Of late, the usual seriousness, barbed wire sarcasm and acrobatic arguments were missing in Bystander’s conversations .He could not explain this sudden change in Bystander’s attitude and this had Muser in a spot of bother. Having likened him as his alter ego, the onus was on his mind to find out the reason for this change. He decided the best way to resolve this enigma was to recollect their recent conversations and ferret out any hidden clue that might lead him to the solution.
The first semblance of a clue emerged from the recent conversation about geometrical objects. Muser patiently rewound that conversation in his mind which at that time gave him a feeling as though his friend was trying out something not natural to him.
“Do you know, the geometrical shapes that we see around us could be constructed using certain other shapes, which I would like to call complementary objects?” said Bystander.
Perplexed, Muser asked, “How come you have developed this sudden interest in constructive geometry? I know during your academic career, you willfully bunked mathematical classes on theorems and drawings!” Where from you got this term ‘complementary shape'?”
Clubbing both the questions, Bystander answered,“You mean to say that I will forever carry on these disinterests from school days? I just borrowed the term normally used to describe colours!”
Muser hastened to add, “I did not mean that but was only trying to understand the basis for this new interest that you have developed about shapes”
“Blame the football, if you want. That only made me to think about different shapes and how one could go about constructing them. For example a football or a sphere can be shaped with pentagons and hexagons”, explained Bystander.
“What is special in that?” asked Muser to keep his friend talking.
“The specialty is that a pentagon itself can be constructed using a trapezium and a triangle. The hexagon is friendlier; a rectangle and two triangles will do the job”.
“If you have done so much thinking, what about enlightening me about the whole process?” asked Muser in a teasing voice.
“Depending on the diameter of the sphere you want, select the number of pentagons and hexagons. There is a relationship between the numbers and the area ratio of pentagon and hexagon that are to be used”
“Assuming that I have the needed numbers of each shape, can you describe how to construct the ball?”, Muser persisted with his line of questioning.
“Place one pentagon at the pole of the proposed sphere; surround it with five hexagonal shapes. Next, position one pentagon, each, in the empty space between two abutting hexagons. Now you have completed 75% of the top hemisphere by using six pentagons and five hexagons, in all. Repeat this process to form the other pole and the unfinished hemisphere”.
Getting impressed and unable to restrain his enthusiasm, Muser continued “How you will close the gap between the two hemispheres that are ready?”
Sensing admiration in his friend’s voice, Bystander continued “Another ten numbers of hexagons are placed, in a slightly zigzag fashion, over an imaginary equatorial line, such that the open spaces between pentagons and hexagons forming the two hemispheres are closed. Muser tried to retrace these steps, in his mind, with twelve pentagons and twenty hexagons. He could easily visualise foot ball or a sphere taking shape.
At the end he had a number of questions to ask Bystander: “why this sudden interest in constructing a foot ball in this fashion? Why this cannot be made using only pentagons or hexagons? What will happen if the selected size of the pentagon is bigger than that of the hexagon used?”
“The answer to your first question is that I recently bought a toy football for my grandson and before he asks me how it is made, I thought, I would find it out now itself! For your second question I have only a partial answer: Using only pentagons, you may get an oval shape like a rugby ball. I don’t know what shape it will take if only hexagons are used.”
Continuing, Bystander said “You may have to wait for some more time to get an answer for the third question. I am not stopping with this but will try to find out a way to deal with other geometrical shapes using triangles, rectangles and squares as simple building blocks.”
It became clear to Muser that Bystander has been bitten by a ‘shapely’ bug and nothing will divert him, at least for some time, from this monomaniacal attention to shapes. May be he has now found out a way to complete that unfinished portion of his learning curve!
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