02 August 2016

Debunking golden ratio architecture (1)

Golden numberists see their beloved ratio

φ = (√5-1)/2 = 0.61803....

or its inverse
Φ =  (√5+1)/2 = 1.61803....

everywhere. Greek mathematicians knew it all right, because it is at the core of the regular pentagon, the regular icosahedron and the regular dodecahedron. Luca Pacioli was so impressed with its mathematical properties—theology is all he considers outside mathematics—that he called it the divine proportion. His contemporaries da Vinci and Dürer didn't share his enthusiasm though, and they avoided φ even in the rare case when they did need it. Both preferred empirical non-correct pentagrams to the geometrically correct ones based on the golden ratio.

Art and architecture are said to abound with φ's. Strangely enough, authorities on architecture have been unaware of this crucial fact for some millennia. The Roman architect Vitruvius wrote a ten volume classic on the trade, without ever referring to the golden number. Neither is φ to be found in any of the many renaissance treatises on architecture. (Overview in this 2002 paper.) No more so in Banister Fletcher's History of Architecture, (here) though it appeared in 1905, when golden numberism (originating in the 19th century) had already started its spectacular career. Sure enough, Le Corbusier and other modern architects did use it deliberately, but this is not another proof of the φ-myth, but just another consequence of it.

We will review the two most popular golden buildings:

The great pyramid

now, and the Parthenon later. Here we go!


What Herodotus did and didn't say.  

The Great Pyramid is the last standing of the wonders of the world, and is claimed to have been designed after the golden ratio. As proof of this, a text by Herodotus has been invoked, which, if it said what was claimed, would indeed imply this. But the only passage in Herodotus resembling what he supposedly said is in Book 2, 124 (here):  
Every way each face of its square is eight plethra, and the height equal.
To verify Herodotus, we will rely on the paper by the Cambridge classics scholar A.W. Verrall:
Herodotus on the dimensions of the pyramids, The Classical Review 12, No 4 (1898), pp.195-199 (here)
A plethron being 100 Greek feet, Herodotus gives the side as 800 Greek feet, which is approximately 776 English feet [Verrall p.196, 2nd column] or 236 m. This is remarkably accurate, because the actual side is something like 230 m [exact dimensions are considered below], and the use of the plethron as unit implies that Herodotus intended to give round figures. The actual height of the pyramid was some 147 m, and Herodotus is either completely wrong with his wild guess of 236—strange, given his accuracy on the side—or understands 'height' differently. Verrall explains that Herodotus understood this as being the 'ascending line of the pyramid', an easily observable physical length. In non-geometrical terms, we are told that the faces of the pyramid are equilateral triangles. In reality they are isoceles triangles with legs of 418 cubits and a base of 440 cubits. The error is some 5%—fair enough for a non-mathematical tourist who wanted to convey a rough idea expressed in plethra.

Measuring stone pyramids

Geometrical pyramids are easy to describe: the base is a square of a certain side, the apex is a certain height above the midpoint of that square. For stone pyramids, things are a little trickier. Their giant blocks of stone make measuring difficult, and they are severely damaged by history. Most have lost their outer casing, and several (among which, the Great Pyramid) have lost their top too. Several have a rectangular (non square) base, and one peculiar pyramid had a midcourse correction: the slope of the faces was changed abruptly, probably for fear it was too steep and might collapse.

The most reliable measurements of (even incomplete) stone pyramids with a square base are: the side a of the base and the angle α of the faces with the base.

From a and α, the (intended) height follows by h=(a/2)tanα.

The Egyptians had a peculiar system of measuring angles. Their seked was the horizontal distance needed to rise 1 cubit, i.e.

For the conversion: 1 cubit = 7 palms, and 1 palm = 4 fingers.

Also, they expressed non-integers in fractions with numerator 1, except 2/3 and 3/4 for which they had specific symbols. A palm being 4 fingers, one would expect a preference for the fractions 1/4, 1/2 and 3/4 in sekeds.

False precision, again

The pyramid of Khufu (Cheops), rightly called the Great Pyramid, is perfectly aligned and built with extreme precision. It has several measurements to offer, and strange relationships are seen to emerge if the right combinations are chosen with the right precision. In this respect, the great pyramid is no different from the human body. Thus, both π and φ have been discovered in the great pyramid. Historians of mathematics agree that this cannot possibly have been intentional, because Egyptian mathematics was not that highly developed. (See here and here.) Yes, magnificent buildings, from pyramids to Gothic cathedrals, have been created by builders using nothing but extreme skill and crude rules of thumb.

As for measures, one should not deduce anything from a single pyramid, no more so than from a single human body. Mark Lehner's comprehensive 1997 book on pyramids contains a table (p.17, here) giving measurements of 35 pyramids.

Five are unfinished (marked purple), four don't have a square base (orange), and nine have dimensions—base and/or height—lacking or not stated with certainty (green). This leaves us with a fine sample of 17 duely measured pyramids (yellow). There are 18 slopes to be considered, because yellow #2 is the pyramid with one slope for the lower part and another slope for the upper part. Our table below shows the remaining ones, ordered after slope. The numbers 1-18 indicate the original places in the yellow sublist above.

It is not clear which of Lehner's data are primary and which have been calculated. Most angles are given up to a second (like 52°7'48''), some up to a minute (like 42°35'), and a few up to a degree (like 52°). This seems to reflect different degrees of precision in measuring the pyramid as it stands today. Stone pyramids, however skilfully constructed, are never mathematically perfect, and the four faces need not have the same slope. The final figure for the Great Pyramid, for instance, is obtained by taking the average of several measurements of several faces. (See here the relevant pages from the magnificent book by Herz-Fischler, 2000).

To deduce the intended seked from the measured slope is still a different problem, and it may even have been when the pyramid was freshly finished. One can easily imagine that a minute difference in some basic template might accumulate to a final difference of a few meters. In the table above, we see that Teti and Pepi II have b=78,75 and h=52.5, while Niuserre has b=78.9 and h=51.68. The first two are identical and the third one, if not also physically identical (with apparent divergences due to the intrinsic imprecision of the measurements) was clearly intended to be identical. Yet the two slopes differ by more than 1°. The same happens with Senwosret III and Amenemhet III: same base, a height difference of just 3 m, yet slopes differing by more than 1°. Hence, it seems unjustified to distinguish slopes that differ by 1° or less. For all we know, they may have been designed after the same seked. Adding to this some millennia of physical deterioration, the conclusion is as follows: 
It is not easy for us to recognize the proportion originally intended by measuring existing buildings whose walls are frequently distorted by setting and destruction, certainly resulting in errors of calculation of one finger or more. [Here, p.12]
This does not, in any way, contradict the fact that the pyramids were built with unbelievable precision. But two architects, separated in time and place, having set their minds on the same seked, will probably end up with pyramids with slightly different slopes and different heights. The orientation of the base square and of the internal shafts must be perfect if the astronomical meaning is to be preserved, but the overall slope, let alone the final height, must not.

Taking this into account, here are our 18 angles (measured with a precision not given) and the seked as it was probably intended. The lower bound for the angle ('from') corresponds to the displayed seked plus 1 finger (i.e., +0.25), the upper bound ('to') to the seked minus 1 finger (i.e., -0.25).

If you think an error of 1 finger per seked is exaggerated, feel free to do your own exercise, for instance allowing an error of 1° in the angles. Note that a seked of 5½ palms gives rise to an angle of 51.842°, very close to the measured angle of Khufu's Great Pyramid which is (give or take a little) 51.844°.

A mystery

Now, where has the mystery gone? If you reject both the limitations of Egyptian mathematics and  the bounds on precision, you may feel inclined (hmm) to compare the measured slope

tan(51.844°) = 1.27278...
4/π = 1.27323...

and if you are in addition of the mystical kind, even to

√Φ = 1.27201...

Mystery at last! Why an architect would want to design templates for his workmen after the proportion 
or after a number (π) that cannot even be written with roots, nor geometrically constructed, now that's a mystery.

This said, even without any added 'mystery', the pyramids are a baffling achievement and a remaining tribute to the human mind as it was some five millennia ago. These days, we're closer to the dark ages again, and we must hope that civilization survives.

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Next stop: the Parthenon!