## Courses and Capstone of the Great Pyramid:

```
The Great Golden Pyramid of Giza
is a square pyramid built in layers,
each layer called a course.

If the bottom layer is called course 1,
the top layer now present is course 203.

If a capstone were present, it would be course 204.

The integers can also form square pyramids:
Square Pyramidal Numbers SPN(k) are
the sums of squares of integers
from 1 to a given integer k.

For example,
SPN(8) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204,
so the courses of the Great Pyramid
correspond to SPN(8).

The courses of the Great Pyramid vary in thickness,
as shown in this chart from The Great Pyramid,
by Peter Lemesurier, Element Books (1987): Here, the units of measurement are
the Primitive Inch (PI = 2.54268 cm)
and the Royal Cubit (RC = 20.6066 PI, or 52.396 cm).

The Royal Cubit was a fundamental unit of measurement
used in ancient times.  The chart shows that
all the courses of the Great Pyramid
are at least 1 RC thick.

The Primitive Inch was introduced as a convenient unit
for correlating the dimensions of the Great Pyramid.
A Primitive Inch is
very close to a British Inch (1 PI = 1.00106 BI).

Another unit shown,
the Sacred Cubit (SC = 63.56725 cm = 25 PI),
was introduced by Isaac Newton
to show relationships among
the Great Pyramid, the Temple of Solomon, and
the size of the Earth (of radius 10,000,000 SC).

If you number the courses starting at the top
rather than the base, and if you count the capstone
as course number 0, then the base is course number 203.

It is clear from the chart that some of the courses are
distinctively thick.  Starting at the top,

these 13 courses are thicker than any preceding course,
and (except for 0 and 1)

Course Number from top = 204 - number from bottom

0
1
2
4
8
24
54
60
86
106        (Upper Shaft outlets are 100(S), 101(N))
160        (Upper Chamber floor is 154)
169        (top of Aurora-parallelogram)
203        (base of Great Pyramid is 203)

These 11 courses are local peaks,
thicker than any other within 3 courses:

32
40
66
74
96
114        (Mid Chamber Outlets would be at 113, 114)
118
130
137
145        (Upper Chamber roof)
185        (Mid Chamber floor is 179)

Jim Branson (e-mail:  knowhow@cyberhighway.net) has noted
that the course thickness variations may not be arbitrary.
Here is a modified version of his idea:

Consider dividing a musical octave into 12 semitone
intervals of the 12th root of 2.
Then use the thickness of the thinnest courses,
1 RC = 20.6066 PI, as the fundamental unit,
and semitone ratios to get a range of thicknesses.
You can then fit the course thicknesses
to the semitone thicknesses.
To illustrate how it works, the following table
gives a fit for the lowest 24 courses.
Other courses can be done similarly.
For example, course number 7 is about 41.21 PI thick,
and so is 1 octave, or 12 semitone steps,
away from the fundamental.
The table also shows the 7 Platonic notes of the octave,
with the Platonic ratio given for those notes.
The table is approximate,
in that calculations have round-off error,
and in that course thicknesses are approximated.

Semi     Plato's
Steps    Ratios     Ratio           Thickness        Courses

0          1      1          1 RC =   20.6066       13
1                 1.0595              21.83
2         9/8     1.125               23.18         20
3                 1.1893              24.51         19
4        81/64    1.266               26.08         17
5         4/3     1.333               27.48
6                 1.4145              29.15         14,15,16,18
7         3/2     1.500               30.91
8                 1.5878              32.72         12,22,23,24
9        27/16    1.688               34.77         9,11,21
10                 1.7824              36.73         8,10
11       243/128   1.898               39.12         5,6
12          2      2          2 RC =   41.21         7
13                 2.1198              43.68         4
14        18/8     2.250               46.36         3
15                 2.3796              49.04
16       162/64    2.531               52.16         2
17         8/3     2.667               54.95
18                 2.8301              58.32         1
19=12+7     3      3          3 RC =   61.82

Since no course is thinner than course 13 at 1 RC = 20.6066 PI
and the bottom course 1 is the thickest at about 58.32 PI,
you can fit all the courses within 18 steps of the fundamental.

Since the bottom 24 steps are thicker than average,
the table is NOT representative of all the courses.
You can go to the gif image near the top of this page
to see thicknesses all the courses in the Great Pyramid.

Here is a similar table for the Chinese 5-note octave,
with the 5-note equal tempered value of the ratio given
after the ratio, thickness calculated by ratio,
but no course numbers given:

Steps      Ratio     Equal Tempered         Thickness

0          1            1             1 RC =  20.6066
1         8/7           1.15                  23.55
2         4/3           1.32                  27.48
3         3/2           1.52                  30.91
4         7/4           1.74                  36.06
5          2            2.00          2 RC =  41.21
6        16/7           2.30                  47.10
7         8/3           2.64                  54.95
8=5+3      3            3.03          3 RC =  61.82

There is also a Chinese 53-note octave,
with 84 steps to the Perfect Fifth in the second octave,
developed by King Fang about 40 BC,
but I have not written a table for it.

Some musical/mathematical references used here are:
The Music of the Spheres, by Jamie James
(Copernicus of Springer-Verlag, 1993); and
Edward Dunne's web page on Pianos and Continued Fractions.

It would be interesting to set the fundamental
and intervals to a musical scale and
play the music of the Great Pyramid courses.

Here is some more interesting geometry:

The Aurora is a unit of area, 10,000 square Royal Cubits.
At the center of the base of the masonry core of each face
of the Great Pyramid is a flat triangle related to the concavity of the face.

/                      \
/________________________\    course 169
/            /\            \
/____________/__\____________\  course 203

The base of the triangle is at course 203, the base course,
and the top of the triangle is at course 169,
the 35th course from the base.
The sides of the triangle are
parallel to the sides of the pyramid,
so the triangle divides the face
into two mirror-image parallelograms
with the center of course 169 as top,
course 203 as bottom,
a triangle side for one side, and
a pyramid side-segment for the other side.
Each parallelogram has altitude 1160.9 PI
and base 3652.4 PI, giving an area
of 56.4 RC x 177.3 RC = 9,999.7 RC^2 = 1 Aurora.

Symbolically:

Since the Great Pyramid is a Square Pyramid,
the 24 courses that are distinctly thick
might represent the Square Pyramidal Number

SPN(24)  =  1 + 4 + 9 + 16 +...+ 24^2 = 4900 = 70^2

the only Square Pyramidal Number
that is itself a square.
In 24 dimensions, it has the elegance of
the 2-dimensional 3,4,5 triangle of 9+16 = 25
that represents the slope of the Second Pyramid.
(Thanks to Herm Sorem (herms@ix.netcom.com)

The capstone, course 0, might represent the Integers;
course  1 might represent the Real Numbers;
course  2 might represent the Complex Numbers;
course  4 might represent the Quaternions;
course  8 might represent the Octonions;
and
course 24 might represent the Leech Lattice.

What was the Capstone of the Great Pyramid?

The capstone of the Great Pyramid has been missing so long
that there are no reliable records of what it looked like.

It is possible that it was a small pyramid, or pyramidion,
on top of an obelisk set on the present square top
of the Great Pyramid about 264.5 RC (one RC = 0.524 meters)
above its base.

If the pyramidion were constructed so that its top
was at the geometric apex of the Golden Ratio Pyramid,
the top of the pyramidion would have been roughly 282 RC
above the base of the Great Pyramid,
with a pyramidion/obelisk height of roughly 17.5 RC,
or roughly 9.2 meters.

However, it is possible that the pyramidion/obelisk
could have been more than 9.2 meters high.
If it were 25.6 meters or about 48.8 RC,
roughly comparable to the height of the obelisk from Luxor,
now in Paris at the Place de la Concorde in Paris,
then the total height from the base of the Great Pyramid
would have been about 313.3 RC,
or 31.3 RC above the geometric apex height of about 282 RC,
or 10/9 of the geometric height of about 282 RC,
or,
from the point of view of the web page of John Miller,
the Golden Ratio geometric apex height would be 10% less
than the height of the pyramidion/obelisk,
and
the top of the pyramidion/obelisk would then be at
the vertex of a half-octahedral pyramid
with slope angle about 54.7356 degrees.

```