Debunking golden ratio shells (1)

Nautilus Pompilius is an octopus living quietly in the deep waters of the southwest Pacific. It floats around with tentacles trailing, which is from right to left in the still below. (Nice videos on Youtube.)

This living fossil is the last cephalopod with an external shell. If you carefully saw an empty shell in two, you see an intricate and most elegant spiral. Here is an artist's impression:

It's a very popular icon, and in the Sacred Geometry of the Universe department second only to da Vinci's Vitruvian Man. The Natural History Museum (here) refers to both at the same time. Of the nautilus shell in Sloane's collection we are told that

Its coil follows the golden ratio used by Leonardo da Vinci to create proportions pleasing to the human eye.

Since we know that da Vinci did not consider the golden ratio at all, there is every reason to distrust the golden claim for the nautilus shell. Did the Divine Architect really design his humble servant Nautilus—among all creatures great and small—with ruler and compasses? In order not to spoil your pleasure, we won't reveal the answer yet, just stay tuned.

The shell in its natural position. The beautiful shell of the Nautilus consists of successive chambers, up to 38 of them, coiled around an axis. We could begin by placing it in the natural upright position, like we would do with the skeleton of a fish, a sea horse or any living creature governed by gravity.

The animal lives in the lower and largest chamber of the many it has successively built while growing to maturity. The aperture of this last chamber deviates from the vertical by an angle α which is some 30° in rest, and oscillates from 15° to 45° in motion. The abandoned chambers in the upper part create buoyancy.

The logarithmic spiral. Most golden numberists consider the Nautilus shell placed in a strange position, and set in a rectangular frame. This is very artificial, because there is nothing rectilinear about the whole Nautilus. The natural framework to use here are polar coordinates: an angle θ (counted in degrees, radians or complete turns, called whorls) and the distance r to the center point. There is every good reason to suppose that the curve is (approximately and for the most part) a logarithmic spiral, given in polar coordinates by

 with a>0 and b>0 constant. 

There is nothing very mysterious about exponentials appearing in natural phenomena. If the rate of change of something is proportional to what there is, the differential equation modelling the phenomenon has the form y' = b y with b constant, and the solution has the form y=a e^bx with a and b constant.

We'll consider a real shell in the next post, but first a few mathematical facts about the logarithmic spiral.

The only non-trivial statement is the fourth; for a proof, see here.

To describe the form of a logarithmic spiral, one can choose between

being: the constant angle between radius and tangent, the factor by which the radius grows on each quarter turn, half turn or complete turn.

(continued in part 2)

Debunking golden ratio architecture (2)

This stop on the Golden Tour is casual and relaxing. You might say funny, because it doesn't get much sillier than this.

Today's claim is, that the façade of the Parthenon fits in a golden rectangle. For proof, one is usually served a front view of the building (with the upper triangle, which is all but missing, completed), with some golden rectangle as overlay. Most of the time, the fit is poor and lines are thick. But, every now and then, some details are provided. The picture below is accompanied by the specification 'that the bottom of the golden rectangle should align with the bottom of the second step into the structure and that the top should align with a peak of the roof that is projected by the remaining sections.'

Even if this claim were true, and even if the golden rectangle had any esthetic value (quod non), no meaningful impression whatsoever can be expected from a view consisting of a triangular roof, a rectangle, and two steps of four that are visible. The building was not even designed to be viewed from there, and the upper step is—by design—not even straight! To disprove the claim, it suffices to obtain reliable measurements of the Parthenon, including details such as the height of the steps. This is what one gets for the horizontal and the vertical dimensions of the proposed rectangle:

h = 30.604 m
v = 18.162 m (upper step to top of triangle) + 0.552 m (upper step) + 0.512 (second step) = 19.226 m

giving a ratio of v/h=0.62821... Wow! Weren't we expecting φ=0.61803...? And to say that this rectangle, no doubt, was selected among many other possible ones because it was the most 'golden'. It's 'approximately φ' all right, but everything is 'approximately φ' within a certain precision.

A secondary claim, often accompanying the primary one, is that the building abounds with (almost) golden rectangles. Here, we need not spend any time verifying, because the claim means nothing. There are so many rectangles to choose from that doubtlessly many of them will be 'approximately golden'. Here a fine selection offered for your entertainment.

When you're done, you might hunt throughout the Parthenon for rectangles displaying a 9:4 proportion. This time, your findings may even be meaningful, because this simple proportion did play a role in the design. (See here.)

Ah, lest I forget! I owe you a reliable diagram with Parthenon measurements. You can't beat Athanasios G. Angelopoulos, author of Metron ariston (unfortunately in Greek), who provides us with measurements up to a mm. Here two detailed views, front:

 and sideways:

Measures are in meters, but the additions in bold refer to a different unit of some 45 cm. Wherever possible, the author expresses his 'bold' measurements as mathematical expressions involving roots, powers, φ (which is (√5+1)/2, what we denote by Φ), π and e. You could do the same (describe real numbers as values of special functions, that is, not measure the Parthenon) by consulting Borwein's A dictionary of real numbers.



Good hunting!

*

Added April 22, 2017. The inventor of Golden Numberism is the 19th century German teacher and writer Adolf Zeising. He says so himself, because the title of his 1854 influential classic translates as

A new theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole of nature and art. 

His weird and sometimes hilarious convictions (deeply rooted in Hegel's idealism) will be discussed separately. Here we restrict ourselves to the two pages (these) where Zeising introduces the Parthenon as the Golden Ratio crown jewel it has remained until today. In his image below we have transferred the vertical measure in red to its horizontal origin.

Zeising claims —the first one to do so— that "the" height (blue) and "the" width (red) of the Parthenon are in a golden proportion, and he provides us with the measures: height=65 German feet of some kind, width=107 feet. The measurements being up to one foot, we only know that the width is from 64.5 to 65.5 feet, and the height from 106.5 to 107.5. Hence, the ratio is between 64.5/107.5=0.600... and 65.5/106.5=0.615... Thanks, Adolf! Whatever the Parthenon ratio is by accident or intention, it's definitely not the golden ratio, because 0.618.... is outside the bounds obtained. Yet Zeising sees only an unimportant fraction of difference between this measurement and his ideal. In spite of the ridiculous number of digits in his models, he's less fussy when reality joins in. On p.310 he considers 11/18 (which is a lousy 0.6111...) as a reasonable attempt of creative nature to realise what he calls our law.

The actual measures given by Angelopoulos are: width=30.604m, height=19.738m if (as in Zeising's drawing) only three steps are taken into account, or 20.038m counting also the fourth step. The ratio is 0.644... resp. 0.654..., both way too big to be golden.

*

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...

to

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.

* *

Next stop: the Parthenon!