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Comment by jstx1 | Hacker News Reader

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.

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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.

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.

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.

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.

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.

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

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 .

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