A mathematician's guided tour through higher dimensions | Hacker News Reader

A mathematician's guided tour through higher dimensions

quantamagazine.org

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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

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.

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

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!

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.

I wonder why he hasn't made many videos recently. Only 4 videos in the past year.

Yes. Obviously higher dimensions in physics require a different kind of intuition, but data folks deal with multidimensional tabular data all the time without ever seeing the underlying structure. Seeking a spatial explanation often hinders rather than helps.

Instead, there’s this notion of a “theory of coordinatized data” [1] where one understands that dimensions (doesn’t matter if they are continuous, discrete, categorical) are essentially coordinates for values. This is a powerful way of thinking about tidy multidimensional tabular data.

Once you realize dimensions are coordinates, a certain mathematical intuition emerges. For instance, most people have a hard time understanding pivot/unpivot operations. But they really are analogous to matrix transposes, but instead on a row/col axis, they rotate on the “coordinate” dimensions which are invariants.

Once somehow understands this, their understanding of SQL and Tableau and of data frames becomes a lot deeper. Aggregations and filtering and window operations take on a new meaning.

[1] https://winvector.github.io/FluidData/RowsAndColumns.html

Sure, encoding an extra dimension in a vector is just an additional element, but for the exception of categorical data this view is very restrictive. If you want to do things like describe embedded-space and projective spaces you can't just add a term to your formulas and expect everything to work. Like an ant walking on a ball in your room on earth in spacetime projected on your computer screen.

In geometric algebra there is a way to encode every element and transformation in such space and those correspond to shuffling around terms in an equation.

The way I think about higher dimensions is just by looking what a sphere of radius r looks like in cartesian coordinates

x^2 + y^2 = r^2 in 2D

x^2 + y^2 + z^2 = r^2 in 3D

x^2 + y^2 + z^2 + t^2 = r^2 in 4D

If that leads to some weird behaviors (spheres are very 'spike-y') then so be it, I don't understand why intuition from 3D is important

Things gets 'weirder' in higher dim manifolds but not really, it's only hard if you want to 'see' it in 3d Euclidean

What do you mean by "spike-y"? That's not how I think of higher dimensional spheres at all.

Oh they are very spike-y, well, my point in the above post is to just solve the eq but easier 'visualization' would be [0]

By the way, this is a similar phenomena to the 'curse of dimensionality' [1]

[0] https://www.youtube.com/watch?v=mceaM2_zQd8

[1] https://en.wikipedia.org/wiki/Curse_of_dimensionality

Most of the volume is near the edge of the sphere in higher dimensions. Closer to soap bubbles than what we consider to be true spheres.

Dimension 2, put 4 circles radius 1 at (1, 1), (-1, 1), (1, -1), (-1, -1). In the centre, put the largest circle you can. Then from the outside, you can't "touch" the inner circle.

Dimension 3, put 8 spheres radius 1 at (1, 1, 1), ... Then from the outside you can touch the inner sphere (and it's a bit bigger).

Once you get to a certain dimension (10 IIRC), the inner sphere is no longer in the convex hull of the outer spheres, it is "poking out" of the arrangement.

Spiky like that.

How does that help you?

It helps me in the sense that if some object is moving (I can artificially make it move for the sake of the argument) then I just change it's coordinates instead o how it 'would look like to m eyes', I don't know, that makes me sleep easier

It's often repeated but describing spheres as "spikey" isn't right at all. The argument made is that they seem spikey in rectangular coordinates. But that's actually a statement about cubes, which are spikey. That's easy to see even just considering 2 and 3 dimensions : the corner of a cube is better for stabbing than a side. But the sphere is rotationally symmetric in all dimensions and is not spikey in the least bit.

This is how predicate transformer semantics views programs. An executing procedure is a walk through the program state space.

For example an instance of a struct with n fields is a point in an n-dimensional state space. A method that modifies that struct is moving that instance through that space. Where this gets cool is that it's possible to prove that for all points in the state space, a given program will reliably establish a defined postcondition.

To give a trivial example, imagine a state space with a few billion variables. Let's suppose one of those variables is called x and we want to establish the postcondition x = 0.

  x := 0

The above program will establish x == 0 regardless of the initial state and we don't need to worry about the several billion other dimensions in the state space. To a mathematician I imagine this is immensely boring, but for a working software developer boring is great, because it's so easy to otherwise build cognitively unmanageable systems.

> To a mathematician I imagine this is immensely boring,

You obviously haven't met many mathematicians; they're building proof assistants that are even more “boring”, and loving it.

Yea I started to come to a similar conclusion after I got a better understanding of data science in general. Also reminded me about role playing games .

That's how it was explained to me 20 years ago and I still think describing columns as dimensions is terribly confusing, even somewhat pretentious

Teenage me had a simple way to describe higher dimensions - you need to enter a higher one compared to current one to escape when completely surrounded.

Are you on the line and blocked from both sides? Use rest of the 2D plane to escape. Using Z dimension to escape from circle. Using 4th to escape from sphere without touching it.

Its true my imagination kind of stopped after 4D but even that was mesmerizing to young me.

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).

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...

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

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.

so knots only exist when there are exactly 3 dimensions? That's interesting.

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.

Even simpler, just take a 3D knot, and pull on it. Where it gets stuck, just lift a strand up in the 4th dimension.

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.

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.

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!

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.

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.

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.

> 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

Hmm yeah that what I’m confusing. I suppose it’s misleading when folks say The Fourth Dimension since it depends on context! But in the case of the article the question becomes how do we visualize four or more dimensions? What the dimensions represent doesn’t necessarily matter unless there’s a specific problem being solved.

The car example works as a vector space, but not an inner-product space. There's no sensible way to rotate the length of the car to gasoline in the tank

That's right! The less pithy answer: objects are the things that stay invariant under a transformation. In that sense then car length and gas are somehow MORE than linearly independent - they are "geometrically independent".

(Note there is a relationship between the length of the car and the gas tank when the car is moving fast (close to c) in your reference frame - but AFAIK there aren't any useful geometric transformations that depend on that fact. :)

Linear independence doesn't guarantee that you can rotate, for that you need to know how to exchange one dimension for another.

Essentially you need to define a Pythagorean theorem for your space. You can do that for the car vector space, but there isn't a natural choice.

Isn't that the definition of rotation, the "exchanging one dimension for another"? In 2-D it keeps a point invariant, in 3-D a line, and in N-D it keeps the N-1 object invariant. (Side note: the more basic operation is reflection, since you can get rotation from two reflections)

>Isn't that the definition of rotation, the "exchanging one dimension for another"?

Maybe? I think it allows solutions not typically considered to be rotations, like the special relativity example.

You can't rotate around an arbitrary vector and have current brake pedal position determine the number of passengers.

There's a fundamental difference between a "set of independent variables" and a physical space with geometric transformations. The latter is far more complex to visualise.

I had a notion for a physical theory of everything, but I got hopelessly lost once I wandered into 6-dimensional knot theory.

This video on how the game Miegakure is being developed helps to break down what is actually happening when a slice of a 4d space is being seen.

https://www.youtube.com/watch?v=vZp0ETdD37E

Same dev who also created 4D Toys toolbox: https://4dtoys.com/

With the 4th dimension, you can cheat a bit and use time itself. For instance, a 4-D sphere is something that starts at a point, grows to a sphere, then shrinks again.

It's not perfect. We can rotate in the normal 3D space but you don't get proper 4D rotation with this technique. Nor does this give you a good intuition about that shadow.

But it's something.

Gives you basically nothing when it comes to the 5th dimension, though, because I'm fresh out of spacetime at that point.

> I'm a highly visual thinker

Most people of average IQ say that.

> 4D object casts a 3D shadow. I can't wrap my head around that

No one can. I don't think you understand what a dimension is.

> > I'm a highly visual thinker

> Most people of average IQ say that.

I seriously question that. Can you point us to any evidence that people of average IQ are most often visual thinkers? (Or most often say that they're visual thinkers, if that's where you were going with this.)

I cannot. I am merely calling out that fact that "I'm a visual thinker" is almost always used as a justification for not being able to keep up. Sort of like saying "My other car is a Porsche."

Is it not possible that different people will have an easier or harder time understanding different kinds of concepts? Maybe IQ favors a particular type of "non-visual" thinking. Or maybe IQ is just another pile of bullshit from the field of physchology.

there are lots of people who are not visual thinkers - see "Aphantasia"

I'm not sure what the intent of this comment is, but it comes off as pompous and condescending.

You may try to imagine 3d objects (shadows of 4d) in consecutive slices. E.g. the foreground 3d scene contains a dot, middle scenes contain bigger and then again smaller spheres down to the background scenewith a dot. By looking at the entire setup from the side you’ll notice that spheres grow and shrink in the same fashion the 2d sphere’s x/y’s do. Similar to this, but in 3d: https://www.google.ru/search?q=sphere+slices&tbm=isch

A 4-cube would look like just few identical 3-cubes from the side, if oriented face-to-3-space. If not, it will be series of prisms or platonic solid alikes.

It doesn’t help much with shapes that have complex 4d-edges, or with higher dimensions, but may give an idea. Also, open two gifs from this link http://www.math.union.edu/~dpvc/math/4D/folding/welcome.html side by side and you’ll quickly get the concept of rotation of a solid around 4th axis.

You can imagine the same room (eg an IKEA show room) but located in different cities. Maybe each city does their show rooms slightly differently.

Here the fourth dimension is discrete, but you can imagine that for each city you get a 3D object.

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?).

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.

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/

[2] https://marctenbosch.com/news/2017/06/4d-toys-a-box-of-four-...

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/

[2] https://www.youtube.com/watch?v=6cpTEPT5i0A&list=PL3C690048E...

[3] https://www.youtube.com/watch?v=vts0YHACsYY&list=PLw2BeOjATq...

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

In the video Neil explains why higher dimensions may explain the queer ways of the quantum world.

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.

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

I love it lol

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."

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/...

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