quickfur wrote:...in spite of its radius being constantly 1.
quickfur wrote:One of the strange things that happen as the number of dimensions increase, is that the vast majority of an n-cube's volume lies close to its vertices rather than its center.
quickfur wrote:Keep in mind that as the number of dimensions increase, the distance of a unit n-cube's vertices from its center increases without bound. Eventually this shifts the majority of its bulk away from its center and towards its vertices.
PatrickPowers wrote:[...]The bulk of the matter is equidistant between the center and the nearest vertex. The average distance to the center is the same as the average distance to the nearest vertex. These two distances increase equally as N grows.
It is true that most of the material is closer to the surface as N increases. But the volume near a vertex is so skinny that even though numerous they in total don't hold much. [...]
quickfur wrote:the number of vertices grow exponentially with increasing dimension -- that the total volume close to the vertices overwhelms the volume around the center.
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This also leads to another weirdness of higher dimensions: at the limit as n→∞, the number of vertices in an n-cube become uncountable whereas the number of facets remains countable.
steelpillow wrote:quickfur wrote:the number of vertices grow exponentially with increasing dimension -- that the total volume close to the vertices overwhelms the volume around the center.
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This also leads to another weirdness of higher dimensions: at the limit as n→∞, the number of vertices in an n-cube become uncountable whereas the number of facets remains countable.
For a measure polytope (n-cube), the number of vertices doubles with each dimension. This is a geometric series, not exponential. The vertices would appear to be countable in an obvious way.
quickfur wrote:The number of vertices doubles with each dimension, meaning that in n dimensions there are 2^n vertices. When n diverges to infinity, you can put each vertex in 1-to-1 correspondence with the binary expansion of a real number between 0 and 1. I.e., the vertex coordinates are <±1, ±1, ...>; map 1 to itself and -1 to 0, and you have an infinite sequence of binary digits, unique per vertex. Since all permutations are taken, that means the binary expansion of every real number between 0 (inclusive) and 1 (exclusive) maps to exactly one vertex of the infinite dimensional cube. Since there are uncountably many real numbers between 0 and 1, there must also be an uncountable number of vertices.
For example we may count them from the lower dimensions up as, say (leaving out the trailing zeroes here), 0,1,00,01,10,11,000,001... and so on.
quickfur wrote:Any countable list of such will only give you a strict subset
mr_e_man wrote:Quickfur knows what he's talking about.
I will repeat Cantor's proof here.
Suppose we have a countably infinite list of vertices. Denote the n'th vertex as v_{n}, with coordinates v_{n,k}, for natural numbers n and k.
Since these are vertices of the cube, their coordinates are v_{n,k} ∈ {+1, -1}.
Consider the point x with coordinates x_{k} = - v_{k,k}.
Since -(+1) = -1 and -(-1) = +1, we have x_{k} ∈ {+1, -1} for all natural numbers k, so this point is also a vertex of the cube.
But since -(+1) ≠ +1 and -(-1) ≠ -1, we have x_{k} ≠ v_{k,k}. Thus, for any natural number n, the vertex v_{n} is not equal to x, since they differ in the n'th coordinate.
So there is a vertex x not in the list. This proves that the cube's vertices are uncountably infinite.
Klitzing wrote:"... with coordinates v_n,k", i.e. k is obviously just the coordinate index.
E.g. let v_1 be the vertex with coordinates 1 throughout, then you'd have v_1,k = 1 for all k.
--- rk
quickfur wrote:Now it's my turn to veer off-topic, but there's an interesting borderline approach that perhaps may appeal more to the likes of steelpillow, who IIRC does not buy Cantor's diagonalization argument.
quickfur wrote:as the number of dimensions increase, the distance of a unit n-cube's vertices from its center increases without bound. Eventually this shifts the majority of its bulk away from its center and towards its vertices.
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