Consider a vector space, over the reals, of arbitrary natural dimension, dim. Allow the presence of a standard (positive-definite symmetric) metric and work consistently in the coordinates associated with a basis which diagonalises this with all diagonal entries 1. This effectively expresses our vector space as U = {({reals}:|dim)}, with the metric as (: (: sum(: v(i).u(i) ←i |dim) ←u |U) ←v |U).
The volume of a sphere of radius r trivially varies as power(dim, r) and the surface area is just the derivative with respect to r, which is d(power(dim, r))/dr = dim.power(dim−1,r) times the volume of the unit sphere. Consequently, we only need to determine the volume of the unit sphere, V(dim), at each dimension; this unit sphere's surface area shall then be A(dim) = dim.V(dim).
Choosing any diameter and integrating the cross-section perpendicular to it, we obtain (with B = {scalar t: −1≤ t ≤1})
in which we can substitute x= sin(t) to obtain
where, taking T = {scalar t: −π/2 < t < π/2}, so that (B|sin:T),
which is π for dim=0, change(:sin:T) = 2 (because sin' = cos) for dim=1 and susceptible to integration by parts when dim > 1:
in which the middle term is zero: dim>1 gives dim−1 > 0 and cos(π/2) = 0 = cos(−π/2), so power(dim−1, cos(t)) is zero at both ends of T.
We can now re-arrange what we have to obtain
or
It is easy enough to obtain, directly, I(0) = π and I(1) = 2: we can then use the recurrence relation to derive
for i = 0, 1, …, n (by induction in i)
and, similarly,
for i = 0, 1, …, n (by induction in i, again)
Multiplying these, we readily obtain
i.e. V(dim)/V(dim−2) = I(dim).I(dim−1) = 2.π/dim for each dim. This, incidentally, implies A(dim+2)/A(dim) = 2.π/dim for each dim; and at least strongly suggests that I(dim) shrinks as 1/√dim for large dim.
One may trivially obtain V(1) = 2 or start from the more familiar V(2) = π (note that I(2) = π/2 = V(2) / V(1) as expected). From here we can work upwards to obtain
since I(1) = 2 = V(1), leading to
i.e. V(2.n) = π^{n}/n!, and
It will readily be seen that both formulae tend to zero as n tends to infinity. They do this faster than 1/n, so the area also tends to zero. Consequently, one should be wary of imagining that integration over the surface of the unit sphere in an infinite-dimensional vector space (such as the Hilbert spaces which many quantum mechanical systems involve) can be commensurate with the metric used in defining the sphere. Or, to rub the point in, integration over all possible states of a quantum mechanical system is a pretty dodgy business.
It can also be shown that the two formulae above are the particular forms of a single formula: V(d) = power(d/2, π) / Γ(1 +d/2). The function Γ used here is Γ(x) = integral(: power(x−1, t).exp(−t) ←t |{positive reals}) and gives Γ(1+n) = n! for natural n; so we can write
and caricature it as extending our earlier V(2.n) = π^{n}/n! to apply even when n isn't an integer. For surface area, this gives A(d) = d.V(d) = 2.power(d/2, π)/Γ(d/2).
We also obtain I(d) = V(d)/V(d−1) = (√π).Γ((1+d)/2)/Γ(1+d/2) whence 2.log(I(d)) is approximately log(π) −g'(1+d/2) with g = log∘Γ; Stirling's formula gives us g'(1+d/2) as 1/d +log(d/2) plus terms of order 1/d/d, so that log(I(d)) = log(2.π/d)/2 plus a polynomial in 1/d, which dies away to zero for large d; thus I(d) is asymptotically √(2.π/d), which shrinks as 1/√d, as anticipated.
If we look for the sphere of unit volume, its radius will have to be r = power(−1/dim, V(dim)); with a little help from Stirling's formula we obtain:
whence r ≈ √(dim/2/e/π) for large dim (and 2.e.π ≈ 17.079). The radius needed to have a surface area of one is power(−1/(dim−1), dim.V(dim)) which, likewise (albeit with smaller correction, lacking any log(dim)/2/dim term), grows as √(dim/2/e/π). For each of these spheres, the RMS (root of the mean of the square of each) co-ordinate on the surface is asymptotically 1/√(2.e.π) ≈ 0.242 (and the unit-area sphere approaches this slightly faster, of the two).
Here's a table of some of the early values for I, V and A:
dim | I(dim) | V(dim) | A(dim) |
---|---|---|---|
−1 | 1/π | −1/π | |
0 | π | 1 | 0 |
1 | 2 | 2 | 2 |
2 | π/2 | π | 2.π |
3 | 4/3 | 4.π/3 | 4.π |
4 | 3.π/8 | π.π/2 | 2.power(2,π) |
5 | 16/15 | 8.π.π/15 | 8.power(2,π)/3 |
6 | 5.π/16 | power(3,π)/6 | power(3,π) |
7 | 32/35 | 16.power(3,π)/105 | 16.power(3,π)/15 |
8 | 35.π/128 | power(4,π)/24 | power(4,π)/3 |
and note that 3.2 = 16/5 > π and 16 > 15 imply
Here's a table giving approximate numerical values to V and A in the above and at some other values of dim (to illustrate the maxima and some critical thresholds being crossed), along with r(dim) = power(−1/dim, V(dim)), the radius of the sphere with unit volume:
dim | V(dim) | A(dim) | r(dim) |
---|---|---|---|
−1 | 0.318 | −0.318 | 0.318 |
0 | 1 | 0 | any |
1 | 2 | 2 | 0.5 |
2 | 3.14 | 6.28 | 0.564 |
3 | 4.19 | 12.57 | 0.620 |
4 | 4.93 | 19.74 | 0.671 |
5 | 5.26 | 26.32 | 0.717 |
6 | 5.17 | 31.01 | 0.761 |
7 | 4.72 | 33.07 | 0.801 |
8 | 4.06 | 32.47 | 0.839 |
12 | 1.34 | 16.023 | 0.976 |
13 | 0.91 | 11.84 | 1.007 |
18 | 0.08 | 1.48 | 1.149 |
19 | 0.047 | 0.885 | 1.175 |
43 | 2.05e-10 | 8.81e-9 | 1.680 |
44 | 7.70e-11 | 3.39e-9 | 1.698 |
47 | 3.83e-12 | 1.798e-10 | 1.750 |
48 | 1.38e-12 | 6.61e-11 | 1.766 |
188 | 4.96e-100 | 9.33e-98 | 3.375 |
189 | 9.04e-101 | 1.71e-98 | 3.383 |
192 | 5.37e-103 | 1.03e-100 | 3.409 |
193 | 9.68e-104 | 1.87e-101 | 3.418 |
1699 | 1.19e-1699 | 2.01e-1696 | 9.999 |
1700 | 7.21e-1701 | 1.23e-1697 | 10.002 |
170781 | 1.28e-341562 | 2.18e-341557 | 99.9999 |
170782 | 7.75e-341565 | 1.32e-341559 | 100.0001 |
∞ | 0 | 0 | ∞ |
At the start of the tables just given, I've used the values of I to iterate
V(dim−1) = V(dim) / I(dim) backwards, yielding: first, with dim = 1, the
almost intelligible V(0) = 1; next with dim = 0, we find the frankly bizarre
V(−1) = 1/π. After that we need I(−1) = I(1).1/0 to obtain
V(−2) = 1/π . 0/I(1) = 0, albeit somewhat suspectly; we can't continue
I's backwards iteration any further. We can continue V's backwards iteration:
V(dim−2) = V(dim).dim/2/π makes V zero at every negative even dimension
and gives us values of alternating sign for negative odd dimension; however,
both this V iteration and the Γ formula it yields are, in principle,
derived from the I iteration, so the link to any notion of sphere
is at
best tenuous below dim = −1. Below, I'll use
the Γ-formula for V to study the values for negative dimension more
closely, but first let's try to make sense of the cases we can get to via I's
reversed iteration.
Now, V(0) is the volume of the unit sphere in {({reals}:|0)} =
{({reals}:|empty)} = {empty} = {0}, the 0-dimensional vector space over the
reals. This vector space has only one member: its length is the square root of
an empty sum, hence zero, which is less than 1, so it is inside the unit sphere;
so the unit sphere is the entire space – likewise, so is the sphere of any
positive radius. We find that this has total measure 1, which seems natural
enough. That it is constant for positive radius again makes sense – we
expected pow(r,0) to be the r-dependency – and naturally implies zero
surface area, consistent with the formulae. Since the vector space of dimension
0 is a natural model for the notion of geometrical point
, this reads as
saying that a point has measure 1 and its surface has measure 0.
Now the surface of the 0-sphere is a curious entity – we expect it to have dimension −1, which is definitely nonsense, so perhaps we're best off just regarding the fact that we've got 0's worth of it as a good excuse for ignoring it. Still, curiosity asks what sense we can make of V(−1) = 1/π, which means discussing the unit sphere in a space of dimension −1. We'd expect the volume of the (−1)-sphere of radius r to be V(−1)/r and its surface area −V(−1)/r/r, so watch out for this possibly mattering to Coulomb-type fields … even if we can't make sense of it.
One approach is to look at simplices (point, line segment, triangle,
tetrahedron, …), each of which has a definite dimension and a standard
representation as simplex(dim) = {({positive reals}: p |dim+1): sum(p) = 1}, a
subset of the (1+dim)-dimensional space (this can be read as the collection of
probability measures on 1+dim). Here we find that simplex(dim) has a boundary
consisting of 1+dim faces
each of which is a simplex(dim−1). So we
have a 2-dimensional triangle with 3 edges, each of which is a 1-dimensional
line, with 2 ends, each of which is a 0-dimensional point, whose boundary is a
single simplex of dimension −1; this, in turn, has a boundary consisting
of 0 simplices of dimension −2. The standard representation gives
simplex(−1) = {({positive reals}: p |0): sum(p) = 1} and we know that any
(:p|0) is empty and so has sum 0, not 1, so there are no such p and
simplex(−1) = {}; it is thus no surprise that, whatever kind of things its
boundary is made up of, there are none of them. Compare simplex(0) =
{({positive reals}: p |1): sum(p) = 1} = {({1}: |{0})} which has one member and
is viewed as a point: this is exactly the 0-sphere, so the 0-sphere has a single
{} as boundary – for which a measure of 0 makes singularly good sense.
Now, let's go back to our 0-dimensional space, {zero} – and distinguish between the natural number 0 = {} = empty and the vector zero = ({0}::). We expect a basis ({zero}: e |dim) and, since 0 = {} = empty is our dimension, we have e = ({zero}: empty :{}). The sum of this is an empty sum, so yields the additive identity of our vector space, {zero}, namely zero. Thus e spans {zero}. Now, to investigate linear independence, we must ask for ({reals}: f |(:e|)) for which sum(: f(i).e(i) ←i :) = zero. There is precisely one ({reals}: f |{}), namely empty (again): and it does yield this zero sum. It's also f= ({reals}: constant(zero) |{}); so the only ({reals}: f |(:e|)) for which the sum is zero is, indeed, constant(zero) and we find that e is, indeed, linearly independent. Since we already know it spans our space, that confirms that it is a basis.
Now, I can make sense of {} as a linear space in the sense that: I can multiply any member of it you give me by any scalar you care for (because you can't give me a member of {}); likewise, I can add arbitrary members of {} – indeed, any equation of form a+b=a+c implies b=c and for any a, b in {} there is a unique x in {} for which a+x=b, so this addition is cancellable and complete. There is only one mapping ({}::), namely empty = {} = (:empty|). Oddly enough, we can't sum it sensibly, because its answer needs to be a member of the target linear space, {}, which has no members. This makes discussion of whether empty spans {} somewhat confused, since we can't sum empty (or the one scaling of it).
For linear independence, notice that the only function ({reals}:|{}) is again empty – which is, none the less, ({reals}: constant(zero) |{}) and so constant with value zero. Thus there are no mappings ({reals}:|{}) which are not constant 0 and, in particular, none such for which sum(: f(i).empty(i) ←i :{}) is zero, so empty can sensibly be viewed as linearly independent. However, since we can't sum constant(zero).empty, I'm uncomfortable with this.
This still doesn't give us {} as a linear space of dimension −1, but it's a weird enough beast that it could relate to our funny sequence. For lower negative dimensions, I can't concoct any kind of an entity to regard as the space whose sphere to discuss. Still …
Earlier, we obtained a formula for V(dim) in terms of the Γ function, as power(dim/2, π) / Γ(1 +dim/2). Now, Γ has a simple pole at each non-positive integer, so V is zero for each negative even dim. The formula also gives values for all odd negative dim. Note that Γ(1−z).Γ(1+z).sinc(z.π) = 1, where sinc = (:sin(x)/x ←x:) with its undefined-at-0 filled in by sinc(0) = 1; thus, substituting the Γ-formula, V(−d).V(d) = sinc(π.d/2), whence V(−even) is always zero and V(−odd) diverges to infinity, alternating negative and positive values. We can arrive at the same answer by reading V(dim)/V(dim−2) = 2.π/dim as V(dim−2) = V(dim).dim/(2.π) and rolling backwards from V(0) – which yields zero for V(−2) and all other negative even dimensions – and from V(1) or, indeed, V(−1). So here's one table giving formulae for a few illustrative cases:
dim | V(dim) | A(dim) | r(dim) |
---|---|---|---|
−even (non-zero) | 0 | 0 | (undefined) |
−1 | 1/π | −1/π | 1/π |
−3 | −1/2/π/π | 3/2/π/π | |
−5 | 6/power(3,2.π) | −30/power(3,2.π) | |
−7 | −30/power(4,2.π) | 210/power(4,2.π) |
and another giving values for odd negative dimension, whatever they may mean:
dim | V(dim) | A(dim) | r(dim) |
---|---|---|---|
−1 | 0.318 | −0.318 | 0.318 |
−3 | −0.0507 | 0.152 | −0.370 |
−5 | 0.0242 | −0.121 | 0.475 |
−7 | −0.0192 | 0.135 | −0.569 |
−9 | 0.0214 | −0.193 | 0.653 |
−11 | −0.0307 | 0.338 | −0.729 |
−13 | 0.0538 | −0.699 | 0.799 |
−15 | −0.1112 | 1.67 | −0.864 |
−17 | 0.266 | −4.52 | 0.925 |
−19 | −0.719 | 13.66 | −0.983 |
−21 | 2.17 | −45.64 | 1.038 |
−45 | 4.94e+8 | −2.22e+10 | 1.560 |
−47 | −3.54e+9 | 1.66e+11 | −1.597 |
−49 | 2.65e+10 | −1.30e+12 | 1.632 |
−189 | 3.73e+97 | −7.045e+99 | 3.283 |
−191 | −1.12e+99 | 2.14e+101 | −3.301 |
−193 | 3.408e+100 | −6.58e+102 | 3.318 |
−1713 | 3.42e+1712 | −5.862e+1715 | 9.994 |
−1715 | −9.34e+1714 | 1.601e+1718 | −10.000 |
−1717 | 2.55e+1717 | −4.38e+1720 | 10.005 |
−170805 | 4.75e341609 | −8.11e341614 | 99.9996 |
−170807 | −1.29e341614 | 2.20e341619 | −100.0001 |
−odd infinity | ∞ | ∞ | ∞ |
Note that r is smallest at dimension −1. While, among positive dimensions, 5 and 7 yielded the maxima for V and A, respectively, dimensions −5 and −7 yield the smallest values for A and V, respectively, among negative odd dimensions.
Written by Eddy.