44.999 326 089 382 855 366 … can only add to our confidence. It is the case that not too much extra analysis would prove it to be so.
In fact, we can see from figure 12.8 , which shows a continuous plot of the cumulative sum, that the convergence is all but accomplished by the twentieth term.
Surface Area in Hyperdimensions
The surface area A n ( R ) of the hypersphere whose volume is
is simply the derivative of the expression with respect to R , so
Figure 12.9. Surface area of the unit hypersphere compared with continuous dimensions.
Table 12.3. Surface areas of hyperspheres.
In particular,
Table 12.3 lists the first few values of the hypersurface areas and indicates a peak at n = 7 for A n (1). figure 12.9 shows a plot of A n (1) against a continuous n and indicates that the surface area of the hypersphere does peak at around n = 7 and that it also tends to 0 as n increases. Entirely similar calculations to the ones previously made show that for maximal A n (1) n satisfieswhich tells us that the maximum is actually achieved when n = 7.256 95 … and takes the value 33.1612 ….
Table 12.4. Radius for maximal surface area.
Almost repeating the argument for volume, if we wish to calculate the radius R of the hypersphere which has maximal surface area in each integral dimension, we use calculus on A n ( R ) and this gives rise to the equation
which generates table 12.4 .
In summary, the unit hypersphere has a maximum volume of 5.277 768 … in 5.256 946 4 …-dimensional space and a maximum surface area of 33.1612 … in 7.256 95 …-dimensional space. Further, a hypersphere of radius 0.696 998 … achieves its maximum volume, and one of radius 0.422 751 … its maximum surface area in two dimensions; one of radius 0.801 888 … achieves its maximum volume, and one of radius 0.574 578 … its maximum surface area in three dimensions, etc.
(Using very similar techniques as before, the sum of the surface areas of hyperspheres can be shown to befor even dimensions andfor odd dimensions, making the total
A similar plot to figure 12.8 again shows that the convergence is all but accomplished by the twentieth term.)
Table 12.5. Distribution of volume in a hypersphere.
The Distribution of Volume
Is any of this useful? A good answer is, who cares? That said, there are implications of some of the strange behaviour of hyper-space to the theory of sampling in large numbers of variables, and the many mathematical ideas which depend on the techniques. We will not discuss them here but we will show an area which causes problems.
The volume of the n -dimensional hypersphere with radius R is, of course,
Now we ask the question, Where is this volume? To answer this, we will initially be particular and ask the question, How much of the volume of the hypersphere is at a distance of 20% from its surface?
The answer, as a percentage to the nearest whole number and for varying dimensions, is given in table 12.5 and the figures clearly show that the volume near the surface is fast approaching 100%.
In general, the amount of volume near the surface of the hyper-sphere of radius R can be measured by the difference between the volume of the hypersphere and the volume of the hyper-sphere of radius R (1 −), whereis taken to be small (in table 12.5 = 0.2). Compare this quantity with the volume ofthe hypersphere itself and we have the fraction
That the asymptotic limit is 1 shows that, whatever volume there is in a high-dimensional hypersphere (and there isn’t much), it is ever more concentrated at its surface. Also, since, for all R ,
inscribe the hypersphere in the n -dimensional hypercube of side 2 R and we see that most of the hypercube’s volume is concentrated at its corners.
Our discussion has concentrated on a few particular areas of the counterintuitive nature of hyperdimensions and we could mention many other manifestations and implications: the full story is big enough to fill books (and has done so). To
In fact, we can see from figure 12.8 , which shows a continuous plot of the cumulative sum, that the convergence is all but accomplished by the twentieth term.
Surface Area in Hyperdimensions
The surface area A n ( R ) of the hypersphere whose volume is
is simply the derivative of the expression with respect to R , so
Figure 12.9. Surface area of the unit hypersphere compared with continuous dimensions.
Table 12.3. Surface areas of hyperspheres.
In particular,
Table 12.3 lists the first few values of the hypersurface areas and indicates a peak at n = 7 for A n (1). figure 12.9 shows a plot of A n (1) against a continuous n and indicates that the surface area of the hypersphere does peak at around n = 7 and that it also tends to 0 as n increases. Entirely similar calculations to the ones previously made show that for maximal A n (1) n satisfieswhich tells us that the maximum is actually achieved when n = 7.256 95 … and takes the value 33.1612 ….
Table 12.4. Radius for maximal surface area.
Almost repeating the argument for volume, if we wish to calculate the radius R of the hypersphere which has maximal surface area in each integral dimension, we use calculus on A n ( R ) and this gives rise to the equation
which generates table 12.4 .
In summary, the unit hypersphere has a maximum volume of 5.277 768 … in 5.256 946 4 …-dimensional space and a maximum surface area of 33.1612 … in 7.256 95 …-dimensional space. Further, a hypersphere of radius 0.696 998 … achieves its maximum volume, and one of radius 0.422 751 … its maximum surface area in two dimensions; one of radius 0.801 888 … achieves its maximum volume, and one of radius 0.574 578 … its maximum surface area in three dimensions, etc.
(Using very similar techniques as before, the sum of the surface areas of hyperspheres can be shown to befor even dimensions andfor odd dimensions, making the total
A similar plot to figure 12.8 again shows that the convergence is all but accomplished by the twentieth term.)
Table 12.5. Distribution of volume in a hypersphere.
The Distribution of Volume
Is any of this useful? A good answer is, who cares? That said, there are implications of some of the strange behaviour of hyper-space to the theory of sampling in large numbers of variables, and the many mathematical ideas which depend on the techniques. We will not discuss them here but we will show an area which causes problems.
The volume of the n -dimensional hypersphere with radius R is, of course,
Now we ask the question, Where is this volume? To answer this, we will initially be particular and ask the question, How much of the volume of the hypersphere is at a distance of 20% from its surface?
The answer, as a percentage to the nearest whole number and for varying dimensions, is given in table 12.5 and the figures clearly show that the volume near the surface is fast approaching 100%.
In general, the amount of volume near the surface of the hyper-sphere of radius R can be measured by the difference between the volume of the hypersphere and the volume of the hyper-sphere of radius R (1 −), whereis taken to be small (in table 12.5 = 0.2). Compare this quantity with the volume ofthe hypersphere itself and we have the fraction
That the asymptotic limit is 1 shows that, whatever volume there is in a high-dimensional hypersphere (and there isn’t much), it is ever more concentrated at its surface. Also, since, for all R ,
inscribe the hypersphere in the n -dimensional hypercube of side 2 R and we see that most of the hypercube’s volume is concentrated at its corners.
Our discussion has concentrated on a few particular areas of the counterintuitive nature of hyperdimensions and we could mention many other manifestations and implications: the full story is big enough to fill books (and has done so). To
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