8 ms·

A mathematician's guided tour through higher dimensions

188 points by Anon84 5y ago

lordnacho 5y ago
I saw a clip from the Big Bang Theory the other day, where it was mentioned that there are no knots in 4 dimensions and higher. I wondered why this is so and found this elegant "proof": https://abel.math.harvard.edu/archive/21a_spring_06/exhibits/unknotting/index.html https://abel.math.harvard.edu/archive/21a_spring_06/exhibits...
- a_ellis 5y ago
  nice proof! usually, "higher dimensional knot" refers to embeddings of n-dim spheres into (n+2)-dimensional spheres (or R^(n+2)). (if the distinction between R^(n+2) and (n+2)-spheres scares you, don't worry about it! it's just one point!) usual knot theory: n = 1, m = 3 OP's proof relates to: n = 1, m = 4 when m - n (the "codimension") is >2, as in the the case from OP's post, there is "so much room" that unknotting can always happen. and at codimension 1, there "isn't enough room". so the interesting theory is codim-2. in fact, there is a well studied theory. here's a book on the subject (disclaimer: I haven't read it): https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf
- kmill 5y ago
  That's how I like to explain it to people, though there's a small caveat that doesn't really affect the argument, but it's worth considering. The only times a knot-with-hues actually corresponds to a knot in Euclidean 4-dimensional space are when you can smoothly modify just the colors to make the knot be monochromatic. For example, if the knot goes through the whole color wheel of hues, 0 to 360 degrees, then that corresponds to a knot in a different space (R^3 x S^1). (A way to avoid this problem is to not use the color wheel, but instead use, say, wavelength of a spectral color.) Although there are no nontrivial circle knots (S^1 knots) in R^4, there are nontrivial sphere knots (S^2 knots). That well-advertised Quanta article about Lisa Piccirillo's work is about this sort of thing.
- CorrectHorseBat 5y ago
  so knots only exist when there are exactly 3 dimensions? That's interesting.
  - mathgenius 5y ago
    The fourth dimension is enough to untangle a knot made out of a one dimensional space (a loop of string). But you can make two dimensional knots in four dimensional space: this is a surface that is knotted with itself in four dimensions. And probably the pattern extends to higher dimensions.
- lupire 5y ago
  Even simpler, just take a 3D knot, and pull on it. Where it gets stuck, just lift a strand up in the 4th dimension.
- yissp 5y ago
  Could this be a reason why space happens to be 3 dimensional? Like, I don't know, maybe knots are somehow important to the structure of the universe and it couldn't exist with some other dimensionality? Whenever these "magic numbers" appear in math it always makes me curious.
  - wizzwizz4 5y ago
    I read that three-dimensional space is the only one where gravitational orbits are possible, because inverse-cube / inverse-linear etc. doesn't have orbits stable to minor perturbations.
- red_trumpet 5y ago
  From you link: > You would not be able to tie a shoe in four dimensional space. I'm not convinced. The thing is that if you tie your shoes with bows at the end, those bows are knot-theoretically trivial, even in 3d-space. Still a nice proof!
scythmic_waves 5y ago
Nice article! I thought they did a great job building up to explaining Hausdorff dimension and the Koch curve. I do wish they'd done a better job discussing time as the 4th dimension, however. It seemed shoehorned in at the end and wasn't really connected to the rest of the writing.
cameronperot 5y ago
Great read. For those who enjoyed reading this, you might also enjoy this [1] short video series on dimensions math (direct link to the YouTube playlist [2]). The videos encompass some of the history of the mathematics, along with a number of animations to help the viewer get an idea how one can visualize a higher dimensional object in a lower dimensional space. The creators also have a series on chaos math [3]. [1] http://www.dimensions-math.org/ http://www.dimensions-math.org/ [2] https://www.youtube.com/watch?v=6cpTEPT5i0A&list=PL3C690048E1531DC7 https://www.youtube.com/watch?v=6cpTEPT5i0A&list=PL3C690048E... [3] https://www.youtube.com/watch?v=vts0YHACsYY&list=PLw2BeOjATqruoac7tS6Clnn-mpxlRkXfV https://www.youtube.com/watch?v=vts0YHACsYY&list=PLw2BeOjATq...
jstx1 5y ago
I think it's easier to approach higher dimensions without talking about spatial dimensions at all. For example - Darts in Higher Dimensions, 3blue1brown and Numberphile - https://www.youtube.com/watch?v=6_yU9eJ0NxA https://www.youtube.com/watch?v=6_yU9eJ0NxA Or even more trivially, you can think of a table where every row is some entity and every column is some attribute associated to it. For example, make a spreadsheet where each row is a person and the columns are age, height, weight, salary, and years to retirement - then you can think of each person as a point in 5-d space. And some properties are intuitively obvious - for example as you keep adding more columns it becomes more difficult to find people who are similar to each other. It's a pretty accessible way to introduce high dimensions without talking about tessaracts.
- ziddoap 5y ago
  Any video with 3blue1brown is worth a watch, in my opinion. I also quite enjoy Numberphile, but Grant (3b1b) has such a fantastic way of introducing and teaching topics. Viewers with even the slightest interest in math, and are not familiar with 3blue1brown, should check out some of his other videos at [1]. Not only is he a great orator, but the visuals he provides have really clarified some of the tougher subjects for me. [1]https://www.youtube.com/c/3blue1brown https://www.youtube.com/c/3blue1brown
  - eyeundersand 5y ago
    I third this recommendation. Have found his expositions to be more comprehensible (and often more intuitive) than most professors’. His visual style of presentation also helps me a lot!
    - dylan604 5y ago
      I really am jealous of students today that have access to supplemental study material like this. Learning from an instructors scribblings on a chalkboard or a single image in a textbook is nothing compared to these types of videos. There was a series on PBS (late 80s/early 90s. If I had to guess, it was a WGBH production) on physics that really helped my understanding of topics that might have been a little fuzzy from classroom lectures and textbook readings. However, even as well as that series was produced, it moved around a lot and was only so many episodes. Today, there's not just one person, but multiple people making videos like this with so many more videos covering so much more ground. You get the advantage of hearing the same thing from different voices that might say it in just the right way that makes it click for you.
noora2000 5y ago
I love Quanta, but recently I encountered a few pieces that were not up to their usual standards. This is one example, and the other example I currently have in mind is also by David S. Richeson - so maybe it’s just him. In the article linked above, there are some glaring omissions (A conceptual overview of the notion of “dimension” that mentions neither the Krull dimension nor matroids? An emphasis on high-dimensionality while ignoring concentration of measure?).
anderson1993 5y ago
Does anyone know what theorems/definitions this paragraph is referring to? "Finally, in 1912, almost half a century after Cantor’s discovery, and after many failed attempts to prove the invariance of dimension, L.E.J. Brouwer succeeded by employing some methods of his own creation. In essence, he proved that it is impossible to put a higher-dimensional object inside one of smaller dimension, or to place one of smaller dimension into one of larger dimension and fill the entire space, without breaking the object into many pieces, as Cantor did, or allowing it to intersect itself, as Peano did."
- jchallis 5y ago
  The Jordan-Brouwer Separation Theorem - which rigorously defines an inside and outside for higher dimensional objects. http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Schmaltz.pdf http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/...
ogogmad 5y ago
Facts about rotations in high dimensions: See this as an introduction to Clifford algebra. Notice that in 4D space, it's possible to have two planes which meet at only one point, and for which every vector on one of the planes is perpendicular to every vector on the other. This implies that for each rotation in n dimensions, it is possible to pick floor(n/2) mutually perpendicular planes which are each invariant under the rotation. This can be proved using eigendecomposition. These sets of floor(n/2) invariant planes, weighed by their angles of rotation, form the "bivectors" in exterior and Clifford algebra. ([EDIT] It's slightly more accurate to say that bivectors are the angular velocities in n dimensions, which means that the angular speeds attached to each plane are not necessarily between [0,2pi] but can be any real.) Also, notice that in even dimensions there is a rotation which sends every vector to a vector perpendicular to it. But in odd dimensions, there isn't even a continuous function which sends every vector to a vector perpendicular to it; this follows from the hairy-ball theorem. However, notice that there is still an algorithm for finding perpendicular vectors in any number of dimensions; one such algorithm is an application of the Gram-Schmidt process (also called QR decomposition).
state_less 5y ago
I’ve worked with higher dimensioned data before with datasets, and have a somewhat intuitive feel for the shape of time from regularly timed events. One can observe a pendulum clock returns to similar spatial coordinates or daily rituals like morning meetings where humans flow along temporal coordinates and then flow back out again, seemingly compelled by time as much as happenstance. If you abstract away a ton of detail, you could almost say you travel back in time each workday. I’ve felt caught in a behavioral loop many a time. The non-integer dimensions are interesting on fractals. How many copies do you get for a given amount of recursion? Neat way to think about dimension.
tenaciousDaniel 5y ago
Every time I try to understand the 4th+ dimension, my brain just completely breaks and I'm left feeling angry. I'm a highly visual thinker so it's difficult for me to grasp concepts like that. Someone once told me that in the same way a 3D object casts a 2D shadow, a 4D object casts a 3D shadow. I just...can't. I can't wrap my head around that no matter how hard I try.
- deleted 5y ago
  [deleted]
- throwawayndim 5y ago
  Here is a much less tricky version: A car’s dimensions can include its length, height, and width, but also the amount of gas in the tank (4th), the position of the throttle pedal (5th) and brake pedal (6th), its weight (7th), the number of passengers (8th), etc etc.
  - deleted 5y ago
    [deleted]
  - wibagusto 5y ago
    I don’t think that works because the N-1 dimension is contained in the N dimension. But all those things described are disjoint. Now you might be able to say, “the mass of all objects in the universe” but is it a dimension? I dunno I’ll leave that to the topologists! Edit: I think you have to have the similar type of units to increase the dimensions. Like the article talks about N dimensional space (e.g. point, line, plane, etc). To consider what mass of an object would be you’d have 1 dimensional mass; 2D mass; …; ND mass whatever that would be.
    - md224 5y ago
      > I don’t think that works because the N-1 dimension is contained in the N dimension. I'm not sure that's right. Each dimension is linearly independent [0] from the others, which means e.g. you can't add up a bunch of width and get height, or add up a bunch of length and get width. So in an important sense, they're not contained within each other. You might be thinking of how a 2D plane contains the first dimension within it, but that's not the 2nd dimension... that's two-dimensional (a combination of two dimensions). [0] https://en.wikipedia.org/wiki/Linear_independence https://en.wikipedia.org/wiki/Linear_independence
hi41 5y ago
Just yesterday, I watched this video by Neil Degrasse Tyson about the fourth dimension. And now we have this article. What a coincidence! https://youtu.be/TapAplvGtQY https://youtu.be/TapAplvGtQY In the video Neil explains why higher dimensions may explain the queer ways of the quantum world.
jellyksong 5y ago
There's an amazing game, 4D Toys [1], that lets you interactively play with 4D objects in a 3D environment. It's great on iPad. I believe the developer had to create a custom physics engine to support 4D space [2]. [1] https://4dtoys.com/ https://4dtoys.com/ [2] https://marctenbosch.com/news/2017/06/4d-toys-a-box-of-four-dimensional-toys/ https://marctenbosch.com/news/2017/06/4d-toys-a-box-of-four-...
excalibur 5y ago
> As n grows, so does the size of the central sphere — it has a radius of [Math Processing Error] − 1. I love it lol

188 points by Anon84 5y ago

I saw a clip from the Big Bang Theory the other day, where it was mentioned that there are no knots in 4 dimensions and higher. I wondered why this is so and found this elegant "proof": https://abel.math.harvard.edu/archive/21a_spring_06/exhibits/unknotting/index.html https://abel.math.harvard.edu/archive/21a_spring_06/exhibits...

a_ellis 5y ago
nice proof! usually, "higher dimensional knot" refers to embeddings of n-dim spheres into (n+2)-dimensional spheres (or R^(n+2)). (if the distinction between R^(n+2) and (n+2)-spheres scares you, don't worry about it! it's just one point!) usual knot theory: n = 1, m = 3 OP's proof relates to: n = 1, m = 4 when m - n (the "codimension") is >2, as in the the case from OP's post, there is "so much room" that unknotting can always happen. and at codimension 1, there "isn't enough room". so the interesting theory is codim-2. in fact, there is a well studied theory. here's a book on the subject (disclaimer: I haven't read it): https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf
kmill 5y ago
That's how I like to explain it to people, though there's a small caveat that doesn't really affect the argument, but it's worth considering. The only times a knot-with-hues actually corresponds to a knot in Euclidean 4-dimensional space are when you can smoothly modify just the colors to make the knot be monochromatic. For example, if the knot goes through the whole color wheel of hues, 0 to 360 degrees, then that corresponds to a knot in a different space (R^3 x S^1). (A way to avoid this problem is to not use the color wheel, but instead use, say, wavelength of a spectral color.) Although there are no nontrivial circle knots (S^1 knots) in R^4, there are nontrivial sphere knots (S^2 knots). That well-advertised Quanta article about Lisa Piccirillo's work is about this sort of thing.
CorrectHorseBat 5y ago
so knots only exist when there are exactly 3 dimensions? That's interesting.
- mathgenius 5y ago
  The fourth dimension is enough to untangle a knot made out of a one dimensional space (a loop of string). But you can make two dimensional knots in four dimensional space: this is a surface that is knotted with itself in four dimensions. And probably the pattern extends to higher dimensions.
lupire 5y ago
Even simpler, just take a 3D knot, and pull on it. Where it gets stuck, just lift a strand up in the 4th dimension.
yissp 5y ago
Could this be a reason why space happens to be 3 dimensional? Like, I don't know, maybe knots are somehow important to the structure of the universe and it couldn't exist with some other dimensionality? Whenever these "magic numbers" appear in math it always makes me curious.
- wizzwizz4 5y ago
  I read that three-dimensional space is the only one where gravitational orbits are possible, because inverse-cube / inverse-linear etc. doesn't have orbits stable to minor perturbations.
red_trumpet 5y ago
From you link: > You would not be able to tie a shoe in four dimensional space. I'm not convinced. The thing is that if you tie your shoes with bows at the end, those bows are knot-theoretically trivial, even in 3d-space. Still a nice proof!

scythmic_waves 5y ago

Nice article! I thought they did a great job building up to explaining Hausdorff dimension and the Koch curve. I do wish they'd done a better job discussing time as the 4th dimension, however. It seemed shoehorned in at the end and wasn't really connected to the rest of the writing.

cameronperot 5y ago

Great read. For those who enjoyed reading this, you might also enjoy this [1] short video series on dimensions math (direct link to the YouTube playlist [2]). The videos encompass some of the history of the mathematics, along with a number of animations to help the viewer get an idea how one can visualize a higher dimensional object in a lower dimensional space. The creators also have a series on chaos math [3]. [1] http://www.dimensions-math.org/ http://www.dimensions-math.org/ [2] https://www.youtube.com/watch?v=6cpTEPT5i0A&list=PL3C690048E1531DC7 https://www.youtube.com/watch?v=6cpTEPT5i0A&list=PL3C690048E... [3] https://www.youtube.com/watch?v=vts0YHACsYY&list=PLw2BeOjATqruoac7tS6Clnn-mpxlRkXfV https://www.youtube.com/watch?v=vts0YHACsYY&list=PLw2BeOjATq...

jstx1 5y ago

I think it's easier to approach higher dimensions without talking about spatial dimensions at all. For example - Darts in Higher Dimensions, 3blue1brown and Numberphile - https://www.youtube.com/watch?v=6_yU9eJ0NxA https://www.youtube.com/watch?v=6_yU9eJ0NxA Or even more trivially, you can think of a table where every row is some entity and every column is some attribute associated to it. For example, make a spreadsheet where each row is a person and the columns are age, height, weight, salary, and years to retirement - then you can think of each person as a point in 5-d space. And some properties are intuitively obvious - for example as you keep adding more columns it becomes more difficult to find people who are similar to each other. It's a pretty accessible way to introduce high dimensions without talking about tessaracts.

ziddoap 5y ago
Any video with 3blue1brown is worth a watch, in my opinion. I also quite enjoy Numberphile, but Grant (3b1b) has such a fantastic way of introducing and teaching topics. Viewers with even the slightest interest in math, and are not familiar with 3blue1brown, should check out some of his other videos at [1]. Not only is he a great orator, but the visuals he provides have really clarified some of the tougher subjects for me. [1]https://www.youtube.com/c/3blue1brown https://www.youtube.com/c/3blue1brown
- eyeundersand 5y ago
  I third this recommendation. Have found his expositions to be more comprehensible (and often more intuitive) than most professors’. His visual style of presentation also helps me a lot!
  - dylan604 5y ago
    I really am jealous of students today that have access to supplemental study material like this. Learning from an instructors scribblings on a chalkboard or a single image in a textbook is nothing compared to these types of videos. There was a series on PBS (late 80s/early 90s. If I had to guess, it was a WGBH production) on physics that really helped my understanding of topics that might have been a little fuzzy from classroom lectures and textbook readings. However, even as well as that series was produced, it moved around a lot and was only so many episodes. Today, there's not just one person, but multiple people making videos like this with so many more videos covering so much more ground. You get the advantage of hearing the same thing from different voices that might say it in just the right way that makes it click for you.

noora2000 5y ago

I love Quanta, but recently I encountered a few pieces that were not up to their usual standards. This is one example, and the other example I currently have in mind is also by David S. Richeson - so maybe it’s just him. In the article linked above, there are some glaring omissions (A conceptual overview of the notion of “dimension” that mentions neither the Krull dimension nor matroids? An emphasis on high-dimensionality while ignoring concentration of measure?).

anderson1993 5y ago

Does anyone know what theorems/definitions this paragraph is referring to? "Finally, in 1912, almost half a century after Cantor’s discovery, and after many failed attempts to prove the invariance of dimension, L.E.J. Brouwer succeeded by employing some methods of his own creation. In essence, he proved that it is impossible to put a higher-dimensional object inside one of smaller dimension, or to place one of smaller dimension into one of larger dimension and fill the entire space, without breaking the object into many pieces, as Cantor did, or allowing it to intersect itself, as Peano did."

jchallis 5y ago
The Jordan-Brouwer Separation Theorem - which rigorously defines an inside and outside for higher dimensional objects. http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Schmaltz.pdf http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/...

ogogmad 5y ago

Facts about rotations in high dimensions: See this as an introduction to Clifford algebra. Notice that in 4D space, it's possible to have two planes which meet at only one point, and for which every vector on one of the planes is perpendicular to every vector on the other. This implies that for each rotation in n dimensions, it is possible to pick floor(n/2) mutually perpendicular planes which are each invariant under the rotation. This can be proved using eigendecomposition. These sets of floor(n/2) invariant planes, weighed by their angles of rotation, form the "bivectors" in exterior and Clifford algebra. ([EDIT] It's slightly more accurate to say that bivectors are the angular velocities in n dimensions, which means that the angular speeds attached to each plane are not necessarily between [0,2pi] but can be any real.) Also, notice that in even dimensions there is a rotation which sends every vector to a vector perpendicular to it. But in odd dimensions, there isn't even a continuous function which sends every vector to a vector perpendicular to it; this follows from the hairy-ball theorem. However, notice that there is still an algorithm for finding perpendicular vectors in any number of dimensions; one such algorithm is an application of the Gram-Schmidt process (also called QR decomposition).

state_less 5y ago

I’ve worked with higher dimensioned data before with datasets, and have a somewhat intuitive feel for the shape of time from regularly timed events. One can observe a pendulum clock returns to similar spatial coordinates or daily rituals like morning meetings where humans flow along temporal coordinates and then flow back out again, seemingly compelled by time as much as happenstance. If you abstract away a ton of detail, you could almost say you travel back in time each workday. I’ve felt caught in a behavioral loop many a time. The non-integer dimensions are interesting on fractals. How many copies do you get for a given amount of recursion? Neat way to think about dimension.

tenaciousDaniel 5y ago

Every time I try to understand the 4th+ dimension, my brain just completely breaks and I'm left feeling angry. I'm a highly visual thinker so it's difficult for me to grasp concepts like that. Someone once told me that in the same way a 3D object casts a 2D shadow, a 4D object casts a 3D shadow. I just...can't. I can't wrap my head around that no matter how hard I try.

deleted 5y ago
[deleted]
throwawayndim 5y ago
Here is a much less tricky version: A car’s dimensions can include its length, height, and width, but also the amount of gas in the tank (4th), the position of the throttle pedal (5th) and brake pedal (6th), its weight (7th), the number of passengers (8th), etc etc.
- deleted 5y ago
  [deleted]
- wibagusto 5y ago
  I don’t think that works because the N-1 dimension is contained in the N dimension. But all those things described are disjoint. Now you might be able to say, “the mass of all objects in the universe” but is it a dimension? I dunno I’ll leave that to the topologists! Edit: I think you have to have the similar type of units to increase the dimensions. Like the article talks about N dimensional space (e.g. point, line, plane, etc). To consider what mass of an object would be you’d have 1 dimensional mass; 2D mass; …; ND mass whatever that would be.
  - md224 5y ago
    > I don’t think that works because the N-1 dimension is contained in the N dimension. I'm not sure that's right. Each dimension is linearly independent [0] from the others, which means e.g. you can't add up a bunch of width and get height, or add up a bunch of length and get width. So in an important sense, they're not contained within each other. You might be thinking of how a 2D plane contains the first dimension within it, but that's not the 2nd dimension... that's two-dimensional (a combination of two dimensions). [0] https://en.wikipedia.org/wiki/Linear_independence https://en.wikipedia.org/wiki/Linear_independence

hi41 5y ago

Just yesterday, I watched this video by Neil Degrasse Tyson about the fourth dimension. And now we have this article. What a coincidence! https://youtu.be/TapAplvGtQY https://youtu.be/TapAplvGtQY In the video Neil explains why higher dimensions may explain the queer ways of the quantum world.

jellyksong 5y ago

There's an amazing game, 4D Toys [1], that lets you interactively play with 4D objects in a 3D environment. It's great on iPad. I believe the developer had to create a custom physics engine to support 4D space [2]. [1] https://4dtoys.com/ https://4dtoys.com/ [2] https://marctenbosch.com/news/2017/06/4d-toys-a-box-of-four-dimensional-toys/ https://marctenbosch.com/news/2017/06/4d-toys-a-box-of-four-...

excalibur 5y ago

> As n grows, so does the size of the central sphere — it has a radius of [Math Processing Error] − 1. I love it lol