Connecting the dots, from geometry to combinatorics and more

Hello mathematician!

I was playing with a square. What if I glue the opposite edges of the square?

If I glue one pair of edges, we get a cylinder. (Ah! Are you wondering about the rules of ‘gluing’?)

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Next, if we glue the two ends of that cylinder we may get a torus or a Klein bottle (depends on how we glue them). This discussion came up in one of our sessions. We wondered about the essence of this ‘gluing’ process.

Once we decide on the rule of gluing, we declare the glued points to be equivalent and get cylinder, torus, Klein bottle at various stages of the process.

Notice that we have so many things happening at the same time:

• gluing leads to the notion of equivalence relations which in turn leads to partition functions in combinatorics (Stirling numbers for example)
• rule of gluing leads us to vector maps and the notion of functions. After all, we have Klein bottle or torus depending on ‘how we glue’.

At once we are prompted to connect geometry and combinatorics. 

Solving problems is one thing. Connecting seemingly different ideas is an entirely different business (though they have dependencies). It is extremely important to pause and let our imagination do its magic every now and then.

In this context let me quote two of the greatest minds of our times.

Thurston in Clay Research Conference (2010), started off with a very curious statement:

“A lot of mathematics is really about how you understand things in your head… we are not just general purpose machines, we are people, we see things, we feel things, we think of things,…, there is something significant about how the representations in your head, changes … profoundly changes how you think.”

Thurston drew different pictures (of 3-manifolds) than what his peers did. That is because he imagined things differently. And he was brave enough to follow his own imagination. It takes courage to not to parrot what the ‘system’ is teaching you.

It takes courage to let your imagination guide you. This is especially true in this age of rat race.

Rabindranath writes in Jibansmriti,

“বাহিরের সংস্রব আমার পক্ষে যতই দুর্লভ থাক্‌, বাহিরের আনন্দ আমার পক্ষে হয়তো সেই কারণেই সহজ ছিল। উপকরণ প্রচুর থাকিলে মনটা কুঁড়ে হইয়া পড়ে; সে কেবলই বাহিরের উপরেই সম্পূর্ণ বরাত দিয়া বসিয়া থাকে, ভুলিয়া যায়, আনন্দের ভোজে বাহিরের চেয়ে অন্তরের অনুষ্ঠানটাই গুরুতর। শিশুকালে মানুষের সর্বপ্রথম শিক্ষাটাই এই। তখন তাহার সম্বল অল্প এবং তুচ্ছ, কিন্তু আনন্দলাভের পক্ষে ইহার চেয়ে বেশি তাহার কিছুই প্রয়োজন নাই। সংসারে যে হতভাগ্য শিশু খেলার জিনিস অপর্যাপ্ত পাইয়া থাকে তাহার খেলা মাটি হইয়া যায়।”

Rabindranath was skeptic about ‘external tools’. Even in his other works in literature and pedagogy, this theme is iterated over and over again: let your internal imagination bring in happiness. Do not worry too much about the tools.

Pause for a moment, Look outside the window. Even the sun does not rise at the same position in the eastern horizon.

Ashani Dasgupta

Passion for Mathematics