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by H. PeterAleff

  

 

 

  

Footnotes :

 

 

 

36 A long but still only partial list of descriptions of this triangle prior to Pascal appears, for instance, in David Fowler: "The Binomial Coefficient Function", The American Mathematical Monthly, January 1996, pages 1 to 17, see particularly pages 10 to 13.

 

 

 

37 As I will argue in the next Volume.

 


 

 

 

  

 

  

  Volume 1: Patterns of prime distribution

 

in "polygonal - number pyramids"     

 
 

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Number "pyramids"

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1.4.2. Polygonal-number "pyramids"

Now take these useful arrangements of the natural numbers and unwrap the bent gnomons into straight layers; then write out the dot- represented entries as numbers, and you obtain what I will call polygonal- number "pyramids".

That name is technically inaccurate because the resulting arrays are triangles, as, for instance, in the well-known and thoroughly misattributed36 "Pascalís Triangle".

However, I refer to them as "pyramids" instead, partly to avoid confusion with the triangular numbers that build up the first of these arrays and help in structuring the prime patterns in the second, and partly to acknowledge the probable priority of the ancient Egyptians who seem to have used the first pair of these two- dimensional number arrays as inspiration for the cross- sections of many among their three- dimensional stone pyramids37.

In the hypothetical scenario I propose, some early number investigator was curious about the distribution of primes in successive segments of the number line.  S/he reasoned that when one cleaves said line to look at this distribution, the lengths between the cuts need to be neither arbitrary nor equal.

To the contrary, the ancientsí emphasis on ratios and proportions, rather than on absolute values as in modern times, would probably have prevented our ancient sage from slashing that line into the Procrustean same- length slabs from which Eratosthenes would later assemble his square.

Our scribe lived in a culture, such as Old Kingdom Egypt, that valued skilled craftsmanship and thoughtful respect for the properties of a material. The appropriate thing to do may therefore have been to observe and take advantage of the natural grain in the material to be worked on, whether it was wood or stone, or even non-material numbers.

To our scribe who looked for patterns in the divisibility of numbers, two of the textures among numbers might have appeared as the natural and most obvious guides along which to order their line into successive stretches that keep growing together with them. This is simply because division is the reversal of multiplication, and multiplication, in turn, is repeated addition.

1.4.3.  The triangular- number "pyramid"

The first of these textures is thus formed by the successive sums of the natural numbers, the inherent benchmarks of their accumulation from putting them together. Each interval between two such consecutive sums includes one more entry than the preceding one. The numbers in that interval form therefore a naturally defined set that differs by only one increment from its neighbors.  They can therefore be systematically compared, with the same common- sense approach of changing one parameter at a time that we now call scientific.

To make the comparing easier, our postulated scribe stacked these so obtained number line segments to view them side by side instead of end- to- end. Accordingly, s/he wrote the numbers, beginning with one entry at the top, two below, the next three numbers in the third row, and so on, one more in each next layer so that each nth layer was n numbers long.

The last entry in each layer, the trailing- edge "cornerstone" of the pyramid "masonry" that completes a new triangle in the array, was and is the same as the quantity of pebbles it took to construct the corresponding dot-figure, so it is a triangular number.

Each of these is the sum of all the preceding lengths, or (n2 + n) / 2 where n is the ordinal number of its layer.

Some scribes may well have arranged those layers in the form of a one-sided stairway to fit all entries into the same simple grid, as shown in Figure 5 below.

Triastai.gif (43505 bytes)

Figure 5: The top of the triangular- number stairway, and its alignments of primes

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view vector version  135 Kb with the free vector viewer plug- in from
autodesk.com/cgi-bin/whipreg.pl


 This array shows several initially continuous strings of primes that go through alternate layers at an angle of two up and one over and repeat their entries again at slopes of two up and three over, then five over, and so on at steadily shallower angles with increasingly stretched- out entries.

Our scribe, however, centered the line segments to better view the strings through the even- numbered layers separately from those through the odd- numbered ones, and maybe also because s/he preferred the quasi- symmetrical look that was a hallmark of ancient Egyptian art.

The result was the array in Figure 6 below

Triapyr1.gif (43648 bytes)

Figure 6: The same number- line segments as in Figure 5, but stacked symmetrically.

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In this arrangement, the steepest of the above prime strings are now columns through alternate layers, and the angles of their repetitions have shifted to become 2/2, then 2/4, and so on. 

Triapyr2.gif (79133 bytes)

The poster at left offers a slightly expanded version of this "pyramid" for the numbers down to 2643.

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view vector version   492 Kb

Because the layer lengths alternate between odd and even numbers, the individual entries in each next layer are offset by half a number width from those in the preceding layer.

This yields the picture of a steep pyramid, built from overlapping blocks, in which every other pair of neighboring columns through the alternating layers contains only even numbers, and then the next pair only odd ones.

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