I wrote this paper and I did not want any of my geometry friends to miss it. This is so simple, yet it escaped me for decades. I truly took the long way home to get to this point, but I am very pleased with its generality. Volumetric Phi Scaling of Any Tetrahedron Any tetrahedron can be sectioned into lessor tetrahedron, actually three different tetrahedra whose edges are scaled by ø^1 = (1.618034) or ø^-1 = (.618034) to progress infinitely larger or smaller, so as the three tetrahedral volumes are then in the phi ratio progression. Given equilateral tetrahedron ABCO, we will give it the volume of 1 unit. The base (x), height ( y) and altitude (z) can be increased by ø^1to generate a similar larger tetrahedron DEFO. Where edge DO is sectioned by A at the golden mean along with edges EO at B and FO at C, so edges DO/AO, EO/BO and FO/CO are ø^1. If ABCO is 1 unit, then DEFO is ø^1*x; ø^1*y; ø^1*z larger or ø^3 = 4.236068. The increase in volume can be described as a frustum of volume ø^3 - ø^0 = 2ø^1 = 3.236068. The altitude of the frustum is (ø^1-ø^0)z or (ø^-1)z = .618034 The frustum can be described as having a top equilateral triangle ABC and its parallel base equilateral triangle DEF and three trapezoids ADEB, BEFC and CFDA By drawing diagonals across all three trapezoids, we form three tetrahedra. Connecting DC across the face of trapezoid CFDA and EC across the face of trapezoid BEFC creates triangle CDE. This triangle along with the larger edge ø^1 equilateral triangle DEF and edge CF form a tetrahedron CDEF, named (U) Connecting AE across the face of trapezoid ADEB creates triangle ACE . This triangle along with the smaller ø^0 equilateral triangle ABC and edge BE form a tetrahedron ABCE, named (W) This leaves the middle tetrahedron ACDE, named (V) Tetrahedrons U,V,W add up to the volume of the frustum = 3.236068 Finding the volume of tetrahedron U, the base is ø^1(x); height ø^1(y) and altitude ø^-1(z), multiplying the three dimensions ø^1*ø^1•ø^-1 = ø^1 or 1.618034 Finding the volume of tetrahedron W, the base is ø^0(x); height ø^0(y) and altitude ø^-1(z), multiplying the three dimensions ø^0*ø^0*ø^-1 = ø^-1 or .618034 U + W = 2.236068. Frustum ABCDEF = 3.236068. 3.236068 - 2.236068 = 1 or the volume of tetrahedron V Volume of tetrahedron V = volume tetrahedron ABCO U + V + W = Volume of frustum = 3.236068 U + V + W + ABCO = 4.236068 = Volume of tetrahedron DEFO Since ABCO is similar to DEFO, a scaling by ø^-1 of tetrahedrons U,V,W can build a frustum within and the next larger frustum of DEFO can be composed of U,V,W tetrahedrons scaled by ø^1. Thus. we have an infinite series defining any tetrahedron in a Phi progression. The tetrahedron is infinitely decomposed or grown of these three volumetrically relative tetrahedral equivalents U,V,W on the pattern . . . ø^-4, ø^-3, ø^-2, ø^-1, ø^0, ø^1, ø^2, ø^3, ø^4 . . . Let U = ø^1, V = ø^0, and W = ø^-1; U3 = ø^4, V3 = ø^3 and W3 = ø^2; U-3 = ø^-2, V-3 = ø^-3 and W-3 = ø^-4 So, W-3 +V-3 = U-3; W + V = U; W3 + V3 = U3 Also V-3 + U-3 = W; U-3 + W = V; V + U = W3 and U + W3 = V3
Posted on: Sun, 24 Nov 2013 09:44:10 +0000
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