Preamble
This paper will prove that at least one ancient Egyptian genius utilised
naturally occurring mathematical phenomena in a particular geometric configuration
to engender:

the ancient Egyptian royal cubit, remen and common (short) cubit measures,
the Roman foot and cubit, the Greek foot and cubit, and the GreekRoman
stade;

the mile, the yard and the inch;

a design strategy for the layout of
the pyramid complex on the Giza plateau.
Transformations within the geometric
configuration produced, amongst other things, the designs for the three
main pyramids at Giza. The pyramids, it will be found, store critical information
in their layouts about the design system that created them. The system
has no counterpart in contemporary society and is a truly wondrous notion.
It is shown that:

the royal cubit was intended to be 20.625
inches exactly;

there were 3072 royal cubits in a mile
of 63,360 inches (3072 is the time number 24 doubled seven times);

the remen measure was 14 7/12 inches;

the common cubit was 17.6 inches;

the "forearm" measure^{1} was 11 2/3 inches
 this measure later became the Roman foot (Zupko, p. 6;
Klein, p. 71);

the Roman cubit was 17.5 inches;

the Greek foot was 12 11/72 inches;

the Greek cubit was 18 11/48 inches;

the improper fraction 22/7 was used
to represent pi;

the improper fraction 99/70, which also
has a sevenbased denominator, was used to represent the square root of
two.
(Note: Fractions in this paper
are shown in vulgar, improper and decimal forms, often in conjunction to
highlight number patterns.)
The common cubit, as stated, was
17.6 inches and there are, therefore, 3600 (60 squared) such cubits in
a mile. Compare this with a mile of 1760 yards and a yard of 36 inches.
Of additional interest is the fact that the common cubit has 24 digit divisions
and consequently there are 86,400 such digits in a mile. It is not unrelated
that there are 86,400 seconds in a day. There is much more on this matter
in Number and Divinity in Ancient Egypt (NADIAE).
The measure values under discussion
are typically described as being:

royal cubit: about 20.62 inches (52.375
cm). The royal cubit had 28 digit divisions (Petrie, IM p. 56;
Petrie,
MW
p. 3; Petrie, TPTG p. 179).

remen: about 14.584 inches. A double
remen is said by Petrie (MW p. 5) to be derived from the diagonal
in a royal cubit square. A remen had 20 digit divisions (Klein, p. 71).

common (short) cubit: about 17.6 inches
and it had 24 digit divisions (Petrie, IM p. 62;
Petrie, MW
p. 4).

Roman foot: 16 remen digits and consequently
about 11.66 inches (Klein, p. 71; Zupko, p. 6 [Zupko wrongly expresses
it as 11.65 inches]; Petrie, MW p. 5.). This measure has sometimes
been described as the ancient Egyptian "forearm" measure: see Endnote 1.

Roman cubit: about 17.48 inches
(Zupko,
p. 6  he should have described it as being about 17.49 inches). The Roman
cubit is equivalent to 1 1/2 Roman feet (Zupko, p. 6).

Greek foot: about 12.15 inches (Zupko,
p. 6; Petrie, MW p. 5).

Greek cubit: about 18.225 inches (Zupko,
p. 6; Petrie, MW p. 5).

Stade: 600 Greek feet or 625 Roman feet,
that is, about 607 British imperial feet (Zupko, p. 6). Note that there
is a 25:24 ratio between the Greek foot and the Roman foot.
One can immediately see that the
present writer has taken no liberties in correcting the above values: the
differences are paper thin.
Here now, for the first time in history
in the public arena, is the geometric configuration that created all the
abovementioned measures and much else besides. Since it is known that
the royal cubit dates back to around 3000 BC, this geometry must have existed
then.
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