by H. PeterAleff |

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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 " For the pursuit of patterns in primes, the number world maze resembles the labyrinth of the netherworld in Virgil’s 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 Aeneas’ further path. Virgil’s 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 primes However, like the handiwork of Daedalus that told his story and also supplied the structure as well as the central image of 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 polygon’s type. You will find some of them illustrated on Figure 4.
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 today’s books on elementary number theory still use these ancient figures to introduce their subject 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 shell’s 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 " The literal meaning of " 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 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 For instance, Sir Thomas Heath says in his " "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 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 square Figure 4 above.These visual proofs are still as intuitively obvious as they were then. Because adding more 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 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 half 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. Advertisement: Save your time and get on time success in http://www.pass4sure.com/certification/network-plus-certification.html and http://www.pass4sure.com/1Y0-A19.html exams by using our latest http://www.actualtests.com/certs/CISSP-training-certification.htm and other superb exam pass resources of http://www.actualtests.com/onlinetest/pmp-training.htm and http://www.actualtests.com/exam-70-640.htm. | |||||||||

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