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May 16

Five-dimensional mathematics

What sort of functions or mathematical theorems would one tend to rely on when doing higher-dimensional geometry (let's say in five dimensions)? What sort of math would come in handy?

Have mercy upon me. I am but a poor English major looking for some buzzwords. :) --Brasswatchman (talk) 15:42, 16 May 2014 (UTC)[reply]

Linear algebra is your friend in analysing the basic behaviour of regular Euclidean space in higher dimensions. If you're interested in regular shapes in those dimensions, polytope and simplex are of interest. AlexTiefling (talk) 16:05, 16 May 2014 (UTC)[reply]

I can give you the formula for the distance between two points in 5 spacial dimensions (no temporal/time dimensions). In 2D we have:

D = sqrt((X₁-X₂)²+(Y₁-Y₂)²)

In 3D we add a term for Z:

D = sqrt((X₁-X₂)²+(Y₁-Y₂)²+(Z₁-Z₂)²)

So, just continue the pattern for 5D, where we use A and B for the two extra dimensions:

D = sqrt((X₁-X₂)²+(Y₁-Y₂)²+(Z₁-Z₂)²+(A₁-A₂)²)+(B₁-B₂)²))

StuRat (talk) 01:38, 17 May 2014 (UTC)[reply]

What exactly are you using these for? Does it need to be 5d geometry, or would any exotic system work? Are you looking to use these "buzzwords" correctly, or do you just need words that convey "exotic mathematically sophisticated geometry" (like words for some scientist style character to drop in a story)? Any information on what exactly you are doing will probably help get you a better answer. :-) Phoenixia1177 (talk) 18:44, 16 May 2014 (UTC)[reply]

Dimension five is special because it is the smallest dimension in which it is possible to realize any surface (isometrically). To understand what this means, consider the torus. Normally we think of the torus as having curvature, but there is another "flat" torus without curvature. This is just a square in the plane, but with opposite sides identified. Since any small part of this surface is indistinguishable from part of the plane, it is "flat". Now this flat torus cannot (smoothly) be realized as a surface in three dimensional Euclidean space. Instead that actually requires four dimensions. But even four dimensions is not enough to realize all possible kinds of surfaces. For that, one must go up to a five dimensional Euclidean space. This is a theorem of Mikhail Gromov. Sławomir Biały (talk) 18:57, 16 May 2014 (UTC)[reply]

If you just need a

C^{1}

isometric embedding, you can do it inside an arbitrarily small ball in 4-space, by the Nash embedding theorem. —Kusma (t·c) 19:26, 16 May 2014 (UTC)[reply]

Which is why I said "smoothly". Sławomir Biały (talk) 22:48, 16 May 2014 (UTC)[reply]

Back to the original question, for rotations in five dimensions I would use geometric algebra. As its name suggests it has feet in both geometry and algebra, so many of its algebraic objects have straightforward geometric interpretations, and it generalises well into higher dimensions. In fact a lot of useful work is done with it in five dimensions via conformal geometric algebra, which uses geometric algebra in five dimensions to generate both rotations and translations in three dimensions.--JohnBlackburne^words_deeds 22:57, 16 May 2014 (UTC)[reply]

A slightly different perspective on the last sentence: the conformal transformations in three dimensions are given by Lorentz transformations in five dimensions. These in turn can be specified by 2x2 quaternion matrices (via the spin representation). Sławomir Biały (talk) 17:21, 17 May 2014 (UTC)[reply]

One mildly interesting thing about 5 dimensional space is that it's the smallest-dimensional space in which there are only three "Platonic" regular polytopes - the regular simplex (equivalent to our tetrahedron), the measure polytope (equivalent to our cube), and the cross polytope (equivalent to our octahedron). In 3-space and 4-space we have other regular polytopes, but it's always struck me as rather interesting that that there are no higher-dimensional analogues of the dodecahedron or the icosahedron. RomanSpa (talk) 15:25, 17 May 2014 (UTC)[reply]

In 4D there are analogues of the dodecahedron and icosahedron. They are the 120-cell and 600-cell respectively. Double sharp (talk) 06:55, 18 May 2014 (UTC)[reply]

Sorry, I meant "higher than 4D", of course (I mentioned that we have other regular polytopes in 4-space above). RomanSpa (talk) 15:20, 18 May 2014 (UTC)[reply]

I guess you could still say they exist in >4D as hyperbolic honeycombs {5, 3, ..., 3} and {3, 3, ..., 3, 5}. Double sharp (talk) 15:33, 18 May 2014 (UTC)[reply]

Of course, if you're just looking for some plausible-sounding SF buzzwords, how about observing that we live in a 3-space, and have one additional dimension: time. Thus we have 4 dimensions in all. Thus our entire universe could be embedded in a 5-space, and alien creatures able to travel along that extra dimension would be able to "see" all our universe, from start to end, as a single object "frozen in time".

Alternatively, if you want a "faster than light" space drive, how about embedding a 3-space bubble in an arbitrarily higher-dimensioned space? Then by a sequence of rotations and translations "move" the bubble to another point in the higher-dimensioned space (carefully arranging for these operations to be performed very fast!). Then undo the bubble again, and you're on the other side of the universe. That sounds just enough like SF gobbledegook to work in those sorts of books! RomanSpa (talk) 15:40, 17 May 2014 (UTC)[reply]

The Kaluza–Klein_theory is a five dimensional theory of relativity. Bo Jacoby (talk) 16:28, 17 May 2014 (UTC).[reply]