Update: 29th December 2005

The Keys of Atlantis

A Study of Ancient Unified Numerical and Metrological Systems

by Peter Wakefield Sault

Copyright Peter Wakefield Sault 1973-2005
All rights reserved worldwide


The Keys of Atlantis

Appendix F.

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


Volume of The Rectangle-Based Axehead

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)


Volume of The Square-Based Axehead

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

Copyright Peter Wakefield Sault 1973-2005
All rights reserved worldwide