Is phi a mystical number as claimed in The Da Vinci Code?

Cecil's Columns/Staff Reports
1

In response to this article: Is phi a mystical number as claimed in The Da Vinci Code? - The Straight Dope

The article says: “phi, like pi, is an irrational number that can’t be expressed as the ratio of whole numbers”. But Pi and Phi are very different constants.

Pi is a transendental number meaning you cannot express it algebriacly. Phi is irrational, but it IS algebriac and I can write it here:

Phi = (1 + root 5) / 2

There is no algebriac equation that can be written for Pi, because all equations for Pi are infinite and would require more space then is available here, on the Internet or the Universe.

This means that Phi is geometric. I can draw Phi as a unit of measure proportional to some other length. Pi cannot do this, because Pi is drawn as a curve as some proportion to a length (like: circumfrence to radius). These are not comparible measures and therefore Pi cannot be drawn as a proportion. ie: it is not “real” in the sense that no one can ever make Pi. Where Phi is real and it is a constructable number.

====

You are right: When people say “Phi is beauty” and speak of proportions in the body or refer to it as mystical… they are blowing smoke. Phi is simply a mathematical constant that appears in all natural growth. Beauty is purely subjective and individual comparisons are not possible with mathematics due to all the extra variables in real life. However, Phi does accurately express perfect growth in a controlled environment. The Fibbonacia sequence expresses this clearly and you can read about. Phi is simply the underlying idealized value of this growth.

====

Speculation about Phi:

What people do not mention about Phi and they should… is that Phi is the reason we use decimal. We use 10 digits in the number system because: cos 36 = Phi / 2. Which is a triangle where 10 of them fit perfectly inside a circle revolved about the center. Becuase 36 degrees = 1/10th of 360. So you should know that Phi is the only constant we know of that divides a circle evenly into 10 parts. We are taught to use Pi to divide a circle with: 2 pi = 360. But if you check wikipedia, you will find that 2 pi is a convension and requires an assumption to use the unit circle (radius 1). 2 pi = 360 is not true at other scales. Whereas, cos 36 = Phi / 2 is scale invariant. This implies it would be more accurate for us to use Phi as the constant of choice for angular division instead of Pi.

One more thing. The “golden triangle” which has sides of Phi and base of “1” has a 36 degree peak as well. It divides the circle into 10 as well.

If you ask a teacher about the source of decimal, she will tell you: we have 10 fingers. First we counted on our fingers and later we used a notation of 10 symbols. But a search of our recent history will show that we actually used sexagesimal (base 60) for some time before switching to decimal. So it is highly unlikely that our society switched a working number system so that children would more easily be able to count on their fingers.

The real reason we use decimal is because Phi divides the circle into 10. Here is why: the history of numbers comes from keeping records of cyclic divisions. The longest history of numbers is to track the stars. The obvious reason for this is to plan for time related events and predict the future. Like anual flooding, crop planting and celestial events like full moons and eclipses. Knowledge of the future would also allow those in “the know” to convince others that they have special abilities… such as communication with gods and therefore allow them to instruct others in the name of their gods. Since numbers are primarily related to dividing circles, it makes perfect sense to divide a circle by a constant propotion… namely Phi. Phi divides the circle into 10, so the number system is divided into 10.

We knew this for a long time… ancient egypt used a base 10 number system like decimal. Ancient egypt divided the day into 10 hours (+10 night hours) and the week into 10 days. Therefore, one week was 100 day hours + 100 night hours. So you can see that not only did they use “10” but they also understood powers of 10. It is not possible to find out exactly how ancient egypt divided angles. But one can assume that it was similar to their time division and also decimal. After all… ancient egypt were masters of angles as seen in some of the oldest, largest and strongests structures ever built on our planet.

You should also note that the sun moves 18 degrees in one ancient Egyptian hour. Ancient egypt had a special triangle called “the sacred triangle”. This was a right angle triangle with a peak of 18 degrees. It is the golden triangle split in half. The peak of this triangle is exactly the amount the sun moves in one eqyptian hour. Since Phi unlike Pi is constructable… the sacred triangle can be easily constructed by anyone. A sacred triangle placed in the sand so that the suns shadow is parallel to one side will take exactly one egyptian hour before the suns shadow is parallel to the other slope. Im saying that “the sacred triangle” of ancient egypt works as a clock measuring 1 hour intervals. This would be very useful for people toiling in the fields.

When the article says: “There’s little evidence that ancient architects used the golden section”… I think the author is completely underestimating the accuracy of Giza. Phi is very apparent in its proportions. To find this you simply need to draw the pyramids slope and extrapilate up to the missing peak. You will find that the angles are extremely accurate in creating a structure using the proportion of Phi. I referring to the entire size and shape of the structure, not some guessing about portions of the structure. Change the angles of the sides by the smallest degree and the proportion of Phi is entirely lost. Based on the values given in wikipedia, the peak is within inches of the perfect value and the difference can simply be accounted for in the limits of the values released to the public. I mean that the angles are not expressed to enough decimal places to be sure, but change the degrees by the smallest increment, and it is no longer Phi proportions. To assume that the proportion of Phi was not intentionally built into the scale of Giza is to ignore the message our ancestors left for us. A message they wanted to remain for thousands of years. The message has nothing to do with Pi. It is this message: “we know about Phi” and “Phi is the constant used to divide circles and leads to decimal”.

Unfortunately, that message is lost on us today. We continue to see the world through a tunnel lens of a Pi centric society.

2

Omega Micron Fi Gamma.

3

I see what you did there.:smiley:

4

And a goo-goo-ga-joob to you, sir. As for the OP, welcome to the Dope! Several mathematicians will be along shortly to tell you how wrong you are. Or right. I, who cannot add a column of figures reliably–with a calculator, wouldn’t know. I hope they can get past your mention of The Da Vinci Code, because that book is a sore point with some here.

5

Certainly phi was known and used by the ancients.
The fact that it can be used to divide a revolution into 10 is one of its neat properties.

I doubt that this particular property, elegant as it is, gave us the base 10 system that we use. I am not aware of any historical evidence that would suggest as much. It is probable that the base 10 system in one form or another was derived independently many times and the most likely motivation was the ease of using fingers to assist calculations in the marketplace. The system(s) became established as a convention as people communicated more in exactly the same manner as every other communication convention we have was established. (Which usually has very little bearing on aesthetics or even ease of use. Witness the metrication fights in the US.)

As for dividing the circle into 10 using an algebraic number – this is derived directly from the fact that we can divide into 5 which is in turn related to the fact that 5 is a prime of the form 2^n+1. Gauss proved that any prime of this form can be used to divide a circle using classical construction methods. We don’t however see much of base 17 or base 257 even though they give rise to a phi equivalent.

As for pi being inconstructable – depends on your definition of constructable. The Greeks limited themselves to straight-edge and compass which effectively meant that they could only solve degree 2 polynomials. The three problems of antiquity (duplication of cube, trisecting an angle and squaring the circle) turn out to require different methods. Two of the problems are equivalent to solving a third order polynomial – which gives an algebraic solution but just not one constructable in the classic sense. Squaring the circle is equivalent to finding pi which is transcendental. Constructable using different curves. (I would pick a cycloid!)

And the connection with the daVinci Code. 'Nuff said.

6

Sorry,
Gaussian primes: 2^(2^n)+1

7

I may not know much math, but I know some history and human nature. The pyramids may have been built using phi, but that message is awful mundane to be built into huge, expensive piles of stone. Instead, their message is “I, Khufu the Great and Powerful, could afford to build this. Be impressed!” Then there is Sneferu’s Bent Pyramid, the angle of which got shallower halfway up, probably to make it more stable. Phi had nothing to do with it. Slump did.

Then there is the hostility towards pi that I don’t get. Pi and phi may have some mystical mumbo-jumbo associated with them to impress the rubes, but what they are is tools to get jobs done, accurately and under budget. Nothing more.

8

And you’d think at some stage I would get it right.

Gaussian primes are complex numbers.
Fermat primes are of the form 2^(2^n)+1. And Gauss did some work with them. There are five known: 3, 5, 17, 257, 65537.

9

Good heavens, no.

And Egypt is the reason we don’t have a day divided into 10 hours.

10

Sorry for the double post, but it took me a minute to track down the quote I wanted.

The supposed connection of phi and the pyramids is ancient quackery that Martin Gardner disposed of decades ago.

Magic Numbers of Dr. Matrix By Martin Gardner, 163 (Also titled The Incredible Dr. Matrix, with the same page numbering.)

11

Yeah, but you can do that pretty easily without phi, which means it’s not really that useful for that purpose. It’s more neat in a mathematical sense than a practical one.

12

Every claim made in the above paragraph is either false or nonsense.