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Information and Dimension - Are They Related?

Information and Dimension - Are They Related?

This is an extreme example, but it shows there may be a fundamental relationship between the availability of information and the concept of dimension.

Sitting on your desk is a sphere. As in mathematics, let's assume an axiom - your sphere is an information void. What's that? It's a closed surface which allows absolutely no knowledge of its interior. The interior of the object is off limits to any form of investigation. Think of its surface as an information barrier that stops any information about the interior from getting out.

So what properties does your sphere have?

You can't break it open to look at the interior, that would violate the axiom, so the object must be infinitely strong.

You can measure its diameter - just use a ruler. You can measure its surface area - just take a small unit square and see how many times you can paste it on the surface.

What about volume? Be careful, knowing volume means you have information about the interior and that's impossible. Of course you can use the formula volume=(4/3)*pi*r^3 where r is the radius. But this is not a measurement of the interior, so the best you can do is call this volume the "external volume". You have no knowledge of the geometry or topology of the interior, so the "internal volume" could be very different!

Now, if you can never have any knowledge of the interior you come to a simple conclusion, the object is totally defined by whatever is on its surface. The object certainly looks 3D but can be fully described as if it was 2D. Interesting object!

Your sphere has no reality inside - just like bubbles in a liquid have no liquid inside. If you own one please handle it carefully!

Content written and posted by Ken Abbott