Re: Strange Maths (was Re: Why not 13 months?)
Andrew Christy (christy@rschp2.anu.edu.au)
25 Jul 1995 06:56:18 GMT
doug@netcom.com (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
>
>Tada. 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 scaleinvariant coincidence (?) involving this angle:
The rightangled 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).
However:
This angle is closely approximated by arctan (sqrt(phi)), where phi is
the wellknown 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 semibase 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* rightangled triangle for which the
three side lengths are in a geometric ratio. Phi, of course, also figures
prominently in the ratios of various interapex 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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