Update: 29th December 2005 # The Keys of Atlantis

## A Study of Ancient Unified Numerical and Metrological Systems

### by Peter Wakefield Sault

The Keys of Atlantis

## The Parthenon Trapeza Figure F-1. The Trapeza within The Composite Tetrahedron

Volume of The Trapeza

The trapeza can be decomposed into nine separate modules, comprising four types identified as V1 through V4 in Figure F-2 below. Figure F-2. Plan View of The Trapeza

The heights, h (normal to the view in Figure F-2), of the modules are all equal. The lengths of the edges are:

 a Short side of the rectangle b Long side of the rectangle c Side of the square of equal area

Where, by definition:

 a/c = c/b \ c2 = ab

The volumes of the module types are:

 Cuboid V1 = ach Triangular prism V2 = ch(c-a)/4 = ah(b-c)/4 Triangular prism V3 = ah(b-c)/4 Tetrahedron V4 = h(b-c)(c-a)/12

from which it can be seen that V2 = V3.

Therefore the total volume of the trapeza:

 VT = V1 + 2V2 + 2V3 + 4V4 = h[ac + a(b-c) + (b-c)(c-a)/3] = h(ab + ac + bc)/3

The rectangle-based axehead can be decomposed into three separate modules, comprising two types identified as V5 and V6 in Figure F-3 below. Figure F-3. Plan View of The Rectangle-Based Axehead

The heights, 2h (normal to the view shown in Figure F-3), of the modules are all equal, where h is the height of the trapeza. The lengths of the edges are:

 a Short side of the rectangle b Long side of the rectangle d Edge of the axehead

Where d = b + c

The volumes of the module types are:

 Triangular prism V5 = abh Tetrahedron V6 = ah(d-b)/6 = ach/6

Therefore the total volume of the axehead:

 VRA = V5 + 2V6 = abh + ach/3 = ah(b + c/3)

The square-based axehead can be decomposed into three separate modules, comprising two types identified as V7 and V8 in Figure F-4 below. Figure F-4. Plan View of The Square-Based Axehead

The heights, 2h (normal to the view shown in Figure F-4), of the modules are all equal, where h is the height of the trapeza. The lengths of the edges are:

 c Side of the square e Edge of the axehead

Where e = a + c

The volumes of the module types are:

 Triangular prism V7 = c2h = abh Tetrahedron V8 = ch(e-c)/6 = ach/6

Therefore the total volume of the axehead:

 VSA = V7 + 2V8 = abh + ach/3 = ah(b + c/3)

from which it becomes clear that the volumes of the two axeheads are equal.

Volume of The Composite Tetrahedron Figure F-5. Plan View of The Composite Tetrahedron

The Keys of Atlantis - Appendix F. The Parthenon Trapeza 