Panagiotis Stefanides 
ROOT GEOMETRIES STRUCTURING THE
ICOSAHEDRON, THE OTHER BASIC POLYHEDRA AND THE GEOMETRIC FORMS RELATED TO THEM 
By *Eur Ing
Panagiotis Chr. Stefanides BSc(Eng)Lon(Hons) MSc(Eng)NTUA CEng MIET 
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 XZ plane, and those of the second are [0,0,0,], [T,0,0], [0,T,0] in the XY 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 coordinates: [0,0,0,], [T,0,0],[0,T,0] and [0,1/T, 1/T^2], having as base, on the XY plane, the Isosceles Orthogonal Triangle mentioned above, with coordinates[0,0,0], [T,0,0] and [0,T,0]. 

Doubling this triangle, in the XY 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^4T^21=0], 

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 coplanar 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 

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. 

*Ex Hellenic Aerospace Ind., R and D Lead Engineer and Directorate Aircraft Engines’ Manufacturing Engineering Methods’ Superintendent. (c.v.) 

Conference: Symmetry Festival 2006 Budapest:


Publication: “Golden Root Symmetries of Geometric Forms” Symmetry:
Culture and Science, The journal of the Symmetri0n ,Volume 17, Numbers
12, 2006, pp 97111. Award: Archimedes Silver Medal for the Invention of a Solar Locating and Tracking System, by the Hellenic Society of Research and Inventions, Athens Hilton 4 Feb 1983. 

©19862011, Panagiotis Chr. Stefanides, Athens 15/04/2011. 