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- Posted November 30, 2012 by
- PCSTEFANIDES

KIFISSIA, Greece

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## CIRCLES, SQUARES TRIANGLES, POLYHEDRA , LOGARITHMS, AND SPIRALS RELATED TO THE THE QUADRATURE TRIANGLE [ P.Stefanides Interpretation]

CIRCLES, SQUARES, TRIANGLES, POLYHEDRA , LOGARITHMS, AND SPIRALS RELATED TO THE THE QUADRATURE TRIANGLE

[ P.Stefanides Interpretation]

**http://www.stefanides.gr/html/Root_Geometries.htm**

**http://www.stefanides.gr/Html/Photo.htm**

http://www.stefanides.gr/html/All_triangles_derive.htm

ABSTRACT

My Work and Artwork involves Symmetries of Geometric Forms such as Circles,Squares, Triangles, Areas, Solids, Spirals and Logarithms, having affinity with the GOLDEN ROOT, which unifies them.

http://www.stefanides.gr/html/Root_Geometries.htm

http://www.stefanides.gr/html/All_triangles_derive.htm

The theory behind this work is based on my, proposed interpretation of Plato’s Timaeus triangles of the structure of matter, “THE MOST BEAUTIFUL” and “THE ISOSCELES” [Pl. Ti 54 B] and also “THE “SOMATOIDES” and “THE STEREOID-MOST BEAUTIFUL BOND” [Pl.Ti.31B, C/32 B] It is additionally based on my special theory, proposed, relating Spirals and Logarithms to various bases.

Of particular interest, to me, is the Form of Nautilus, the kind of which, after measurements of a Shell, found to comply with the logarithm having as base that of the GOLDEN ROOT. By GOLDEN ROOT it is implied the SQUARE ROOT OF THE GOLDEN RATIO.

The Classical Greek word for symmetry, “SYMMETRIA”, means “WITH

MEASURE”, ”IN MEASURE WITH,” “DUE PROPORTION”, “COMMON MEASURE”, “HARMONY”. The concept of “SYMMETRY” contrasts the conditions prevailed before the World was Created, while all elements [FIRE, AIR, EARTH and WATER] were “WITHOUT PROPORTION”[alogos] and “WITHOUT MEASURE”[ametros], and only “TRACES” of them existed ,as all things ,naturally exist in God’s absence. God ,under these conditions, transformed them via “IDEAS” and “NUMBERS”, for them to become “MOST BEAUTIFUL” and “BEST” as possible, contrary to their previous state

[ Pl. Ti. 53B].

http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf

http://www.stefanides.gr/pdf/PROPOSED_GEOMETRY_OF_THE_PLATONIC_TIMAEUS_GREEK.pdf.pdf

© Panagiotis Chr. Stefanides,

ROOT GEOMETRIES STRUCTURING THE ICOSAHEDRON,

THE OTHER BASIC POLYHEDRA AND THE GEOMETRIC FORMS RELATED TO THEM

http://www.stefanides.gr/html/Root_Geometries.htm

http://www.stefanides.gr/Html/Photo.htm

Under Root Geometries Structuring the Icosahedron, the other Basic Polyhedra and the Geometric Forms Related to Them, we refer to the basic geometric configurations which, as this theory contemplates, are necessary for the progressive mode of formation of the five polyhedra, via lines, areas and volumes.

The Root Geometries are two configurations of triangles. A Special one, the Quadrature Scalene Orthogonal Triangle [Author’s interpretation of the Timaeic Most Beautiful Triangle] with sides [T^3], [T^2] and [T^1] in geometric ratio T, which is the square root of the golden ratio[ Φ], and the Isosceles Orthogonal Triangle, with its equal sides [T]. The surface areas of these triangles are taken perpendicular to each other and in such, naturally, defining an X, Y, and Z system of coordinate axes. In so, the coordinates of the first are [0,0,0,], [0,0,T^2], [T,0,0] in the X-Z plane, and those of the second are [0,0,0,], [T,0,0], [0,T,0] in the X-Y plane. A line from [0,T,0] to [0,0,T^2],creates the same Scalene Triangle in the Y, Z plane.

Arctan [T] is the Scalene angle [θ] of the Special Triangle with the property that the product of its small side by its hypotenuse is equal to the square of its bigger side: [T^1]*[T^3] equal [T^2]^2 [Quadrature].

Using a pair of the Special Scalene Triangle, and a pair of a Similar Kind of Triangle [Constituent of the Special] with sides 1,T and T^2 [Kepler's (Magirus) Triangle with sides 1, sqrt(Φ), and Φ] a Tetrahedron [dicta Form 1] is obtained, by appropriately joining the edges of the four triangles, with coordinates: [0,0,0,], [0,0,T^2], [T,0,0] and [0, 1/T, 1/T^2].

By joining, a line, from point [0,T,0] to point [T,0,0], a Second Tetrahedron [dicta Form 2] is obtained [ as a natural extension of Form 1], with co-ordinates: [0,0,0,], [T,0,0],[0,T,0] and [0,1/T, 1/T^2], having as base, on the X-Y plane, the Isosceles Orthogonal Triangle mentioned above, with coordinates[0,0,0], [T,0,0] and [0,T,0].

Doubling this triangle, in the X-Y plane, a square is obtained of side [T], with coordinates [T,T,0], [T,0,0], [0,0,0], and [0,T,0].

By connecting a line from point [T,T,0] to point [0,0,T^2] a third Tetrahedron [dicta Form 3] is obtained with coordinates:[T,T,0],[T,0,0], [0,T,0] and [0,0,T^2],

having also as base, the Isosceles Orthogonal Triangle with same dimensions [mirror image] as that of [Form 2]. The three Forms are wedged firmly together, leaving no empty space between them. Their volume ratios

Form 3: Form 1: Form 2 equal to [1/6]*[T*T*T^2]: [1/6]*[1*T*T]: [1/6]*[T*T*(1/T^2)] is the golden ratio [T^2], and the sum of volumes of Form 1 and Form 2 equals to [1/6]*[1*T*T ] + [1/6]*[T*T*(1/T^2)] equals to [1/6]*[T^2+1] equal [1/6]*T^4[SINCE T^4-T^2-1=0],

the volume of Form 3.The volumes of the three Forms sum up to [(2/6)T^4 equal to (1/3)T^4].

Two of the four bases of Form 3, are symmetrical orthogonal triangles, with coordinates [T,0,0], [T,T,0], [0,0,T^2] and [T,T,0], [0,T,0], [0,0,T^2], each of which has an angle [ φ], equal to arctan[T^2].

Two such triangles joined coplanarily and symmetrically along their bigger vertical sides, create one of the four triangular faces of a great pyramid model with coordinates [T,T,0], [0,0,T^2] and [T,(-T),0].

The Structure of the three Forms bound together [dicta Form 4] with Volume [1/3]*T^4 is one quarter of the volume of the great pyramid model, which has a square base of side 2T, height T^2 and Volume [4/3]*T^4.

Splitting one of this model’s triangular face into two orthogonal co-planar triangles to form a parallelogramme [with sides T^1 and T^3], we have constructed the basic skeleton of the Icosahedron, since three such parallgrammes, orthogonal to each other, determine its twenty equilateral bases, by joining adjacent corners in groups of three, by lines.

Similarly, we proceed to the construction of the

Dodecahedron, the tetrahedron, the octahedron and the cube, together with their related forms such as squares, circles, triangles, circumscribed circles to the parallelogrammes of the polyhedra skeletons, circumscribed spheres and logarithmic spirals.

Reversing the whole process, the volumes decompose to the areas of the triangle surfaces structuring them which, in turn, resolve to four line traces harmonically codified in space.

© Panagiotis Chr. Stefanides,

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