1University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, U. S. A.
2N. A. S. A. White Sands Test Facility, New Mexico, U. S. A.
3Olney, United Kingdom
In ancient Egypt the priests of God Ra, or the God of Sun, encoded within the structure of the Cheops Pyramid, and in particular within the design of the pharaoh tomb, certain ``cosmic'' numbers such as the golden section ratio $\phi(= 0.5(1 + \sqrt{5}))$ and number $\pi$. The latter was related to $\phi$. Indeed, the geometry of the tomb suggested the simple relation, $\pi = (6/5) \phi^2$ . This gives $\pi$ within the accuracy of a fraction of the percent: $1.52\times 10^{-3}\%$ to be exact. So, it appears that the Egyptians knew both numbers well. Here we intend to look a little deeper into the relation between $\phi$ and $\pi$ First we. invoke the Fibonacci series and propose its variation in form of a reciprocal series, the so- called Carl series. By manipulating the sum of this new series (a finite number-contrary to the sum of the original series in which a finite number does {\bf not} exist) we arrive at upper and lower bounds for it. The interval between these bounds can be narrowed down with an arbitrarily high accuracy. Thus, using calculus and dealing with the infinite series, we are able to improve on the Egyptian equations. Finally, employing number $\phi$, encoded both in the pyramid and in the Fibonacci series, along with certain more recent notions such as affine transformation, the Sierpinski triangle and the Mandelbrot set (all useful in quantifying Chaos), we generate a fractal space which -- as it turns out -- resembles the pyramid itself.