We all know that the volume of a cube with each side = 1
is equal to 1. That's because volume = 1^3 = 1.

In fact, the volume of any higher dimensional hypercube
is also 1 because in any dimension n the volume = 1^n = 1.

Example, the volume of a hypercube of 100 dimensions is
1^100 = 1.

But what is the volume of a 100D hypersphere of radius = 1
that is inscribed within this 100D hypercube?

Answer: π^50/30414093201713378043612608166064768844377641568960512000000000000

= 2.3682021018828293*10^-40

= 0.00000000000000000000000000000000000000023682021018828293

Why so fucking small, when the volume of the containing hypercube
is 1.0?

In fact, as the dimensions increase further, a hypersphere of radius=1
has a volume that approaches zero.

How can this be?

Consider a hypersphere of radius = 1 mile. As the dimensions increase
the volume will approach 0 cubic miles (i.e. nothing, zip, nada).

But the containing hypercube will always have a volume of 1 cubic mile.


Oblinux:

Maxima CAS was used to do the exact calculations.


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On Thu, 2 Jan 2025 16:55:28 -0600, Physfitfreak wrote:

Sounds like something one gets tempted to ask AI about it.

IMO, that's a very bad attitude to take. One should not
always immediately run to AI as if it is a panacea.

But that's just what the commercial vendors want you to do.





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On Thu, 02 Jan 2025 16:55:28 -0600, Physfitfreak wrote:

Sounds like something one gets tempted to ask AI about it.

There's no need for AI (even it could give an answer).

The answer is in the "corners."

A hypercube of n dimensions has 2^n corners, and it is these corners
that contain most of the volume.

An inscribed hypersphere will always touch each face of the cube but
it always curves away from the corners.

Furthermore, the distance from the center of the hypercube (and its
inscribed hypersphere) to each corner is sqrt(n) * r, where n is the
number of dimensions and r is the radius of the hypersphere and also
the length of 1/2 side of the hypercube.

Thus as n increases, so too does the distance to each corner from
the center of the hypersphere.

N-D spaces are fascinating places.

For n=100, the cube has 2^100 corners with each corner located
10 units from the center of a hypersphere of radius = 1 unit.

That's LOT of hypervolume outside of the hypersphere but still
inside the hypercube.


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On Fri, 3 Jan 2025 20:21:12 -0600, Physfitfreak wrote:

Yes, makes sense. There'll be more corners and therefore a smaller
volume for the hypersphere. Interesting. But for some reason I feel I've
got other, much more down to earth, facts of life entangled with me.

The space in which we exist is 4-D.

But for some idiotic reason, the educational system still instructs
students that we exist in 3-D space. Only those choose to become
scientific specialists later in life actually learn the truth.

For instance, when I boot and out of blue, nothing comes up on the
screen, but computer lights indicate normal activities, is it the fault of:

Monitors don't need input to be diagnosed. Monitors have
buttons or switches located somewhere to allow configuration.
If the monitor is functional, those buttons should produce
a display on the screen.

Maybe it is just that the wrong input has been selected,
e.g. VGA instead of HDMI.

Get a copy of the monitor manual and find out how to enter
the configuration.

Things of this nature. Hehe :)

Doesn't go well with n-dimensional hyperstuff.

There is more to life and living than laboring for
immediate practical ends -- or at least there should
be. Otherwise we are no more than slaves.

Recreational mathematics is a fantastic pastime, and
GNU/Linux provides the tools for glorious exploration.





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On Sat, 4 Jan 2025 07:39:32 -0000 (UTC), RonB wrote:

What part of this is NOT off topic?

The part in which you inserted your boorish, ignoramus ass.

That's what part.



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