The King's Evil Test

Kids are using the Internet to chat and play games endlessly.  There is, however, a vast treasure trove of information that can be a fine teacher for youngsters.  Some of the best teaching aids can be found in geometry.  Since Geometry is so visual, it lends itself to the spectrum of the world as contained on the Net.

One of the most interesting shapes that we use in our everyday life is the Hexagon.   It is used in chicken wire for example. We've all seen it in the honeycomb.  Somehow bees over evolutionary time have adapted to using only the hexagon for their hives.

Other examples occur all over.  Shown above are crystals of basalt that grow in hexagonal cylinders.

Of course the Hexagon is the basis for the famous 3 dimensional Geodesic Dome shown above that is so strong.  It is constructed by joining the edges to form triangles as is shown below

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So what's so magical about this shape and why does nature so like it?  The reasons are more complex than you might imagine.  So let us imagine!

The Mason's Task

Our story begins 3000 years ago in the realm of a great and eccentric King who lived in the Badakhshan region of Afghanistan.  He and his ancestors were ruthless.  He wanted to have the best and most intelligent stonemasons to continue to build the Kingdom of his fathers and their fathers.  The King devised a clever test for his masons.

The King said:  "Dear honoured Mason, please make me a tile pattern that covers the floor of my  bed chamber.  Make it out of Lapis Lazuli from my precious mines.  The tiles should be about the size of the palm of my hand, dear craftsman.  I hold it out to you, so measure it well."

I want you to use solid gold from my treasure room instead of grout.  Use enough gold to separate the tiles uniformly to a thickness of 1/8 inch. 

Whatever gold you have leftover, you can keep.  I've placed enough in my treasure room for your needs, if you select well.  If you  are wise, you will leave my service as a rich man.  If you do not select the right pattern, you will be cast out of my Kingdom and made to climb the great mountain range to the east known as the Karakoram

The mason consults his friend the magical and wise Queen Bee and he uses the pattern shown below that uses the least gold

The best floor covering pattern

The conjecture that this was the pattern that uses the least material was proven only recently.  So in our fable, how did the King know how much gold he should leave in the room to produce the right pattern? 

Well, you see like the Queen Bee the King's ancestors had been applying this test again and again over eons.  As the Masons guessed better and better, the King would reduce the gold and banish the masons.  In the end he got the wisest masons, because before the imperfect masons left for the Karakoram, they told their apprentices, what they had tried and how much gold they lacked to produce the floor of the King's Chamber.  Just like the Bees, they learned over time.

It took many years to prove that this was indeed the perfect pattern to enclose infinitely many unit areas of the same size and type. No other shape will do it.
 

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