Quote:
Originally Posted by CraigS
Rob;
I'm havin' trouble with this one, too.
There's some tricky wording around how to do this …
So what exactly is an infinite scattering of points if:
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Craig,
The concept challenges the way we think about volume. When we calculate volume, we give it in terms of so many m^3, cm^3 or mm^3. But why stop there? We could give it as so many nanometres^3 or picometres^3. In fact we could give it as the number of atoms of hydrogen gas that make up that volume.
I think the idea is that the sphere is considered to be made up of an infinite number of lattice points rather than a finite number of individual atoms. Analogous to the concept that an interval consists of an infinite number of points rather than a finite chain of atoms.
In that sense, the sphere could not be physically divided into 4 pieces (ignoring the centre) that can be reassembled into two identical spheres as the original. Simply because the sphere cannot be physically divided down past the size of an atom. It is probably best to think of each divided piece as a cloud of an infinite number of points scattered in the lattice of the sphere.
From my understanding, the paradox says nothing about the shape of the pieces. They may appear as a random framework of an infinite number of points that are loosely connected. However, remember also that physically atomic nuclei are actually strung out in a lattice with spaces between them.
As a theoretical possibility, the concept of division of a solid sphere into a finite number of pieces and then reassembly into two copies of itself is still extraordinary.
Regards, Rob