# Is phi a mystical number? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [July 19, 2010, 8:59pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272 "2010-07-19T20:59:54Z")\ **Posts on this page:** 18\ **Page:** 3
**Author:** ![Metrolog](https://avatars.discourse-cdn.com/v4/letter/m/5e9695/32.png) [@Metrolog](https://boards.straightdope.com/u/Metrolog)\ **Post date:** [July 20, 2010, 10:15pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/41 "2010-07-20T22:15:35Z")
> [@Indistinguishable](#): > > I have no idea what the significance of these numbers is… What’s so special about 9130.8? The original \*\*architectural \*\*base-breadth was **9131.05** inches ..9131,05 - 9130.8 * * * …0.25 inches = 0.635 cm ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 20, 2010, 10:17pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/42 "2010-07-20T22:17:44Z")
Um, ok. I still don’t get it. What’s so special about 0.25 inches? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [July 20, 2010, 10:57pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/43 "2010-07-20T22:57:01Z")
> [@Metrolog](#): > > The original **architectural** base-breadth was **9131.05** inches. Where in the book does Flinders Petrie say this? ---
**Author:** ![XT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/xt/32/456_2.png) [@XT](https://boards.straightdope.com/u/XT)\ **Post date:** [July 21, 2010, 12:06am UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/44 "2010-07-21T00:06:04Z")
[QUOTE=Indistinguishable] Um, ok. I still don’t get it. What’s so special about 0.25 inches? [/QUOTE] It’s not the length, but the girth…so to speak… -XT ---
**Author:** ![Metrolog](https://avatars.discourse-cdn.com/v4/letter/m/5e9695/32.png) [@Metrolog](https://boards.straightdope.com/u/Metrolog)\ **Post date:** [July 21, 2010, 1:32am UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/45 "2010-07-21T01:32:39Z")
I don’t want to bother you, sorry! Peace be with you! 😉 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 21, 2010, 2:12am UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/46 "2010-07-21T02:12:36Z")
Noooo! Now I’ll never know! ---
**Author:** ![qazwart](https://avatars.discourse-cdn.com/v4/letter/q/5fc32e/32.png) [@qazwart](https://boards.straightdope.com/u/qazwart)\ **Post date:** [July 21, 2010, 4:28am UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/47 "2010-07-21T04:28:08Z")
> [@Metrolog](#): > > \*\*Is phi a mystical number? > \*\* > [Is phi a mystical number as claimed in The Da Vinci Code? - The Straight Dope](http://www.straightdope.com/columns/read/2513/is-phi-a-mystical-number-as-claimed-in-em-the-da-vinci-code-em) > > Is phi a mystical number? Yes, it is! > > Height of the Great Pyramid = 5813.011885 inches > > 5813.011885 x 1.618033988 = 9405.650803 inches > > Length of the Grand Gallery = 1881.130161 inches Let’s go through this again. I had to check your measurements, so I checked this thing called “The Internets” and lo and behold, found [the actual height and length](http://www.theglobaleducationproject.org/egypt/studyguide/gpmath.php): Height (including capstone): 480.69 feet (5768 inches) Your measurement is off by 45 inches. Length of Grand Gallery: 156.9 ft (1882.8) Your measurement is off by an inch Now: 5768 inches \* 1.618033988 = 9,332.820042784 9,332.8200 / 5 = 1,866.5640085568 Riddle me this: Why take the height of the pyramid and the length of the grand chamber? Why not the width and height of the pyramid? Or, the height of the grand chamber? Or, maybe the hundreds of other possible dimensions of the pyramid? Most importantly, why the division by 5? If this was truly a mystical number that the Egyptians were after, you’d think they’d make it a bit more obvious. After all, almost all Ancient Greek architecture is in a ratio of phi because the Greeks thought it was an important ratio that denoted true beauty. Any why measurements in millionths of an inch? Could the Egyptians measure that precisely? If they could, then why aren’t the lengths of the sides off by almost half a foot? Since your measurements are off, and then in an accuracy that’s almost impossible today, it appears that someone fudged the data to get what they want. The person who came up with this looked at all the various measurements of the pyramid looking for phi, but the best they could come up with was the height of the pyramid and the length of the great chamber, but only if they first divided the height by five. That came close, but not quite, so they munged the numbers just a bit. Add a bit of height and make the great chamber just a tad longer, and by almost amazing coincidence, the mystical phi amazingly appears! That’s sheer proof! And, proof can’t get much more sheer than that! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 21, 2010, 6:23am UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/48 "2010-07-21T06:23:37Z")
> [@Chronos](#): > > But another method is to get each point by adding phi to the previous point and then chopping off the integer part. So your set of points might start off {0, phi-1, 2phi-3, 3phi-4, 4phi-6, 5phi-8…}. Of course, you could do this with any irrational number, not just phi, but with any other number, you’ll get clumping in your points. With phi, your data points will always be approximately uniformly distributed, no matter where you cut off the process. I guess this probably follows somehow from the continued fraction 1 + 1/(1 + 1/(1 + 1/(1 + … for phi, but I can’t quite see how the argument goes (not that I know exactly what theorem to establish). Can you expand on the argument? ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [July 21, 2010, 7:07pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/49 "2010-07-21T19:07:19Z")
> [@Czarcasm](#): > > Attempts to make arithmetic look mystical make the baby Einstein cry. Yeah, well, it makes the baby Julian of Norwich cry, too. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [July 21, 2010, 7:08pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/50 "2010-07-21T19:08:31Z")
> [@qazwart](#): > > Any why measurements in millionths of an inch? Don’t forget to add the length of the coastline of Britain. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [July 21, 2010, 7:10pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/51 "2010-07-21T19:10:33Z")
> [@FriarTed](#): > > Anyone who knows anything about the history of numerology soon realizes one thing- in that view of things, ALL numbers are mystical numbers. The debates are- what each number’s mystical significance is and does it have any bearing on reality? Shall we take the proof that all numbers are interesting as read? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 21, 2010, 8:07pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/52 "2010-07-21T20:07:02Z")
> [@](#): > > I guess this probably follows somehow from the continued fraction 1 + 1/(1 + 1/(1 + 1/(1 + … for phi, but I can’t quite see how the argument goes (not that I know exactly what theorem to establish). Can you expand on the argument? No, I can’t. As a physicist, it’s enough for me to know it works, not necessarily how, and it was a very long time ago indeed that I first encountered it (before college, certainly). But I seem to vaguely recall it being related to the Fibonacci thing, not to the continued fraction (of course, there are probably multiple independent lines of proof for it). You’ve got me curious now, though, so I’ll do some looking and see if I can turn anything up. IIRC, the context in which I first saw it was that some plants have leaves spaced that way, so as to minimize shading from other leaves. ---
**Author:** ![Yllaria](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yllaria/32/3452_2.png) [@Yllaria](https://boards.straightdope.com/u/Yllaria)\ **Post date:** [July 21, 2010, 9:09pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/53 "2010-07-21T21:09:29Z")
> [@qazwart](#): > > . . . proof can’t get much more sheer than that! I like this. It says so much. I think I’ll make a sticky note and attach it to my screen. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 21, 2010, 9:14pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/54 "2010-07-21T21:14:55Z")
OK, I found a [good discussion](http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html) of the use of phi in sampling (this is for the plant leaf context, but it should be generally-applicable). It looks like the first proof of it was by Douady and Couder in 1993 (which was probably after I had first read about it, so apparently it was believed correct before that), and that page does give a non-rigorous argument based on the continued fraction. ---
**Author:** ![Cheshire\_Human](https://avatars.discourse-cdn.com/v4/letter/c/5f8ce5/32.png) [@Cheshire\_Human](https://boards.straightdope.com/u/Cheshire_Human)\ **Post date:** [July 21, 2010, 11:29pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/55 "2010-07-21T23:29:17Z")
> [@Metrolog](#): > > Sorry, the numbers speak! Unfortunately, they do tend to mumble a lot. ☹ ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 26, 2010, 7:37pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/56 "2010-07-26T19:37:36Z")
> [@Chronos](#): > > Let’s say you want to select points from, say, the interval [0,1), in such a way that they’re approximately uniformly-spaced. … > [A] method is to get each point by adding phi to the previous point and then chopping off the integer part. With phi, your data points will always be approximately uniformly distributed, no matter where you cut off the process. This approach is used in certain computerized algorithms for scheduling and hashing. (I’m not completely sure the following is completely correct, but I’d give myself at least even-money 🆒 ) First note that φ is, in a sense, the “most irrational” of numbers because its continued-fraction approximants converge so slowly. Unless I am mistaken, the special virtue **Chronos** speaks of can be considered to have two components: A virtue when the number of points is smallish, and a virtue when the number of points is largish. The virtuousness of φ for the few-point case obtains in part, I think, because it is the **Golden Mean**. The virtuousness of φ for the many-point case obtains, I think, because it is so “very irrational.” You can verify this fact with simple experimentation using, for example, 0.610000 (a rational approximation to the **Golden Mean** ) and 0.723607 = .5+sqrt(5)/10 (“very irrational” number, not the **G.M.** ) With ten points, the first number outperforms the second; with a hundred points, this is reversed. That φ is _both_ the **Golden Mean** _and_ very irrational and thus optimizes _both_ virtues for the problem **Chronos** describes does seem somewhat “magical” if not “mystical”! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2010, 7:41pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/57 "2010-07-26T19:41:56Z")
I’m having a little difficulty following you, **septimus**. What do you mean by “number of parts”? ETA: Ok, I think you mean the number of discrete bins you split [0, 1) into when counting collisions. Is that right? ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 26, 2010, 8:24pm UTC](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272/58 "2010-07-26T20:24:07Z")
> [@Indistinguishable](#): > > I’m having a little difficulty following you, **septimus**. What do you mean by “number of parts”? > > ETA: Ok, I think you mean the number of discrete bins you split [0, 1) into when counting collisions. Is that right? Yes, although I minimize a segment-length statistic, rather than just “counting collisions,” segment-length being the distance between adjacent selected points. (You must have clicked too soon because the “Edited by septimus” was my changing every “parts” to “points”! I usually ignore the “Reason for edit” box – with my luck, filling in a Reason would cause me to miss the 5-minute window.) What do you think, was my post clear? Better yet, was it correct? :dubious: [Previous page](https://boards.straightdope.com/t/is-phi-a-mystical-number/547272.md?page=2)