Crystal Spirit




Blue eyes hide a myriad of worlds.                                                                                                                                                                  Indian artifacts flow from nimble hands.                                                                                                                                                     Proud spirits of ancient chiefs keep watch.

She is a lady:  tall, noble, elegant.                                                                                                                                                                 She loves words, the magic of their sounds.                                                                                                                                             Colors are her joy:  red, blue, green…

A crystal of light shimmers                                                                                                                                                                         and traverses space on a cloud of thought.



How A Handful Of Pebbles Contained The Universe, Part 1.

For thousands and thousands of years, people had only two words for quantities:  “one” and “many”.  Even ancient languages reveal this fact.  When people saw an aggregate of stones, they referred to them as “many”.  And when they picked up a single stone they used the word “one”.  “I have one stone not many stones”, they would say.  It must have seemed obvious to them.  Little did they know that by using these terms they were blocking off the awesome universe which surrounded them.

At some mysterious point in time, someone must have picked up the stones individually and wondered how to designate them.  Perhaps, one person wanted to exchange his “many” for another person’s “many” in a business matter.  How could each determine that he wasn’t cheated except by counting.  Stones, then, were no longer simply a collection of “many”, but had special identities or designations.  Thus, from such primitive beginnings the world of number was born.  The simplest numbers uncovered were the “counting numbers” or “natural numbers”.  The numbers 2, 3, 4, 5, 6, 7, 8, 9 came into being, which followed 1.

The Babylonians and Egyptians needed numbers to measure fields and the shapes of the fields necessitated some rudimentary elements of geometry for buying and selling properties.  However, it was the ancient Greeks that studied number as something that had its own existence independent of human needs.  Indeed, to Pythagoras, all things were numbers.  Men and women were numbers and he assumed there were only ten heavenly bodies, since 10 had special significance.  In fact, Pythagoreans worshipped the “tetrakys”, a triangle composed of four rows of dots.  The first row had one dot, the second two dots, the third three dots and the fourth four dots.  When the four rows were added, the sacred 10 was the result.  Everything appeared to represent perfect balance and harmony until…  Yes, even in this well-constructed world, irrationality stuck out it’s ugly snout.  Someone constructed a square with sides one.  When a diagonal was added, the length of the diagonal had to be the square root of 2.  Pythagoras’s own theorem led to the unhappy result.  The irrational could be dealt with later.  But there was still one important number missing:  0.

The Greeks never could find 0, and this fact imposed strict limits on what they could do with numbers.  For zero, we have to go to another country, India.  The Indians had long had a concept of nullity.  It came from their philosophy.  It came from their religion.  When “0” joined the counting numbers, a major step was put in place for solving equations, the construction of the Cartestian plane, and, in today’s world, the binary system which is the basis of computer circuitry.

In the Middle Ages, the first algebraic equations were born, arising from Arabic countries.  The mysterious x and y of algebra represented an abstract way of thinking hitherto unknown in mathematics.  The Greeks may have been the philosophers of number, but the Arabs were not only philosophers, but active participants in extending the range of number to greater practical and theoretical heights.  However, algebra and geometry were still separated.  It required a major step to bring them together.