Is there a good layman’s explanation of higher dimensions as they relate to this type of problem? I’m trying to envision what that means, which is probably a wrong approach…
An n-dimensional space is just a collection of points, each defined uniquely by a set of n-numbers. The semantic meaning of those numbers doesn't really matter. It might be like actual physical space, but it could just as well be something like "time" and "the price of big macs". We have a bunch of mathematical operations that work well on 2 or 3 dimensional space that correlate nicely with our physical intuitions of 'curvature' and 'holes', and that still work perfectly well in more generalized forms in higher dimensions.
I'm not really sure it's that useful to try and visualize what it means on higher dimensions, to be honest.
Given your response, is it fair to say time as the 4th dimension is just a sci-fi concoction?
But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).
there's an old joke about a mathematician teaching an engineer about thirteen-dimensional spaces. "What do you think," the mathematician asks. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?"
"Well, it's not so hard. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13."
"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it."
So what kind of intuition could you use instead then? Or what exactly do you mean with "work perfectly well"?
The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.
From what I've heard, a fair number of the mathematicians doing research on 4+-dimensional things seem to have developed very good intuition about them, and okay-ish ability to "visualize" them. Those abilities falls off as you add dimensions, or try applying them to more-complex shapes...which is hardly surprising, considering how an average person's ability to intuit and visualize (in dimensions 0 through 3) falls off when complexity and dimensions are added.
In particular the video "Conceptualizing the Christoffel Symbols". Also look at content on the Metric Tensor
Additionally, there is content from other sources (albeit less produced) on describing projective geometry which is also related
But what about fractional dimensions (not to be confused with fractal dimensions)? Any advice about reasoning about the geometry of lets say something 0.6309297535... dimensional? It seems so easy, I mean it is somewhere between 0 and 1 dimensions, both of which have trivial geometric interpretations.
Closest I could think of is doing augmentations into the next highest integer dimension. That would be similar to how we often use projections to lower integer dimensions to think about higher integer dimensions, but in reverse.
And yes, fractional dimensions do exist, just like fractional derivatives or fractional Fourier transform, etc.
1. create a copy of the point and move it a fixed distance along a specific direction (which we will call the X-axis) and join the two points together -- this is a 1D line;
2. create a copy of the line and move it along a specific direction that is orthogonal (at 90 degrees to) to the direction the line is facing (which we will call the Y-axis) and join the two end points together -- this is a 2D square;
3. create a copy of the square and move it along a specific direction that is orthogonal (at 90 degrees to) to the other (X and Y) directions (which we will call the Z-axis) and join the four end points together -- this is a 3D cube;
4. for 4D and higher dimensions you can repeat this process to form other hypercubes [1]. The 4D version is a tesseract [2] on whih you can see this construction.
The general approach of visualising these is to use a projection [3] in a way similar to how a cube is displayed on a 2D screen or image. The idea is to cast a shadow to the next dimension down. For a hypercube you can project this to 3D space and then to 2D space.
Dimensions are typically defined in terms of unit vectors. These are vectors pointing in the X, Y, Z, ... directions with a length of 1. I.e. all coordinates are 0 except for the direction which is 1. For 4 dimensions they will have the values x̂ = (1,0,0,0), ŷ = (0,1,0,0), etc. Thus, you can express coordinates as multiples of these unit vectors. (This is similar to the x + iy notation for complex numbers, where 1 is the real unit vector and i is the imaginary unit vector.)
A hypersphere is an n-dimensional object where the points on the surface are a fixed distance (radius) away from the hypersphere's origin. This is typically defined as sum(s_n^2) = 0 -- for a circle (2D) this becomes x^2 + y^2 = 0; for a sphere (3D) this becomes x^2 + y^2 + z^2 = 0.
A manifold is just a generalized closed n-dimensional surface, such as the surface of a cube, circle, donut, or other object [4]. It is defined as a set (collection) of the points on that surface. These points can be defined as an equation, such as the equation of a hypersphere, or more generally. For example, you could define each face of a hypercube separately.
[1] https://en.wikipedia.org/wiki/Hypercube
[2] https://en.wikipedia.org/wiki/Tesseract