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    GOLDEN ROOT SYMMETRIES OF GEOMETRIC FORMS Interpretation by Panagiotis Stefanides


    Interpretation by Panagiotis Stefanides


    The publication of this book, under the title “GOLDEN ROOT SYMMETRIES OF GEOMETRIC FORMS”, presented to the International conference “SYMMETRY FESTIVAL 2006” in Budapest Hungary, mainly, has intention to present a novel theory-exposed also to two National Conferences in Greece- to the Scientific Community and to the wider public with interests in particular knowledge.
    This theory proposes a new Theorisis and Thesis of PLATO’S TIMAEUS “MOST BEAUTIFUL TRIANGLE.”

    The proposed geometric form of this triangle is based on a mathematical model, which has been analyzed, in connection with the relevant PLATONIC topics of TIMAEUS. The phrase “ΤΡΙΠΛΗΝ ΚΑΤΑ ΔΥΝΑΜΙΝ” is being taken as “THIRD POWER” and not “THREE TIMES THE SQUARE” as is the current practice of its interpretation in the topic 54 of TIMAEUS.

    LIDDEL & SCOTT dictionary states: “math…, power, κατά μεταφοράν η εν γεωμετρίας λέγεται δ., …usu (usually) second power, square, κατά δύναμιν in square, PI.T: 54b, …square root of a number which is not a perfect square, surd, opp. Μήκος, PI. Tht. 147d.

    In THEAITETUS, 147d, PLATO speaks in general about “POWERS” (περί δυνάμεων τι ημίν Θεόδωρος όδε έγραφε…”).
    That’s what Theodorus told us about “powers”:
    “The lines which form an equilateral and plane number we called them length, those the scalene one we called them powers because they are not symmetric, as to the length, with those lines, but only with the areas they form. We did the same with the solids.”

    Here, it is concluded that Plato by the phrase “…we did the same with the solids” (και περί τα στερεά άλλο τοιούτον”), correlates solids (their volumes) with “POWERS” (“ΔΥΝΑΜΕΙΣ”) as well (περί δυνάμεων …όδε έγραφε).

    In PLATO’S POLITEIA, topic 528B (Book Z), and Plato states: “After the plane form, I said, we got the solid which is in circular motion, before examining the solid itself. The just thing is after examining the two dimensional to examine those having three dimensions. I think, this exists, in the Cubes and the bodies, which have depth. This is true, he said, but Socrates these ones have not yet been invented.


    In section 53/54, of PLATO'S "TIMAEUS", PLATO speaks about the triangular shapes of the Four Elemental Bodies, of their kinds and their combinations:
    These Bodies are the Fire (Tetrahedron) the Earth (Cube), the Water (Icosahedron), and the Air (Octahedron). These are bodies and have depth. The depth necessarily, contains the flat surface and the perpendicular to this surface is a side of a triangle and all the triangles are generated by two kinds of orthogonal triangles: the "ISOSCELES" Orthogonal and the "SCALΕΝΕ" Orthogonal. From the two kinds of triangles the "Isosceles" Orthogonal has one nature. (i.e. one rectangular angle and two acute angles of 45 degrees), whereas the "scalene" has infinite (i.e. it has one rectangular angle and two acute angles of variable values having, these two acute angles, the sum of 90 degrees). From these infinite natures we choose one triangle "THE MOST BEAUTIFUL".
    Thus, from the many triangles, we accept that there is one of them "THE MOST BEAUTIFUL.
    Let us choose then, two triangles, which are the basis of constructing the Fire and the other Bodies : "Το μεν ισοσκελές, το δε τριπλήν κατά δύναμιν έχον της ελάττονος την μείζω πλευράν αεί."

    Proposed New Interpretation:

    One of these two is the "ISOSCELES" orthogonal triangle, the other is the "SCALENE" orthogonal triangle, its hypotenuse having a value equal to the "CUBE" of the value of its horizontal smaller side and having its vertical bigger side the value of the "SQUARE" of its smaller horizontal side. The value of the smaller horizontal side is equal to the square root of the Golden Number, the ratio of the sides is equal, again, to the Square Root of the Golden Number (geometrical ratio) and the Tangent of the angle between the hypotenuse and the smaller horizontal side is also equal to the Square Root of the Golden Number (Θ =51 49-38-15-9-17-19-54-37-26-24-0 degrees). The product of the smaller horizontal side and that of the hypotenuse is equal to the "SQUARE" of the bigger vertical side, and the following equation holds:
    T^4-T^2-1=0 , T = SQRT [(SQRT.(5) + 1)/2].

    N.B. Kepler [ Magirus] Triangle is a SIMILAR one, but NOT the SAME and NOT as BEAUTIFUL AS THIS PROPOSED TRIANGLE:
    Panagiotis Stefanides

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