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by H. PeterAleff

 

 

  

Footnotes :

 

 

 

26 Richard K. Guy: "Unsolved Problems in Number Theory", Springer- Verlag, New York, 1994, page v.

 

 

 

27 That portal was at the entrance to the temple of the Sybil which was the first stop on Aeneasí visit to the realm of the dead. See Virgil, Aeneid VI,30-35.

 

 

 

28 John H. Conway did provide a formula that yields all the primes in sequence, but it involves prohibitively lengthy calculations to be useful.

For a description, see Dominic Olivastro: "Ancient Puzzles", Bantam Books, New York, 1993, pages 20 to 27, and John H. Conway & Richard K. Guy: "The Book of Numbers", Springer Verlag, New York, 1996, see "Fourteen Fruitful Fractions", pp. 147-148.

 

 

 

29 At least according to one scholar, Penelope Reed Doob: "The Idea of the Labyrinth from Classical Antiquity through the Middle Ages", Cornell University Press, Ithaca, 1990, see chapter 8: Virgilís Aeneid, pages 227 to 253.

 

 

 

30 For instance, Melvyn B. Nathanson: "Additive Number Theory -- The Classical Bases", Springer Verlag, New York, 1996, see Chapter 1.1: "Polygonal Numbers", pages 4 and 5. Also: John H. Conway and Richard K. Guy: "The Book of Numbers", Copernicus - Springer Verlag, New York, 1996, pages 27 to 59.

 

 

 

31 Eric Partridge: "Origins -- A Short Etymological Dictionary of Modern English", Greenwich House (MacMillan), New York, 1983 edition, entry for "gnome (2)", page 258 right..

 

 

 

32 Sir Thomas Heath: "A History of Greek Mathematics",  Clarendon Press, Oxford, 1921, edition consulted Dover, New York, 1981, Volume 1, page 79 top.

 

 

 

33 Sir Thomas Heath: "A History of Greek Mathematics", cited above, Volume 1, page109.

 

 

 

34 Sir Thomas Heath: "A History of Greek Mathematics", cited above, Volume 1, pages 76, 77, and 83, 84.

 

 

 

35 Timaeus 55D to 56B, and 55C, as cited by Sir Thomas Heath in "A History of Greek Mathematics", cited above, Volume 1, pages 295 to 297.

 


 

 

 

  

 

  

  Volume 1: Patterns of prime distribution

 

in "polygonal - number pyramids"    

 
 

You are on page

Polygonal numbers  

    0  1   2   3   4   5   6   7   8     + 10   11  12  13  14  15  16  17  18

1.4. Polygonal numbers

On the other hand, Richard K. Guy, a modern number theorist who also chronicles the advances and challenges in his field, reminds us in the preface to his book on "Unsolved Problems in Number Theory" that "Ďunsolvedí problems may not be unsolved at all, or may be much more tractable than was at first thought"26.

For the pursuit of patterns in primes, the number world maze resembles the labyrinth of the netherworld in Virgilís Aeneid.  The portal of the temple at its entrance displayed the entire story of the labyrinth and its tangled design27, carved there by the labyrinth- builder Daedalus himself on a series of panels. 

Similarly, the patterns formed by the primes are most clearly visible at the entrance to the number world, among the first few thousand numbers.  They display there the blueprint for how they will appear in the distance, once the fog of cumulative factors obscures them in part, like the mists and shadows on Aeneasí further path.

Virgilís Sibyl called those carvings mere spectacles, and their numerical counterparts, too, may not give number theorists much help in many parts of their quest: they do not prove Riemann right or produce a simple list of all consecutive primes28, nor do they settle any other great pending conjectures about that elusive group.

However, like the handiwork of Daedalus that told his story and also supplied the structure as well as the central image of the Aeneid29, these spectacles invite contemplation because their series of panels displays the order among the basic components of the number world the way the Periodic Table of Chemical Elements illustrates the structure of the material world.

Moreover, the patterns in those panels at the entrance to the number world link some otherwise isolated prime- producing functions into a coherent system of artfully interlaced paths through their maze.

To observe these patterns, one simply writes the natural numbers, beginning with 1 at the top, in successively stacked layers.  Each of these layers goes from the entry just after a polygonal number to the next polygonal number of the same kind. Let me briefly explain those polygonal numbers before I describe the arrays based on them.

The polygonal numbers are the triangular numbers, squares, pentagonal numbers, and so on.  They take their names from arrangements of number- representing dots in the geometric shapes of the successive regular polygons, or "forms with many sides", that the Pythagoreans introduced to Classical Greece in the sixth century BCE.

Each entry that completes a layer added to such a polygon is a polygonal number of that polygonís type.  You will find some of them illustrated on Figure 4.

Dotfigu1.gif (12768 bytes)

Figure 4: Polygonal numbers as
dot figures, and some simple
visual proofs of their relationships
Gif = 13 Kb

The dot figures of the polygonal numbers show that adding two successive triangular numbers always yields a square, and that a square also results from eight times any triangular number, plus one.

Visualizing the numbers by means of these regularly shaped arrangements is so helpful for learning about them and the relationships between them that many of todayís books on elementary number theory still use these ancient figures to introduce their subject30. Their layered shapes are so simple and so logically constructed, and they speak so well to our intuition, that our speciesí initial introduction to the mysteries of mathematics could well have included them, too.

It is easy to imagine some prehistoric philosophers pondering how to place pebbles so that these formed regular shapes with equally long sides, and then trying to predict how many stones it would take each time to increase that same shape by another layer. Some of them must have solved these puzzles long before anyone recorded their orderly arrangements for our history books.

A possible trace from the probable pebble- layout origin of the Classical polygonal numbers may be that their successive accretions get wrapped around only that part of the starting shape that is away from the initial 1, like the successive growth rings of a mussel that do not reach back to the hinge but expand the shellís shape away from it.

This is by no means the only way to represent numbers as polygons: there is, for instance, the separate and different series of concentric polygonal numbers; the designs for these might occur just as intuitively to people working on a piece of paper that does not oblige them to move away when growth layers come between them and the starting point.

The Classical shape, however, is what our postulated pebble- placers would have obtained by sitting or squatting at the starting point and adding more pebbles in arcs before them as they stayed in place and reached each time a little farther away.

1.4.1.  Gnomons and triangle chemistry

The Greek word for these incremental growth layers of polygonal number shapes was "gnomon", from the same root as "gnome" = "intelligence" and "gnosis" = "knowledge" which we still use to designate its lack in "ignorance"31.

The literal meaning of "gnomon" was "knower". By extension, the upright pointer of a sundial was a gnomon, a "knower of time", and the carpenterís square that has the shape of the gnomon for square numbers and allows its user to know right angles was also called gnomon.

These uses of that root could well preserve some ancient association of such pebble mathematics with thinking and knowing, back when the precursor languages to Greek formed their vocabulary or even before. That linguistic evidence fits again the hypothesis of a pre-Pythagorean and very early origin for the organization of quantities into such figures.

In mathematical terms, the engineer and inventor of a steam engine Heron of Alexandria (ca. first century CE) defined the gnomon as "that which, when added to anything, number or figure, makes the whole similar to that to which it is added"32.

Strictly speaking, this definition applies equally well to concentric growth rings, but usually the term is reserved for sea-shell-like accretions that do not fully surround the shape they enlarge.

In any case, the gnomonís property of preserving the shape, highlighted by the writing of dots instead of actual numbers, allowed the Classical Greeks to visually  demonstrate various number laws with those polygonal arrays. 

For instance, Sir Thomas Heath says in his "History of Greek Mathematics" that

"the Greeks were from the time of the early Pythagoreans accustomed to summing the series of odd numbers by placing 3, 5, 7, etc., successively as gnomons round 1; they knew that the result, whatever the number of gnomons, was always a square."33

The dot- manipulators showed further, among other laws, that the sum of any two consecutive triangular numbers adds up to a square with sides as long as those of the larger triangle, and that eight times a triangular number, plus one, always yields another square34, as also illustrated in Figure 4 above.

These visual proofs are still as intuitively obvious as they were then. Because adding more gnomons to the figures in these juxtapositions does not change their shapes, the relationships remain always the same for all examples. Those figurate numbers made and make it therefore easy to appreciate how the triangles are related to their siblings in the family of polygons, the squares and higher-numbered kin which are all composed of as many triangles as they have polygon sides, less two.

These relationships were extremely important to ancient number mystics because, as the Greek philosopher and mathematics- enthusiast Plato (427 to 347 BCE) explained their views in his Pythagorean- influenced Timaeus, they believed that everything was composed from four elements.

They identified these four elements with four of the five regular solids: the true pyramid, or tetrahedron, had the shape of fire, the octahedron embodied air, the icosahedron was water, and the cube corresponded to earth. The fifth solid, the dodecahedron, represented the universe itself.

The faces of the first three solids are all made up of equilateral triangles, and those of the other two can be decomposed into the right- angled triangles which one obtains by cutting the equilateral one in half35.

These mathematical relationships made it therefore possible to transmute the ideal shapes of the elements into each other by rearranging the triangles from which they were made.

By analogy, such transmutations were deemed to be equally feasible among the actual elements in the tangible world; this mystic belief inspired the efforts of many alchemists and so ultimately led to the chemistry without which our modern life would be hard to imagine.

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