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May 27[edit]

On the subject of dimensions[edit]

Yesterday I was having a civil lunchtime discussion with my peers when the following question arose: could a "sub-dimension" exist? It may be hard to explain, but I’ll try. I’m talking about dimensions such as 2.5D (though that’s more of a gaming topic than an academic one). Could, say, 3.14D exist? Or 1.6D? The fate of my, or perhaps my friends', five dollars depends on your answer. — Preceding unsigned comment added by 166.127.1.10 (talk) 14:35, 27 May 2022 (UTC)[reply]

What type of dimension would be partway between area and volume? --←Baseball Bugs ^{What's up, Doc?} carrots→ 16:48, 27 May 2022 (UTC)[reply]

See Fractal dimension. CodeTalker (talk) 17:32, 27 May 2022 (UTC)[reply]

Every illustration in that article is either 2-dimensional or 3-dimensional. --←Baseball Bugs ^{What's up, Doc?} carrots→ 18:18, 27 May 2022 (UTC)[reply]

This is going to get really technical really fast, but sure, they all have integer topological dimension. In fact topological dimension is always a natural number.

However Hausdorff dimension can have a fractional part. Original Poster, if you're interested in that topic, it would probably go better on the math reference desk, but I could explain it here if you like. Just so you're aware, it's a bit complicated. --Trovatore (talk) 18:23, 27 May 2022 (UTC)[reply]

To answer some of the questions explicitly, for any

n

-dimensional Euclidean space and any value of

D

such that

0 \leq D \leq n

, there are subsets of this space that have Hausdorff dimension equal to

D

. For the mountainous surfaces in Figure 7 of the article Fractal dimension, we have

2 < D < 3

. It is possible to define a random process that generates such a surface for which the Hausdorff dimension is exactly equal to

2.5

; likewise, Hausdorff dimension

3.14

– or, for that matter,

π

– can be realized, but not for a subset of 3D space; we need to introduce a fourth dimension. --Lambiam 19:31, 27 May 2022 (UTC)[reply]

The items in Figure 7 are three-dimensional. --←Baseball Bugs ^{What's up, Doc?} carrots→ 11:52, 28 May 2022 (UTC)[reply]

Actually they're two-dimensional, topologically, because they're just the surface, not the interior. The same way the Earth is three-dimensional, but the surface of the Earth is two-dimensional. Or like a circular disk is two-dimensional, but the circle (just the curve around the outside) is one-dimensional.

But there are other notions of dimension according to which the surfaces in Figure 7 have dimension between 2 and 3. --Trovatore (talk) 17:54, 28 May 2022 (UTC)[reply]

It comes down to what the OP is really talking about: mathematical theory, or reality? --←Baseball Bugs ^{What's up, Doc?} carrots→ 23:09, 28 May 2022 (UTC)[reply]

False dichotomy, but in any case, the two notions I explain below — number of numbers needed to describe a point, and scaling behavior with linear size — are both very much "real-world" notions of dimension with real-life consequences.

If it were up to me to judge the outcome of the wager, I'd call it a push, on the grounds that it was never stated clearly enough to have a single definite answer. Fractals and fractal dimension is probably not what they really had in mind, but it is a defensible and important notion of dimension that is genuinely in use. --Trovatore (talk) 02:34, 29 May 2022 (UTC)[reply]

The surface of the earth is three-dimensional, not two-dimensional. However, it would be interesting to be a fly on the wall for whatever discussion the OP plans to have with colleagues. --←Baseball Bugs ^{What's up, Doc?} carrots→ 02:37, 29 May 2022 (UTC)[reply]

Sorry, you're just wrong about that. The surface of the Earth is two-dimensional. --Trovatore (talk) 05:06, 29 May 2022 (UTC)[reply]

Since when are mountains two-dimensional? --←Baseball Bugs ^{What's up, Doc?} carrots→ 05:27, 29 May 2022 (UTC)[reply]

Bugs, perhaps reading Surface (mathematics) will help you to understand this. There's a difference between the dimensionality of an object and the dimensionality of the space in which it exists. A mountain is 3-dimensional but the surface of the mountain is 2-dimensional. CodeTalker (talk) 06:15, 29 May 2022 (UTC)[reply]

Or to put it another way, the fact that the Earth's surface is 2D is why any point on the surface can be identified with only two numbers (latitude and longitude). CodeTalker (talk) 06:24, 29 May 2022 (UTC)[reply]

Bugs, take it one dimension down. Just think of a curve, just a line that wiggles. You wouldn't call it two-dimensional just because it's wiggly, would you? It's still one-dimensional; that doesn't change just because you bend it. --Trovatore (talk) 06:51, 29 May 2022 (UTC)[reply]

What you're describing is two-dimensional. Meanwhile, it would be interesting if the OP would show up here and assess the various responses. --←Baseball Bugs ^{What's up, Doc?} carrots→ 07:45, 29 May 2022 (UTC)[reply]

It is a one-dimensional object embedded in a two-dimensional space. —Kusma (talk) 08:58, 29 May 2022 (UTC)[reply]

Ha. --←Baseball Bugs ^{What's up, Doc?} carrots→ 16:00, 29 May 2022 (UTC)[reply]

Look, Bugs, you yourself acknowledged this implicitly above, asking "[w]hat type of dimension would be partway between area and volume?" Surely you agree that a wiggly line has only length, not area? More precisely it does have an area, but that area is zero. Or if you don't, then please explain how you'd go about finding the area of a wiggly line. --Trovatore (talk) 17:50, 30 May 2022 (UTC)[reply]

It occurred to me that maybe I can simplify this enough to give some of the idea without too much math.

Very, very roughly, with lots of caveats, you can think of topological dimension as telling you how many numbers you need to find a point on the object. For example, to find a point on the surface of the Earth, you just need latitude and longitude; that's why we say the surface of the Earth is two-dimensional. The solid Earth is three-dimensional, because you need three numbers -- say, latitude, longitude, and distance below the ground.

For this sort of dimension, fractional parts don't make sense. There is no way to use two and a half numbers to describe a position, because there's no such thing as two and a half numbers in the first place.

But you can also treat dimension as telling you how things scale with size. For example, if I take a spherical concrete ball one meter across, and another one ten meters across, and I smash them up into little bits and put them in buckets of the same size, it would take 10³ = 1000 times as many buckets for the bigger ball. That's because the ball has dimension 3. The surface of the ball has dimension 2; if I want to make a blow-up plastic ball ten meters across, it will take 10² = 100 times as much plastic as to make one one meter across (assuming the plastic is the same thickness).

Instead of changing the size of the ball, what if we change the size of the buckets? If I have a solid ball one meter across, and I want to cut it up into cubelets one millimeter on an edge, it'll take 1000 = 10³ times as many as if I make them one centimeter on an edge (ten times larger). Again, because it has dimension 3. But if I just want to cut up the surface, the ratio is only 100 = 10², because the surface has dimension 2.

Well, this last sort of dimension is one version of fractal dimension (specifically box dimension). And it can be non-integer. If you want to cover the boundary of the Koch snowflake with boxes of size s, the number that you need is approximately inversely proportional to s^1.26, so the boundary has box dimension about 1.26.

Does that help? --Trovatore (talk) 20:08, 28 May 2022 (UTC)[reply]

Dimension is one of those things that depends on which definition of dimension you are using. One way to think about dimension is to think about how many numbers are necessary to describe a location in that dimension; as noted several times above, a line (straight or otherwise) is considered 1-dimensional because a single number can be used to describe a location along it. A surface (flat or otherwise) is 2-dimensional because two numbers are all that is needed to describe a location on it; the surface of the earth is not flat, but it is 2-dimensional because every point on the surface can be fully expressed with only 2 numbers. The earth itself (which is to say, the entire earth, not just the surface) is three dimensional because you need 3 numbers to describe a location within the earth (say, a latitude, a longitude, and a depth). One of the things that mathematics does is to generalize concepts, especially taking concepts which are discrete and generalize them to include the continuous; we do this when we expand the definition of say, exponentiation to include concepts broader than repeated multiplication, but allow for things like negative, fractional, and imaginary exponents. We do this because it is useful. Now, the new expanded definition cannot contradict the simpler definition, so just like defining exponentiation in the new method (which uses calculus to define it as the function which is proportional to its own derivative) also includes the old definition (exponentiation with whole numbers still represents repeated multiplication). The notion of fractal dimensions (of which there are several distinct definitions) actually includes simple definitions; but is expanded to include definitions that encapsulate "roughness" and the fractal nature of some objects. Smooth objects (which is to say, things which have a continuous derivative) still have whole number dimensions. But objects which are, by their nature, fractal, which is to say that they have infinite complexity, can still have dimensions, they just have dimensions between the numbers. It's an expansion on the simpler idea which includes the simpler idea but which also allows us to analyze and understand objects for which the simpler definition doesn't make sense. Fractal dimension is discussed excellently in This video by 3blue1brown. I recommend to anyone (especially Bugs and the OP) who are unfamiliar with this concept to watch that video. --Jayron 32 12:03, 31 May 2022 (UTC)[reply]