Is Mecca at a special place on the earth's surface, dictated by the Golden Mean?

Question

Mecca considered by Muslims to be the holiest city of Islam.

This video about the Mecca makes the following claims:

• Mecca is at a special point on the Earth's surface in the sense that its proportion in distance to the South and North pole, and also the proportion of eastern and western elongation, are equal to the Golden Mean.

Another example here.

Are these claims true?

Answer 1

Given the following definitions:

• S = distance from Mecca to South Pole
• N = distance from Mecca to North Pole
• L = distance from left side of rectangular map to Mecca
• R = distance from Mecca to right side of rectangular map

I will make the following statements:

1. The fact that both S/N and (N+S)/S are the golden ratio is not a surprise. That is one of the well-known features of the golden ratio and works for any two numbers¹. Try it. This is in fact the reason why the ratios of successive numbers in the Fibonacci series converges to the golden ratio².
2. The same can be said regarding L/R and (L+R)/L
3. The fact that L/R is the golden ratio is a function of the arbitrary selection of the starting point of the map projection. With the appropriate starting point of the left side of the map, any point along the same latitude line as Mecca will have the same L/R golden ratio. The video mentions the projection starting at the solstice line, which makes no sense because a solstice is a time of the year, not a geographic location³. The convention of world maps centered around Greenwich Meridian as the 0° longitude line is itself an arbitrary convention established in 1884 to standardize locations⁴, long after the city of Mecca was established, and therefore the L/R golden mean location is an accident of history.
4. Similarly, the final point about the distance from bottom-left corner to upper-right is just as uninteresting because the ratios of hypotenuses of triangles will be the same as the ratios of the legs - an elementary Geometry fact⁵.
5. Note that we could also reverse south/north to get another latitude line on an arbitrary world map that have all the properties discussed in the video (except that it would be the ratio from north pole distance to south pole distance instead). Given that the actual location of Mecca is roughly 21.423-21.247=0.176° off from the actual location of the golden mean (according to answer by @userknown), which is an error of 0.176°/360°*40,075.16 km ~ 20 km⁶, if we thus allow an error tolerance of 20 km we get a total surface area of 569,000*2/510,000,000=0.00224^{7 8}, or 0.22%, of the world map that fulfills the requirements in the video. This region would include such major cities as Honolulu, Hawaii and Cancún, Mexico on the north side and Francistown, Botswana on the south side⁹.

Therefore, the only interesting concept in the entire video is the very first one, that the ratio of the distance from the South Pole to Mecca has a golden ratio proportion to the North Pole from Mecca (which itself is in significant error). Every other point is arbitrary or a simple mathematical deriviation. That there should be a relationship between an arbitrary number and the significance of a geographical place is a total non-sequitur (and also not unique to Mecca).

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References:

¹ http://mathworld.wolfram.com/GoldenRatio.html: Euclid ca. 300 BC gave an equivalent definition of by defining it in terms of the so-called "extreme and mean ratios" on a line segment, i.e., such that Φ = AC/CB = AB/AC

² http://mathworld.wolfram.com/FibonacciNumber.html: The ratios of successive Fibonacci numbers approaches the golden ratio as approaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62).

³ http://en.wikipedia.org/wiki/Solstice: A solstice is an astronomical event that happens twice each year when the Sun's apparent position in the sky, as viewed from Earth, reaches its northernmost or southernmost extremes.

⁴ http://wwp.millennium-dome.com/info/conference.htm: Why does the Prime Meridian (Zero Longitude) pass through Greenwich? It dates back to October 1884. At the behest of the President of the United States of America 41 delegates from 25 nations met in Washington, DC, USA for the International Meridian Conference...

⁵ http://mathworld.wolfram.com/Triangle.html: If a line is drawn parallel to one side of a triangle so that it intersects the other two sides, it divides them proportionally, i.e., AX/XC = BY/YC

⁶ http://geography.about.com/library/faq/blqzcircumference.htm: The circumference of the earth at the equator is 24,901.55 miles (40,075.16 kilometers).

⁷ http://en.wikipedia.org/wiki/Sphere - I used this formula to integrate the area of the band from 21.071° to 21.423° on a sphere to estimate the size of the region of tolerance that would satisfy the requirements presented in the video for Mecca.

⁸ http://www.universetoday.com/25756/surface-area-of-the-earth/: The surface area of the Earth is 510 million square kilometers or 5.1×10^8 km^2.

⁹ http://en.wikipedia.org/wiki/List_of_cities_by_latitude

Answer 2

A user on YouTube, called TheRationalizer has examined the claim that Kaaba, in Mecca, lies upon (one of) the Golden Ratio points between the North and South pole, and between Greenwich Meridian.

His three minute YouTube video reproduces the calculations, using the freely available tools of Google Earth and Wolfram Alpha. He shows each of his calculations on-screen, with justifications.

His conclusion is that the geographical point that has the properties claimed (approximately 21.24611°N, 42.49223°E) is 276.92km (about 172 miles) from Kaaba, in Mecca.

So, even if you accept:

• The Golden Ratio is spiritually significant.
• The location of the arbitrary East/West divide of the Greenwich Meridian is spiritually significant, and
• The inherently North-centric and East-centric view implicitly assumed.

there is still a large error in the claim.

This is not a peer-reviewed source. I can see no errors in his approach.

Answer 3

also the proportion of eastern and western elongation, are equal to the Golden Mean.

First of all, Al-Masjid al-Ḥarām (The Grand Mosque) was established in Mecca in 638. That's thousand years before Greenwich observatory was established.

Secondly initial meridian is completely arbitrary. Historically there have been dozens of them. Greenwich has been established as initial meridian only in 1851 and internationally accepted in 1884. And that still wasn't universal adoption.

It's sounds ridiculous to affirm that location of The Grand Mosque is based on coordinate system that didn't exist until 1250 years later.

Answer 4

From OpenStreetMap I derived the Kabbah to be at 21.423° north. Here is an schematic image, how the formula is derived:

echo "scale=8;a=21.423;180/(90+a);(90+a)/(90-a)" | bc 
1.61546538
1.62478673

A value of 1.618 is reached for 21.247°, which is even in the south of the slightly more south city 'Al Taif'.

If you pan with the mouse over the map of OpenStreetMap, you can read the position in the lower right corner.