and so is the dimension in which the unit hypersphere has a maximum volume and, substituting this value back into the formula, gives that maximum volume as 5.277 768 ….
So, a hypersphere of radius 1 achieves its maximum volume in 5.256 946 4 …-dimensional space.
What must the radius of the sphere be to achieve its maximal volume in precisely five-dimensional space, or for that matter what must the radius of the sphere be to achieve its maximal volume in any other integral dimensional space? To answer these questions we need to consider the general V n ( R ) and differentiate it with respect to n just as we have done previously. The almost identical calculations are
Table 12.2. Radius for maximal volume.
and d V n ( R )/d n = 0 requires that
and so
and
We have the general condition that
Table 12.2 shows the values of R for which a hypersphere of that radius achieves its maximal volume in low integral dimensional space. That is, a hypersphere of radius 0.696 998 … achieves its maximum volume in two dimensions, one of radius 0.801 888 … achieves its maximum volume in three dimensions, etc.
Sums of Volumes
The transcendental Gelfond Constant, e π , appears naturally when we probe a little further into unit hypersphere volumesand investigate (ignoring units of measurement) the total volume of the infinite sequence of them. Since
there is at least a chance that there is a value toand, if we evaluate the finite sum for a large number of terms, our optimism seems to be well founded with that sum equal to 44.999 326 089 382 855 366 ….
To find a closed form for this we will again need to consider even and odd dimensions separately.
Recall that, for n even, we may writefor n = 2,4,6,… and, if n = 2 m , we have that V 2 m (1) = π m / m ! for m = 1,2,3,…. This means that
and we have the promised appearance of Gelfond’s constant.
Matters are far more complicated if n is odd. Now we have thatfor n = 1, 3, 5,… , and, if n = 2 m − 1, we havefor m = 1, 2, 3,… and the sum of the volumes is now
which is altogether more challenging.
In fact, we can eliminate the Gamma function from the expression by using a result which connects it to another exotic function, the double factorial N !!, which is defined by
Using the standard properties of the Gamma function, it is not too hard to show thatand thismakes
and it is easier still to show that (2 m − 1)!! = (2 m )!/(2 m m !), and this makes
Now that we have the sum expressed in more elementary terms it is still far from obvious whether or not this series for odd n has a closed form, as the much simpler one did for even n . If we begin to write out the series explicitly, we have
and those coefficients do not look particularly promising: a look in a standard mathematical handbook reveals nothing. In fact, the series does have a closed form, and to approach it we will engage in a common mathematical technique: the optimistic guess. Since e π appears in the expression for even n , it might just do so here and if it does the most reasonable form of its appearance would be
where S ( π ) is an infinite series in π . To find the form that this series must have, we need to rewrite the expression and expand both sides to compare coefficients:
which leads to the sequence of coeffcients,and our new series is
Even these coefficients promise little, but a second look in that mathematical handbook reveals the error function, Erf( x ), with its series form
In fact, it is defined by
and comes from the theory of the normal distribution in statistics. Using the series expansion of e -t 2 and integrating term by term results in the series form. Evaluate atand we have precisely
Our S ( π ) isand therefore we have
Figure 12.8. Cumulative volumes of the unit hypercubes.
And this makes
And the total volume
Of course, we have not rigorously proved this but from what we have seen it is at least feasible and evaluation of this exact expression to
So, a hypersphere of radius 1 achieves its maximum volume in 5.256 946 4 …-dimensional space.
What must the radius of the sphere be to achieve its maximal volume in precisely five-dimensional space, or for that matter what must the radius of the sphere be to achieve its maximal volume in any other integral dimensional space? To answer these questions we need to consider the general V n ( R ) and differentiate it with respect to n just as we have done previously. The almost identical calculations are
Table 12.2. Radius for maximal volume.
and d V n ( R )/d n = 0 requires that
and so
and
We have the general condition that
Table 12.2 shows the values of R for which a hypersphere of that radius achieves its maximal volume in low integral dimensional space. That is, a hypersphere of radius 0.696 998 … achieves its maximum volume in two dimensions, one of radius 0.801 888 … achieves its maximum volume in three dimensions, etc.
Sums of Volumes
The transcendental Gelfond Constant, e π , appears naturally when we probe a little further into unit hypersphere volumesand investigate (ignoring units of measurement) the total volume of the infinite sequence of them. Since
there is at least a chance that there is a value toand, if we evaluate the finite sum for a large number of terms, our optimism seems to be well founded with that sum equal to 44.999 326 089 382 855 366 ….
To find a closed form for this we will again need to consider even and odd dimensions separately.
Recall that, for n even, we may writefor n = 2,4,6,… and, if n = 2 m , we have that V 2 m (1) = π m / m ! for m = 1,2,3,…. This means that
and we have the promised appearance of Gelfond’s constant.
Matters are far more complicated if n is odd. Now we have thatfor n = 1, 3, 5,… , and, if n = 2 m − 1, we havefor m = 1, 2, 3,… and the sum of the volumes is now
which is altogether more challenging.
In fact, we can eliminate the Gamma function from the expression by using a result which connects it to another exotic function, the double factorial N !!, which is defined by
Using the standard properties of the Gamma function, it is not too hard to show thatand thismakes
and it is easier still to show that (2 m − 1)!! = (2 m )!/(2 m m !), and this makes
Now that we have the sum expressed in more elementary terms it is still far from obvious whether or not this series for odd n has a closed form, as the much simpler one did for even n . If we begin to write out the series explicitly, we have
and those coefficients do not look particularly promising: a look in a standard mathematical handbook reveals nothing. In fact, the series does have a closed form, and to approach it we will engage in a common mathematical technique: the optimistic guess. Since e π appears in the expression for even n , it might just do so here and if it does the most reasonable form of its appearance would be
where S ( π ) is an infinite series in π . To find the form that this series must have, we need to rewrite the expression and expand both sides to compare coefficients:
which leads to the sequence of coeffcients,and our new series is
Even these coefficients promise little, but a second look in that mathematical handbook reveals the error function, Erf( x ), with its series form
In fact, it is defined by
and comes from the theory of the normal distribution in statistics. Using the series expansion of e -t 2 and integrating term by term results in the series form. Evaluate atand we have precisely
Our S ( π ) isand therefore we have
Figure 12.8. Cumulative volumes of the unit hypercubes.
And this makes
And the total volume
Of course, we have not rigorously proved this but from what we have seen it is at least feasible and evaluation of this exact expression to
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