My thanks to Tish Farrell who (innocently enough, please don’t blame Tish) reminded me of the golden proportion, phi. Last week Tish was discussing and illustrating (with a number of amazing photos) the photographer’s “one-third” rule. I thought about the Renaissance artists’ use of phi, the golden proportion. And before you could say Leonardo Da Vinci, I was again reading about phi and the construction of the Great Pyramid.

In my mind I wandered the halls of the pharoah king’s Grand Gallery, pondering …..

*“Oh no, Bumba! Are you going to serve us one of those sticky, long-winded posts about the golden proportion again?”*

“You bet your sweet you-know-what I am!”

*“Please Bumba, have mercy! We already heard about George Packard and the Golden Proportion, and then there was one called Take Five , and one called Mathematical Discoveries, which at least was short, and then there was one about……Well it was about a couple of hundred words too long…”*

“Never mind. It’s good for you to think about things transcendent. Contemplation of the phi proportion acts to clear up acne as well as to soothe painful arthritis, not to mention alleviating symptoms of upper-tract gastritis, eliminating (oops) lower-tract irregularities, and de-railing off-the-track run away thoughts. And did I mention that phi helps with feelings of social alienation and lack of global perspective?”

*“Huh?”*

The Great Pyramid of Cheops was clearly built with the phi proportion. Or was it? It’s an interesting question.

When you measure the greatest (and still standing 44 stories tall) of the Seven Wonders of the Ancient World, the phi proportion quickly reveals itself as the blueprint for the arranging of all those 5 and 10 ton building stones – all of which, of course, were perfectly cut, fitted, and exquisitely aligned to the cardinal points and to the constellation Orion. As a tomb, the Great Pyramid can’t be beat. If you’re a pharoah and want to get on in the afterlife, (and what pharoah/god on earth *doesn’t* want to get on in the afterlife?) then proper alignment – indeed a pathway via the pyramid’s “ventilation ducts” – to the celestial god Orion can come in quite handy. Spiritually speaking, a good pyramid can give a pharoah quite a boost.

The angle of inclination of the Great Pyramid (and of most all the Egyptian pyramids) is 51 degrees and 52 minutes: this is the phi angle, described by the tangent of the square root of phi and 1, where the vertical height is the square root of phi and where 1 is half the base. As a consequence, the height times 2 pi equals the perimeter of the base. Perfectly! (error less than a hundredth of a percent). So the early Egyptians already knew the values of pi and phi.

Apparently they knew their maths.

“Not necessarily so,” says Kurt Mendelssohn in his *Riddle of the Pyramids*. Mendelssohn states that the Egyptians were not necessarily great mathematicians; however they were *great* stonemasons. The appearance of the phi and pi proportions, he proposes, was perhaps a fortuitous result of how they measured. Mendelssohn, citing a suggestion by an engineer named T.E. Connolly, proposes that the Egyptians used a stone drum to mark off distances – a drum on which a royal cubit – or a multiple thereof – had been notched. Twine and rope will stretch. A drum is more accurate. Since the drum had a circumference of exactly one cubit or 25.01 inches, and since the circumference of any circle is 2r π, then the base as measuring out by the drum is a multiple of pi. The pyramid’s pi proportions, as well as its phi proportions, are simply a fortuitous and elegant consequence of the measuring system. says Mendelssohn. Thus the height of the pyramid is in nearly perfect pi proportion to the perimeter of the base.

However, when the height in cubits is multiplied by 10,000,00, the diameter of the earth is revealed. Now that’s geometry! Geo=earth, metrics = to measure. The interior chambers, of course, are also characterized by the same amazing accuracy of measurement, correspondences with actual earth measurements, and concordance with the “golden” phi proportions.

Egyptology and Pyramidology are fascinating subjects for sure. The hows and whys of the pyramids have long eluded researchers. After 4,500 years and all the earthquakes, tomb raiding, quarrying, and weathering, it’s difficult to get good info. Archeologists and scientists and the entire world have long been fascinated by the Great Pyramid of Cheops, this last remaining of the Seven Wonders and repository of ancient knowledge. Much of the literature on the pyramids, though, is less than scientific. However, there are many fine and noble books: *The Great Pyramid* by John Romer, *Secrets of the Great Pyramid* by Peter Tomkins, *The Great Pyramid Decoded* by E. Raymond Capt. There are some fine websites too. Check ’em out.

Tell ’em Groucho sent ‘cha.

Well thank you kindly for that tribute, Bumba. Am indeed honoured to have offered even the smallest iota of inspiration for this piece. What a cracking essay: how to laugh your socks off, learn maths and ancient history all at once. You have a gift, sir, and I really mean that.

I am very glad that you enjoyed it. I struggle with the explanations about the geometry and wish I could explain it more clearly. So I keep working at it. As for the jokes, no apologies. And here I must comment that you have been punning shamelessly of late.

Here’s some food for thought, Bumba. More at http://www.geopolymer.org/archaeology/pyramids/are-pyramids-made-out-of-concrete-1

It would make a good fit with the golden proportion (?)

I don’t provide links on my blog. Initially I was careful with copyright worries, and now I’m just too lazy. If you google great pyramid and phi proportion – you’ll see tons of good stuff. You do have to wade through it, screen out what’s serious or not.

I believe you, honestly!

Humbly requesting an entire blog series on calculus! Tangent – those juicy little oranges I get at Christmas time! How do I get a square root of an angle with degrees and minutes? And did they make a musical instrument drum to use as a measuring tool? I knew I should have paid more attention in math class, it really was “useful” for something!

That’s right. Had you listened in math class, you might have built a decent pyramid by now. And when you understand what a square root really is, let me know.

OK square root was the one thing I thought I understood! 2 is the square root of 4. 7 is the square root of 49. Something tells me you do more with it than that, don’t you smarty pants??????

OK. If you have a line of 2 inches and you square it you have a square of area 4 sq in. Cube it and you have 3 dimensions. So what’s the square root of a line look like?

Weren’t the maths the gift of visiting aliens? Or perhaps it was just the drum.

Great post! The golden ration is everywhere especially in art and architecture. I remembered awhile ago, I read a book called “Golden Ratio and Fibonacci Numbers” by Dunlap and it was a very informative book. Phi and Fibonacci numbers have so many connections with each other after all.

Yes, I find it tremendously interesting too.