(BEING CONTINUED FROM 15/08/2016)
The Pythagorean Pentagram
Assume the square ABCE has side length a. Bisecting DC at E construct the diagonal AE, and extend the segment ED to EF, so that EF=AE. Construct the square DFGH. The line AHD is divided into extrema and mean ratio.
The key to the compass and ruler construction of the pentagon is the construction of the isoceles triangle with angles and . We begin this construction from the line AC. Note the figure
Divide a line AC into the `section’ with respect to both endpoints. So PC:AC=AP:PC; also AQ:AC=QC:AQ. Bisect the line AC and construct on the perpendicular at this midpoint the point B so that AP=PB=QB=QC.
Define PAB and QPB. Then . This implies , and hence . Solving for we, get . Since PBQ is isoceles, the angle QBP . Now complete the line BE=AC and the line BD=AC and connect edges AE, ED and DC. Apply similarity of triangles to show that all edges have the same length. This completes the proof.
The only regular polygons known to the Greeks were the equilaterial triangle and the pentagon. It was not until about 1800 that C. F. Guass added to the list of constructable regular polyons by showing that there are three more, of 17, 257, and 65,537 sides respectively. Precisely, he showed that the constructable regular polygons must have
sides where the are distinct Fermat primes. Recall, a Fermat prime is a prime having the form
Fermat ( 1630) conjectured that all numbers of this kind are prime.
Pierre Fermat (1601-1665), was a court attorney in Toulouse (France). He was an avid mathematician and even participated in the fashion of the day which was to reconstruct the masterpieces of Greek mathematics. He generally refused to publish, but communicated his results by letter.
Are there any other Fermat primes? Here’s what’s known to date.
By the theorem of Gauss, there are constructions of regular polygons of only 3, 5 ,15 , 257, and 65537 sides, plus multiples,
sides where the are distinct Fermat primes.
The Pythagorean Theory of Proportion
Besides discovering the five regular solids, Pythagoras also discovered the theory of proportion. Pythagoras had probably learned in Babylon the three basic means, the arithmetic, the geometric, and thesubcontrary (later to be called the harmonic).
Beginning with a>b>c and denoting b as the –mean of a and c, they are:
The Pythagorean Theory of Proportion
In fact, Pythagoras or more probably the Pythagorean s added seven more proportions. Here are a few, given in modern notation:
Note: each of these is expressible in the notation of proportion, as above.
Allowing A , G, and H denote the arithmetic, geometric and harmonic means, the Pythagorean s called the proportion
the perfect proportion. The proportion
was called the musical proportion.
The Discovery of Incommensurables
This discovery is usually given to Hippasus of Metapontum ( cent B.C.). One account gives that the Pythagorean s were at sea at the time and when Hippasus produced an element which denied virtually all of Pythagorean doctrine, he was thrown overboard.
Theorem. is incommensurable with 1.
Proof. Suppose that , with no common factors. Then
whence by the same reasoning yields that . This is a contradiction. width .1in height .1in depth 0pt
But is this the actual proof known to the Pythagorean s? Note: Unlike the Babylonnians or Egyptians, the Pythagorean s recognized that this class of numbers was wholly different from the rationals.
“Properly speaking, we may date the very beginnings of “theoretical” mathematics to the first proof of irrationality, for in “practical” (or applied) mathematics there can exist no irrational numbers.” Here a problem arose that is analogous to the one whose solution initiated theoretical natural science: it was necessary to ascertain something that it was absolutely impossible to observe (in this case, the incommensurability of a square’s diagonal with its side).
The discovery of incommensurability was attended by the introduction of indirect proof and, apparently in this connection, by the development of the definitional system of mathematics. In general, the proof of irrationality promoted a stricter approach to geometry, for it showed that the evident and the trustworthy do not necessarily coincide.
Other Pythagorean Geometry
Quadrature of certain lunes was performed by Hippocrates of Chios. He is also credited with the arrangement of theorems in an order so that one may be proved from a previous one (as we see in Euclid).
The large lune ABD is similar to the smaller lune (with base on one leg of the right isosceles triangle . Proposition:The area of the large luneABD is the area of the semi-circle less the area of the triangle .