Re: Strange Maths (was Re: Why not 13 months?)

Andrew Christy (
25 Jul 1995 06:56:18 GMT (Doug Merritt) writes:
>Take the pyramid height to be 100 units and the wheel to be 1/2 unit
>in diamater. Then its circumference is pi/4. Now measure out 100
>rotations of the circle from the center of the pyramid, giving
>a triangle with base y = (100*pi/4) and height 100. The hypotenuse
>is then
> r = sqrt( (100*pi/4)^2 + 100^2)
> r = sqrt( 6168.5 + 10000 )
> r = 127.155
>The angle formed by the hypotenuse and the base is
> sin theta = y/r
> sin theta = 100/127.155
> sin theta = 0.7864
> theta = 51.85 degrees
> theta = 51 degrees 51 minutes
>Ta-da. Simple, and easily within reach of Egyptian technology. All
>they have to do is measure 100 units for the height and 100 rotations
>of the wheel for the base. (Obviously one gets the same answer no
>matter whether this number is 100 or 55 or 723, as long as it's
>the same for the height and base.)
> Doug

And now for an amusing scale-invariant coincidence (?) involving this angle:

The right-angled triangle calculated by Doug above (midpoint of base edge,
centre of base and apex of pyramid as vertices) shows an inclination
of the side edges of arctan (4/pi) = 51 deg 51.24' . Fine. As Doug shows,
it would have been easy for the Egyptians to measure out the base and
height of such a pyramid, and there is an obvious motivation: *symbolic*
squaring of the circle (or showing that they could use wheels to generate
a square with the same area as a given circle).


This angle is closely approximated by arctan (sqrt(phi)), where phi is
the well-known Golden Ratio, (sqrt(5)+1)/2 = 1.6180339..

This angle is 51 deg 49.64'. The tangents , and hence pyramid heights,
would differ by only 0.096%. The important feature of this angle is
that the semi-base of the pyramid and the height (the orthogonal sides
of the triangle) and the sloping face length (hypotenuse) are in the ratio
1:sqrt(phi):phi. This is the *only* right-angled triangle for which the
three side lengths are in a geometric ratio. Phi, of course, also figures
prominently in the ratios of various inter-apex distances in the pentagon,
dodecahedron & icosahedron, and is the limiting ratio between successive
terms of the Fibonacci series 1,1,2,3,5,8,13,21,34,55,89... The Ancient
Greeks were fascinated by it .

AFAIR Phi was used in architecture by the Greeks in classical times. I
am not aware of any evidence supporting similar mathematical sophistication
for the Egyptians, but the coincidence of these two angles would have
doubtless been exciting if they *had* been into phi.

Just an idea...

Andy C


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