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Originally Posted by Robh
Hi Craig and all,
For counter-intuitive, you can't go past this one from the mathematicians.
It is possible to take a solid sphere S and divide it up into a finite number of pieces and then reassemble them (using only translations and rotations) into two identical copies of the original sphere. This is known as the Banach-Tarski Paradox.
In fact, ignoring the original sphere's centre, it can be divided into precisely 4 pieces S1/S2/S3/S4 that can be re-assembled into two spheres S1+S2 and S3+S4 each of the same volume as the original.
A more impressive statement of the theorem says that it is possible to take a solid sphere the size of a pea and divide it into a finite number of pieces that can be reassembled into a sphere the size of the Sun.
Regards, Rob.
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Rob;
I'm havin' trouble with this one, too.
There's some tricky wording around how to do this …
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Originally Posted by Wiki
The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points.
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So what exactly is an infinite scattering of points if:
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Originally Posted by Wiki
... the original ball is decomposed into a finite number of point sets
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I guess I need a refresher on my set theory definitions …
Cheers
PS: Clearly a pea and the Sun have different masses. So, there's some trickiness around what these 'points' actually are. Seems to me, this maybe one which exists in set theory, but may not necessarily have a one-to-one mapping with the physical world