Panagiotis Stefanides 
The Most Beautiful Triangle  Timaeus Plato's 
HELLENIC
MATHEMATICAL SOCIETY INTERSCIENTIFIC RESEARCH GROUP SOCIETY FOR THE
HISTORY OF MATHEMATICS Panhellenic
Conference for the History and (Dedicated to
the Anniversary for the Completion of 2,200 Years Subject : History and Philosophy of Classical Greek Mathematics Conference
Proceedings Paper Issued by : National Research
Center 31 March
1996 "The Most Beautiful Triangle  Timaeus Plato's " 
Πρώτον μεν δη πυρ και γη και ύδωρ και αήρ, ότι σώματά εστί............................................... τρίγωνα πάντα εκ δυοίν άρχεται τριγώνοιν.... προαιρετέον ούν αύ των απείρων το ΚΑΛΛΙΣΤΟΝ..... ΤΡΙΠΛΗΝ ΚΑΤΑ ΔΥΝΑΜΙΝ ΕΧΟΝ ΤΗΣ ΕΛΑΤΤΟΝΟΣ ΤΗΝ ΜΕΙΖΩ ΠΛΕΥΡΑΝ ΑΕI". In section 53, 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. This is the probability of the beginning of the creation of the Fire and of the other Bodies. These bodies are four, are dissimilar but capable of each one of them being created from the other and also dissolved into others. If it is happening so we have the truth about the creation of the Earth, Fire and their other proportional media. We do not agree that there will be other Bodies more beautiful than those ones, each one of them with its own gender. 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", and we leave those by which the equilateral triangle is constructed (i.e. by using six "scalene" orthogonal triangles, having 30 and 60 degrees their acute angles). 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." 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 49381591719543726240 degrees). This angle reaches the PYRAMIDAL one. The product of the smaller horizontal side and that of the hypotenuse is equal to the "SQUARE" of the bigger vertical side, of this triangle, and at the same time the "PYTHAGOREAN THEOREM" is valid. The values of the sides of this triangle are given by surd numbers, (solution of a fourth degree equation). Reorganizing this triangle, we get another one with the same angle values, which has its bigger vertical side equal to FOUR (4), its smaller horizontal side equal to FOUR divided by the SQUARE ROOT of the GOLDEN NUMBER, and its hypotenuse equal to FOUR multiplied by the SQUARE ROOT of the GOLDEN NUMBER. (Four divided by the SQUARE ROOT of the GOLDEN NUMBER is equal to 3.14460551) On the 3091986, it was sent by the writer to W.I.P.O. in Geneva a text of one page, which contained the elements of one triangle having its vertical side equal to 4, its horizontal side equal to 1/0.3180049125 (equal to 3.14460551), its hypotenuse equal to 5.08878597 (4x1.27201965), and its angle between horizontal and hypotenuse equal to: θ = 51 49381591719543726240 with the name of "Tο ειδικό ορθογώνιο τρίγωνο" (i.e. The "Special Orthogonal Triangle") this "Special Orthogonal Triangle", was the result of studying TOM VALENTINE's book "THE MYSTERY OF THE GREAT PYRAMID" (Greek translation published by ORORA  1981), in which it is stated that JOHN TAYLOR reached into the conclusion that the area of each of the four sides of the Great Pyramid, is equal to the square of its height (in the conference, it was stated that this relationship was given by "HERODOTUS"). Also, according to MAX TOTH, Best Seller 1988, "The Prophesies of the Pyramids"Greek translation  and the pieces of information from "HERODOTUS" that the area of each side of the Great Pyramid is equal to the square of its height, the mathematicians have derived a triangle with vertical side equal to the square root of the GOLDEN NUMBER, the horizontal side equal to unity and the hypotenuse equal to the GOLDEN NUMBER". Also they derived a relationship which states that Pi is equal to four divided by the square root of the GOLDEN NUMBER and it is stated as being a useful formula for it (however, it is not derived as a proof). This study was undertaken by the writer after discussing with a certain researcher in Athens, certain philosophical mathematics and it was suggested to him to study the Great Pyramid. After this study and deriving the "Special Orthogonal Triangle" and further after discussions with a very good friend and researcher of philosophy of the University two (2), Paris, Dr. STAMATIOS TZITZIS, (DIRECTEUR DE RECHERCHE AU C.N.R.S.), it was suggested to the writer to study and search the text of PLATO'S TIMAEUS. The basic classical text (in the original language at the critical points) was studied and analysed. On 26101987, the writer sent again to W.I.P.O. (World Intellectual Property Organization) a text with the title: "PLATO'S TIMAEUS"  "THE MOST BEAUTIFUL TRIANGLE". The text, was one page and it contained the triangle presented to this conference. Further down the Mathematical analysis is presented. It must be noted, also, that "PLATO" speaks about the construction of the Earth (cube), from the "ISOSCLELES" triangle, and the other three bodies from the "SCALENE". The base of the cube (Plane surface) from four (4) "ISOSCELI" orthogonal triangles, whereas the base of the other bodies from six (6) "SCALENE" of the 30 and 60 degree types. Also, "PLATO" speaks of a further fifth, structure which God used it for the design of the UNIVERSE. It is concluded here that by "THE MOST BEAUTIFUL TRIANGLE", PLATO correlates the four elements (UNIFIED THEORY) through the General Analogies of their sides (Fire, Air, Earth and Water), i.e. Fire/Air is equal to Air/Water is equal to Water/Earth, to T, where T is equal to the SQUARE ROOT of the GOLDEN NUMBER. T = SQR ((SQR.(5) + 1)/2) (ό τι περ πύρ προς αέρα, τούτο αέρα προς ύδωρ, και ό τι αήρ προς ύδωρ, ύδωρ προς γήν, ξυνέδησε...............ουρανόν, PLATO'S TIMAEUS SECTION 32). 
