post by saiya-jin »

ilitirit wrote: post #52
dvh wrote: post #4

Just today on numberphile channel they showed how to pass circle through smaller square hole by bending it in higher (3rd) dimension: https://m.youtube.com/watch?v=AvFNCNOyZeE

There's a (sort of) intuitive way of understanding why things like these are possible without even seeing it visually. How can you make two solid 2D squares occupy the space? They can't travel through each other, so...? Easy. Lift one of the squares into the 3rd Dimension, move it to the same X,Y coords, then drop it on the other (think of putting stacking 2 pieces of paper). Similarly, you can have two 3D objects oc…

I do similar mental exercise with higher dimensions, but for me with easier situation - escaping from certain area without crossing boundaries.

1 dimensional - an endless line with 2 spots marking boundaries to some subpart X. If you're inside, you need to enter 2nd dimension to escape that subpart X without crossing boundaries (and ie come back to original dimension).

2 dimensional - circle, you're inside, you need to enter 3rd dimension to escape without touching the circle.

3 dimensional - sphere, entering 4th dimension to escape without touching the sphere.

I can't go with my simpleton mind much further but it should scale indefinitely :)

Post reply on HN