bumbling ball in a box and humbling math…

Let us say that we want to measure the volume of a sphere, fitting snugly  in to (or ‘bounded by’) a cube. It is like – we are putting a tennis ball in a ‘cubical’ box – wherein, the diameter of the ball is almost the same as the length of one edge of the box.  Simple, eh? Very easy to visualize…

But is amazing that, the volume of the ball could only be a very very very verrry small fraction of volume of the cube.

You may ask how,  you silly ol’ man?

But apparently it is so in dimensions higher than 3. Ta Da!

Brian Hayes has written a delightful essay on the subject – called ‘An adventure in the Nth Dimension’ – please savour it, if you can!

The area enclosed by a circle is πr2. The volume inside a sphere is 4/3πr3. These are formulas I learned too early in life. Having committed them to memory as a schoolboy, I ceased to ask questions about their origin or meaning. In particular, it never occurred to me to wonder how the two formulas are related, or whether they could be extended beyond the familiar world of two- and three-dimensional objects to the geometry of higher-dimensional spaces. What’s the volume bounded by a four-dimensional sphere? Is there some master formula that gives the measure of a round object in n dimensions?

Some 50 years after my first exposure to the formulas for area and volume, I have finally had occasion to look into these broader questions. Finding the master formula for n-dimensional volumes was easy; a few minutes with Google and Wikipedia was all it took. But I’ve had many a brow-furrowing moment since then trying to make sense of what the formula is telling me. The relation between volume and dimension is not at all what I expected; indeed, it’s one of the zaniest things I’ve ever come upon in mathematics. I’m appalled to realize that I have passed so much of my life in ignorance of this curious phenomenon. I write about it here in case anyone else also missed school on the day the class learned n-dimensional geometry.

Go to thearticle, and become ecstatic… Really.

This is how any stuff about popularizing sciences should be.

It is amazing that – kindling our imagination and provoking the curious minds  – are still happening in spite of the best efforts of the Discovery  and National Geographic Channels – not to speak of the other unspeakable channels …

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Comments

  • sunder and sonati  On December 12, 2011 at 8:53 am

    Great article. And however far one reaches, one has learnt something new. I have shared it on Facebook 🙂

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