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 π

r^{2}. The volume inside a sphere is 4/3πr^{3}. 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 inndimensions?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 learnedn-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 …

doubteronomy and numbersThis is a reflective piece written on ‘doubt’ by a NammaShaale parent and adult, Rama.

Thanks Rama, and keep ’em articles/essays coming the blog way…

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Doubt(Rama)I had been meaning to write for a long time now. When I did mention the idea of writing to some they always said, “But where do you have the time!” and that’s just what I want to hear. Anyways, here I go. I plan to keep to it but let me see how long.

Yesterday my sister and I watched the film Doubt. As part of the post film discussions we realized that

Doubtcan be a powerful emotion.Doubt is a good thing I’m sure because much enquiry comes from doubt. Men (and women) have once upon a time sinfully doubted if the earth was the center of the Universe.

Only last week in class I gave bunch of 9 and 10 year olds the presentation of measuring the internal angles of triangles, quadrilaterals and polygons. We measured the angles of an equilateral triangle and saw that they added up to 180 degrees. Now, I was surely not going to give away the secret here but even if I did it would be completely “doubted”. So the children saying, “I doubt if it would be so for an isosceles triangle or a scalene triangle!”, “what if the triangle had an obtuse angle?”, “what if it was a larger equilateral triangle?” set out to measure the angles of many, many triangles and other shapes as well. The results are yet to be arrived at.

But I have many times in the past seen on their faces the joy of discovery, the joy of clearing a doubt.

The joy of seeing that the sum of internal angles of a triangle is always 180 degrees! There are always a pi number of diameters in the circumference of a circle! An inscribed square is always half a circumscribed square (I doubt if this works for all quadrilaterals, need to check out!)

In an elementary class the discoveries go on to – multiples of 9 always add up to 9, the square of a decanomial is the sum of its cubes, hot air always rises; light always travels in straight lines; words that end with ‘c’ and are occupations or hobbies are always end with the suffix –cian, monocotyledonous plants always have parallel veins and flower parts in threes and multiples of three… I could add one everyday!

The knowledge acquired is impressive but what matters to the child is the joy each of these discoveries gives him because he builds his very personality with these discoveries. As Mario Montessori says,

“When the elementary child is given a vision of the order of the universe he constructs the inner order of his personality through experiences in a structured world. Inner order is necessary to be able to see meaning in one’s existence, to find one’s identity, to achieve independence, and to act in a meaningful way.”Last Saturday I spent some blissful hours doing a few higher algebra activities with the cubing material. I was doing (x + 2) (x + 1) and I did see in the book that it should result in x

^{2}+ 3x + 2. But I doubted it! I did a good ten variations of x – 4, 7, 8… and saw that it worked always! Believe me it was most joyful!!!Doubts and disbeliefs are plenty but predictions and certainties are way more! What can be more joyful than ¼ always being 0.25! (But in one of the presentations a child did say, “I doubt if this would be so in base 6…)

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Rama also happens to be the grand duchess of the school, in case you have

. Surprised? Please note that there is even a quote in the text, by the sonnyboy ofdoubtsla grande mademoiselleMontessori herself, to prove the point! QED.ps: sorry about the labouredpfunon some ‘old testament’ stuff – in the title of the post…