Category Archives: mathematics

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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paul lockhart: a mathematician’s lament

I would admit upfront my biases (or reflections on my personal experiences, if you will).

I am of the opinion that almost all Christian missionary schools (or for that matter, schools belonging to any other denomination or abomination that is centrally organized) are really a bad idea, at best. And, of course, I must have gotten ‘educated’ at a particularly bad missionary school – where, how actually Christ lived (by example and with compassion) was far removed from the context of the new testamentish conversion fervour with which the school was run. In fact, Jesus would have actually run away from such pretentious schools which seek to belittle other cultures, faiths and individuals. I also learned that, while these schools did all the damage they could with gay abandon, they were also getting considerable financial support /  aid  from the Government, to boot! This should have been in addition to the foreign funding that they routinely got to harvest more souls.

Till that time, I thought – only white Europeans were contemptuous of the black and brown natives & were endlessly condescending (‘The whiteman’s burden’). But it was with a sense of shock that I realized that a few of our own folks were belittling our own folks! It was a good 5 years later (after I finished my 10th std) that I got to know about the great TB Macaulay and allied colonial machinations. Anyway…

The mathematics that was ‘taught’ in my school was actually really silly – but then our textbooks were also silly so I cannot blame the teachers or the school. But I still recall with pleasure, the fact that a few of my classmates and I would huddle together in the back benches and feverishly solve the math text (of Tamilnadu and NCERT books) of the next few academic years and this was lovely. Of course whenever we were unfortunate enough to get caught in the act of not sticking to the 7th standard book by our math teacher (‘kanakku master’), our ears were boxed and our hides were tanned… But doing interesting math sums was an exhilarating idea, even though the canings were very painful. And ah, sometimes someone would bring Martin Gardnerish puzzles and solving them would be heavenly. Remembrance of things past and passé, what else…

Coming from this slightly shady background and having an endless angst about the way Science, Math, History etc etc are handled in most schools, I am convinced that the children have to get exposed to math (and of course every other ‘subject’) in the loveliest possible, fascinating and connection-rich way. I have faith in the children that they would instinctively gravitate towards the best things in life (given a set of meaningful choices) and may be at least some of the nammashaale erdkinder will continue to find math beautiful, well into their adulthoods… The hope!

And so, it was with pleasure that I got this document from Sunder & Sonati and read it again! I recall that this document was doing the rounds in 2003 or thereabouts on Internet math forums and I had chanced upon it earlier over IRC.  It was lovely. But I thought it was mainly a diatribe against US schooling system. And, now I reread it and it continues to be lovely and a very well formed document – and I think it is applicable to the most of the whole world, except perhaps the Russian schools…

Some extracts from the text:

“There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract.”

“TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about.”

chuckle, chuckle…

Please read the EXCELLENT document of Paul Lockhart:  ‘A Mathematician’s Lament’ off the Mathematical Association of America website:

http://www.maa.org/devlin/LockhartsLament.pdf

Enjoy! It is well worth the investment…

martin gardner, rip & calculus made easy!

The Annotated Alice‘ of Lewis Carrol and Martin Gardner’ was (finally) returned a couple of weeks back by Rama and I was fondly leafing through it, before sentimentally returning it to the library shelves. It is currently rubbing shoulders with the books of the likes of  Isaac Asimov, JBS Haldane, Erwin Schrödinger, Enrico Fermi, Paul Dirac et al and should be feeling happy now; what a work of deep scholarship!

Rest in peace, Martin. You lived to a ripe old age of 96 and also did a great job of living, all the while!

Having thoroughly enjoyed (actually a lame word like ‘enjoyment’ does begin to describe the pure exhilaration one feels studying a Martin Gardner or a Douglas Hofstadter or a Richard Feynman) ‘The Annotated Alice’ among many other works of Martin, I am reminded of that 1910  gem ‘Calculus Made Easy‘ of Silvanus Thompson which was later updated and edited by Martin in 1998. ( I just realized that this classic, a real classic at that, has completed hundred years of its existence!)

Now, what is great about the book? One may feel, after all, the phantoms of differential and integral calculus  don’t trouble me anymore – so what’s the point? Besides, I got a good grade in Math 101 (also in Math 505) – I am in a cushy job with an MNC as an ‘engineer extraordinaire’ spending my time (and earning my megabucks) in daylong meetings, boring conference calls & excruciating powerpoint presentations –   and so, why the hell do I even need to go through that drivel again…

I would say that  you have to read this because as the book says (and delivers on the promise, faithfully):

Calculus Made Easy: Being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the Differential Calculus and Integral Calculus

I would say that the book is indeed beautiful – it restores your faith in the pursuit of knowledge. That Science and Math are not pointless. That they are creative. That they are actually fine arts. That they also happen to have real life applications – gazillions of them!

Now, I ‘studied’ in one of the well-known schools/colleges (which ought to know better, siddhir bhavati karmaja (chapter #4 of the bhagavat gita and all that), but I really wonder as to how this book was not used at all in our undergraduate years! Not even a passing mention of the book was made!! (But I should remember with gratitude that the physics department of my school indeed used the delightful Feynman Lectures on Physics – so it was not all gloom)

I really feel that Mathematics HAS to be approached via ecstatic books such as these.

I chanced upon the Thompson book on calculus when I was trying to desperately to understand & solve some practical problems of heat transfer in the wasted days of my entrepreneurship – and I was thoroughly bowled over by this incredible book. Really. There were also other books (by Piskunov et al) that I really began to appreciate subsequently – but all this was some 10 years after I graduated(!) from my alma mater.

Believe me, this book would make mighty sense to a reasonable 12 year old or even younger ones – if the mind is prepared. Hence, given half-a-chance, I would plan to sneak this in to the erdkinder’s minds. Wish me good luck.

Here’s a scanned picture of a page of the book!

This is the title page of the St Martin Press edition (1998)

The original Macmillan version of the book without Martin’s contribution is available in the public domain. While it is not the same as the later  St Martin’s version – it is STILL a great work.

Enjoy! Math is actually fun! Calculus definitely IS.

Children are like sponges. Their concept of beauty is still unspoilt. Their cognitive capabilities are still good, in spite of TV, pointlessly obscene birthday bashes and Helicopter parents. They normally & instinctively would gravitate towards (and absorb/internalize) fine things in life, given a set of meaningful choices. Faith? Hope?? Let us see…

doubteronomy and numbers

This 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…

o-o-o-o-o-o-o-o-o-o-o-o-o

 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 Doubt can 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 x2 + 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…) 

o-o-o-o-o-o-o-o-o-o-o-o-o

Rama also happens to be the grand duchess of the school, in case you have doubts. Surprised? Please note that there is even a quote in the text, by the sonnyboy of la grande mademoiselle Montessori herself, to prove the point! QED.

ps: sorry about the laboured pfun on some ‘old testament’ stuff – in the title of the post…