## Thursday, November 24, 2011

### The Power of Sums: Arithmetic

The first secrets of operative masons were those of arithmetic and geometry.  Today, these hardly seem like subjects that could generate secrets, but when you begin to see all the practical applications that are possible through mathematics, and when you realize that most people at the time only understood basic sums, they do begin to appear nearly magical.  Through the use of arithmetic masons could lay out plots of land, raise huge arches, domes, and flying buttresses, and bridge wide rivers.  The methods they used were closely guarded secrets:  Knowledge was truly powerful.  As the knowledge of mathematics spread, the physical applications lost the aura of mystical secrecy.  But in their place, more cerebral mystical uses of arithmetic sprung up, such as magic and mystic squares.
While many of these arithmetical arrangements have a sort of whimsical aspect to them, at the same time they can be used to draw astonishing conclusions and patterns.  One of the more interesting arrangements is called Pascal's Triangle.

To create a Pascal's Triangle (or a Khayyam Triangle, if you happen to live in Iran), you begin with the number 1 at the top.  The numbers in the succeeding rows are each a sum of the number above and to the left, and the number above and to the right.  In this way the triangle can grow infinitely large.

1
1      1
1      2     1
1      3      3     1
1      4      6     4     1
1      5     10    10    5     1
1      6     15    20    15    6     1
1      7     21    35    35    21    7     1
1      8     28    56    70    56    28    8     1
1      9     36    84   126  126   84    36    9     1
1      10    45  129  210  252  210  129   45   10    1
1    11    55   174  339  462   462  339  174  55   11    1
1    12     66   229  513  801  924  801  513  229  66   12    1
1     13   78   295  742 1314 1725 1725 1314 742  295  78  13   1
1    14   91  371  1037 2056 3039 3450 3039 2056 1037  371 91 14  1
1   15  105 462 1408 3093 5090 6489 6489 5090 3093 1408 462 105 15  1

All sorts of interesting patterns start to emerge from this exercise; some of them simple, some of them very complex and rather difficult to understand.  For example, if you were to sum all the digits in any given row, the sum would be twice the sum of the digits of the preceding row.
The external diagonals contain only 1s.  The next diagonal in contains the natural numbers in order.  The next one is made of the triangular numbers in order.  (Triangular numbers are the numbers of objects that could be stacked to form a perfectly equilateral triangle).   The next diagonal contains the tetrahedral numbers. (Numbers of objects that can be stacked to form a tetrahedron, or three-sided pyramid).  The next diagonal is made up of the pentatope numbers.  In this way, the Pascal Triangle becomes a representation of the increasing complexity of dimensions:  The outermost diagonal represents the 0 dimension - the point.  Moving inward, you see a line - the 1st dimension - a 2nd dimensional shape, a 3rd dimensional form, and onward an upward.  Thus the triangle - a Masonic symbol of completeness and perfection, used by the Egyptians to signify the origin of all things, and the Christians to represent God - represents the whole of creation; what we can see, and what is far beyond the scope of our senses.
Also interesting is the fact that if you were to colour in all of the odd numbers, the resulting pattern is a fractal known as the Sierpinski Triangle.  A fractal like this is another perfect symbol of infinite growth and capacity.

Another interesting attribute of the Pascal's Triangle is that if you were to draw shallow diagonals through the triangle, at about 30°, and sum the digits on these lines, you arrive at the Fibonacci sequence:  A series of numbers that are related to the Golden Mean, and are found ubiquitously throughout the natural world.  In fact the Fibonacci numbers have often been used as an example of the theoretical underlying mathematical structure that describes and builds the entire universe.
One triangle provides a wealth of symbols and insights, and provides more with greater depth of study and insight.  It is no wonder that the ancients saw arithmetic as one of the great Keys and Sciences.