Room Acoustics question:

With all due respects to Euclid et.al. the Golden Section goes back....Way Back. Many Egyptian monuments, including the near 5000yr old Great Pyramid use this ratio as a design element.

There IS a mathmatical relationship between phi and pi.
Phi is the only number that squared = itself +1. (2.618)
other relationships exist.

It goes back father than that. Nature has been using the "Golden Ratio" to design things pleasing to the eye, and to function their best for aeons. ie: As mentioned, the Nautilus shell, what we consider the most beautiful faces, the human hand's bone structure, your ear's cochlea, the DNA helix, flowers, etc.(http://cuip.uchicago.edu/~dlnarain/golden/activity7.htm)

Magfan, writers/historians studying the Pyramids disagree on whether PHI was used in the design. According to Livio, it is possible either PHI or PI was used and more likely it was PI (see his book for the explanation). Just because there is a relationship between Pyramid elements that approximates PHI doesn't prove it was used. There is lots of controversy surrounding PHI.

A regular pentagon is made up of 5 isosceles triangles. The ratio of the longer side to the base is PHI. It is an irrational number, that is, it is a number that cannot be expressed as a ratio of two whole numbers. This really bothered early geometrical researchers.

PHI can also be constructed by taking the limit of the ratio of adjacent numbers in the series formed by summing integers:

1+1=2
1+2=3
2+3=5
3+5=8
5+8=13
8+13=21
etc.

Series is 1, 2, 3, 5, 8, 13, 21 etc.

2/1=2
3/2=1.5
5/3=1.666667
8/5=1.6
13/8=1.625
21/13=1.6153846...
etc.

A logarithmic spiral has the property of not changing its shape as it grows larger. The radius keeps increasing as the spiral grows just at the right rate and this is governed by PHI. This spiral is the logo for Cardas. Does PHI work in the design of cables and rooms? George clearly believes it does.

Just a slight correction........
The RATIO is 1:1.618. It is NOT the NUMBER 1.618.
The RATIO is often called the Golden Mean and it is found both Geometrically (The diagonal of 1/2 a square added to the square), and mathematically as Zargon has demonstrated. This mathematical progression is called the Fibonacci Series after the Latin mathematician who discovered it. It works by adding the 2 preceeding numbers to obtain the next in the series and if you move far enough into the series (55/89) you get 1:1.618.

Zargon,
I wish I was near my reference materials rite now! I think it was Strabo or Herodotus who mentioned that the ratio of side to area of base was 1/2......I simply can't remember.
Bottom line? Phi was used in calculation:

Also, Pi=6/5 phi2.

Yes, there is Substantial disagreement. However, and this is WAY beyond a Hi-Fi posting, the numbers/ratios of such monumental constructions simply don't lie.
Way TOO much to go into here. PM me for more information, some of which is non-standard. I have met several professional Egyptologists and they typically, though very smart, would have trouble balancing there own checkbook.