Following on from the previous post on a measure of √2/√3 inches put forward by David Kenworthy, I'd like to look further into why the inch itself may be a significant base unit in which to express important ratios, how it might work at the Great Pyramid of Giza, especially with √3, and what this ratio itself might signify.
There are many interpretations of the Great Pyramid of Giza's intended dimensions. Why should it matter what the architects of this ancient structure adopted as their units of measure, and in what quantities? Finding a pleasing theory of of precise units of measure used in precise numbers and proportions can reveal something, albeit theoretically, about the civilisation that created it, in terms of mathematical and engineering sophistication, and the structure of a society able to produce these. No texts documenting the design or build of the Great Pyramid, contemporary with its construction, survive, as is to be expected from such a long time ago. Whether the architects used some sort of cubit, foot, inch, digit, or metre, or any other unit, it is all, for us, a question of interpreting the evidence of the dimensions as we find them today.
Below are some pages from Petrie's work on measuring the Great Pyramid of Giza.
(4)
Petrie's value for the mean base side of the pyramid is thus 9068.8 inches, and the height is 5776 inches, with lee way either side of 5 inches for the base and 7 inches for the height. It's been known for a long time that there is a pi ratio to be found between the base and the height. Indeed, 5776 x π / 2 = 9072.9196, which is in close accordance with Petrie's measures. Using 22/7 for π produces a closer match still, 9076.5714 inches.
One interpretation of the Great Pyramid's dimensions that I find intriguing and potentially significant involves the square root of three. I was surprised to find that 10,000/√3 is very close to the height of this pyramid in inches. 10,000/√3 works out in decimal notation as 5773.50269, which is within Petrie's parameters for the height. Using pi, the resulting figure for the base side also fits within the parameters. 10,000/√3 x π / 2 = 9068.9968. This is one possible interpretation of the architects's design: 10,000 inches as a starting point from which to derive a measure for the height, 10,000 divided by the square root of three, and multiplied by π/2 for the base. The imperial units such as the inch and foot are not necessarily English in origin. According to Mauss, the yard is in fact 5/3 of a two Persian measures, the Amman cubit and Zyad cubit, and the Persian Royal cubit is 72/100 yards, or 25.92 inches. The number 25920 is a significant one, being 12 x 12 x 180, and is found in ancient texts. This in itself would suggest that when an important unit can be expressed in this number of inches, that the inch itself is important as a base unit. What Mauss calls the cubit of the architects of Egypt is 30.24 inches, (and 3024 is 12 x 12 x 21). which times three hundred gives 9072", close to Petrie's measure for the mean base of the Great Pyramid.
One way to interpret a 10,000 inch measure divided by the square root of three for the height of the pyramid is to envisage a cube, with sides of 10,000 inches. The diagonal of this cube corresponds to the height of the Great Pyramid.
Another way is to think of the square root of three is as part of a hexagon: the height of a hexagon is, with sides of 1, the height is 1/√3.
Or quite simply, you can imagine an equilateral triangle positioned in relation to the Great Pyramid.
The number of metres in the height of the Great Pyramid is also potentially signiicant: take 146.94945 metres for the height, equivalent to 5773.5 inches, which is within Flinders Petrie's parameters, multiply by the sqaure root of 3 and the number of days in a year, 365.242199, divide by 1,000 and you have the number of days in a lunation, 29.53059. This reflects a quirk of the relationship between metre and imperial inch.
Mythmatics
There are several curious accounts of mathematical relationships between major deities, incorporating the square root of three, that may be relevant here. Michael S. Schneider, in his fabulous book A Beginner's Guide to Constructing the Universe, writes about how "ancient myths were represented by geometric constructions. With a deity as numerator and another as denominator, Apollo / Zeus = 1060 / 612 = 1.732 = the square root of three, the relationship of the axes of the vesica piscis, the crossing opposites through which the musical scale is born. " (page 242). He calls this "mythmatics", a great word. There are two main features to this approach. Firstly, it is not about arithmetic, in that it is purely about proportion and ratio. Secondly, where arithmetic does come into it, so does gematria: the names of these deities are associated with certain values, in that each letter carries a number, and so the name is a sort of mathematical (or "mythmatical") code. Michael Schneider writes:
In the alphanumeric system of gematria, the name Apollo has a value of 11061. The name Zeus, the central Olympian deity, has a value of 612. The name Hermes, the messenger who invented the lyre, has a value of 353. The relationship of Apollo to Zeus can be expressed mathematically as 1060 / 612 = 1.732, the value pf the square root of three (1.73215...) And the relationship between Zeus and Hermes is virtually the same (612 / 353 = 1.7236) Root three also describes this relationship between the two axes of the vesica piscis, which is the proportion within the triangle. (5)
Michael Schneider has a diagram that I've recreated here demonstrating the square root of three ratio between "Apollo" and "Zeus".
Or it could also be expressed in this way:
Another researcher has a slightly different approach but the square root of three ratio is still the same between these two deities.
Daniel Gleason writes:
On a higher level, Zeus, Apollo, and Hermes were also mathematical metaphors. The diagram below illustrates how the Greek spelling of each god's name results in a gematria value that can be used to unite the gods in a single Sacred Geometry diagram.
Daniel Gleason has slightly different values in terms of the gematria of the names, and also includes a third god, Hermes.
Zeus: 612
Apollo: 1061
Hermes: 353
However, while gematria is an interesting and somewhat mysterious field of study, numerical values are not necessary to correspond to Apollo and Zeus, or any other deity, for the relationship between these deities to be the square root of three. This "number", if it is a number, √3, cannot be represented accurately by any fraction. If otherwordly beings are going to be invoked in relation to the square root of three, it must be because nowhere in our earthly world does such a number actually exist. In all the diagrams above, of course we can measure the sides and heights of the various shapes mentioned, even when the square root of three unites them, but we measure them as approximations. Michael Schneider's concept of mythmatics opens the door to a different way of making sense of irrational numbers and incommensurability. (6)
Associating lengths or numerical values with gods was an Egyptian practice too. Indeed as Lepsius remarked (1866, 18) the subdivisions of the cubit are sometimes unequal and the subdivisions of the digit are not the same length. For example, in the cubit of the museum of Torino, each digit is associated with a god whose name is written above it.
Irrationality
The square root of three ratio linked to the gods of the sky and the sun offers an intriguing window into a mode of thought that seems quite odd. Gods are related to each other by an irrational number. Curiously, the son, Apollo, is greater than the father, Zeus. But the more you think about it, the more it makes sense. There is no number that can be squared to make three. Yet, the square root of three ratio can be observed in an equilateral triangle, a hexagon, or a vesica piscis. Perhaps we are wrong to think of the square root of three as a number. Rather, it should be through of as a concept, a mathematical, philosophical, and perhaps mythmatical concept. We cannot measure the side of the equilateral triangle and its height precisely in the same unit. When we go from one to the other, we necessarily move from one framework to another, one might say from one plane to another. When researchers claim that "the Ancients" didn't know about square roots, what exactly do they mean? We tend to think of square roots as the numbers that appear on our calculators when we enter the square root symbol and a number. We tend to think of the square root of three as roughly 1.73205, or the square root of two as roughly 1.414, as they appear in text books or on our calculators. Perhaps this leads us to believe that the square root of a number is also necessarily a number. But when a square root is not an integer, such as the square root of four being two, we rely on approximations such as the ones our calculators give us. We may even believe that we "know" the square root of two or three, because it's what our calculator tells us. But are we deceiving ourselves? When we say that long ago, even sophisticated mathematicians, such as those from Ancient Greece, Mesapotamia, India and Egypt, didn't know that a square root could be irrational, or didn't know the value of the square root of two or three, we position ourselves on very shaky ground. There is no number which, squared, gives two. there is no number which squared gives three. We may be looking down on our ancestors and predecessors for not knowing something that is fundamentally unknowable. This puts our own contemporary philosophical and mathematical position into potentially an absurd position. We pretend that knowing the value of the square roots of two and three is not an insurmountable problem, and go about measuring geometrical shapes as if it were possible to do so precisely, in decimal notation, using one single unit of measure for the whole shape. Surely, if we recognised that certain square roots cannnot really easily be thought of as numbers at all, that the various parts of a shape that relate to each other by these square root ratios do not belong within the same measuring system, we could begin to reflect on the meaning of irrationality a little better. Perhaps it is us who are irrational in conferring a concept of irrationality onto the measure of a geometrical shape, yet pretending that the square root of two or three or five can be measured in the same unit as the other parts of a shape. While it makes sense to say the side of a hexagon is 1 inch or cm, and it's height is approximately 1.732 inches or cm, it makes less sense to say it measures √3 inches or cm. We might redeem ourselves if we had a different unit of measure for the side from the one for the height of this hexagon. This applies equally to a square (side and diagonal), to various triangles (sides, height, hypotenuse), circles (diameter, circumference), and polygons, where the square root of a number is a so called "irrational" number, or where pi or Phi, the golden ratio, are involved.
Plato
THEAETETUS: Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit: he selected other examples up to seventeen—there he stopped. Now as there are innumerable roots, the notion occurred to us of attempting to include them all under one name or class.
SOCRATES: And did you find such a class?
THEAETETUS: I think that we did; but I should like to have your opinion.(2)
In Plato's dialogue Theaetetus, there is a discussion on the problem of the various parts of a shape which cannot be measured in the same unit. It's the problem of incommensurability. A square's sides and diagonal are linked by a ratio of root two. If the one is measured in a particular unit, the other then cannot be exactly measured in the same unit. The same would apply to a circle, or example, which has the its diameter and circumference linked by pi.
The discussion is really about knowledge in general, but this little interlude provides insight into a problem that is often overlooked: what are the implications of irrationality in a geometric shape when it comes to measure? It seems like an impossible situation. How can we get over the problem of not being able to measure precisely the various parts of shapes such as a square or circle in the same unit?
It's possible that the problem of incommensurability was partly addressed (if not solved), at some time in the distant past, by designing a system of measure that provided different units for these various parts that are in metrological conflict. This would require a two or three (or more) tier system, and these tiers should be related to each other by the very irrational ratios that characterise a square, a circle, or any other shape that encapsulates irrational numbers. If that were the case, the irrational relationship between the various parts of the square or the circle would simply be transferred to the relationship between the various units used to measure them. It might seem pointless, but it would allow for precise measurement of all parts of a circle or square in rational numbers. An irrational number of parts to a linear measure is problematic with regard to precision and ease of calculation, and perhaps it also seems counter to an ideal of being able to measure everything in the world. To a certain extent, the problem of measuring these parts would be overcome. After all, today we have different units to measure length and area and volume. Why don't we have a different unit to measure the various lengths of a figure that are irreconcilable through irrational ratios?
Theoretically, units of Phi metres make good sense as part of a system of ancient metrology. For example, a Royal Egyptian Cubit of 20.618181818" is really just a metre of 39.375" (so 54 Egyptian or Roman digits of 0.7291666667") multiplied by an applied version of Phi squared (in it's Fibonacci number guise of 144/55) and then multiplied by 2/10. Why would you need such a unit relating to another by either an applied Phi or a theoretical exact Phi? Could the metre (or a unit close to our current metre) be used to measure one part of a shape, or even derived from one part of a shape, and the 'Phi metre' or Royal Egyptian Cubit another part of the same shape?
The Egyptian unit named remen is related to the Egyptian royal cubit by the square root of two. Petrie, in the Wisdom of the Egyptians (London, 1940 p. 71) wrote:
The half diagonal of this [royal cubit of 20.6 ins.] was the remen, a second unit of 14.6 ins., which was divided in 20 digits of .73 [ins.]. Thus, by the use of the diagonal, the half of any square area could be readily formed and defined. That this was fully recognized is shown by the half of the area of 100 x 100 cubits being also called remen in land measure... The remen means an arm, or the branch of a tree, and agrees with the fore-arm down to the clenched knuckles, still a favourite mode of measuring in Egypt.
The question of measuring the various parts of a circle is raised by Geoff Bath's excellent book on stone circles.(1) Geoff writes about diametric and perimetric units. This means that one unit is for the diameter and another the perimeter of a circle. But if a circle with a diameter of one meter has a circumference of six royal cubits, why is the royal cubit not related to the metre by pi instead of Phi squared? The 20.618181818" Royal Egyptian Cubit is 39.375" x 2/10 x 2.618181818, which is Phi squared approximated by the two Fibonacci numbers 144 and 55. Pi is roughly Phi squared, pi being Phi squared x 6/5, and this works perfectly with the Fibonacci number approximations: 144/55 x 6/5 = 3.1418181818. You can think of the Egyptian Royal Cubit in fact as a Phi squared metre x 2/10.
Is there a unit of 1 39.375" metre x 2/10 x 3.1418181818 = 24.7418181818" ?
So in fact, in an applied sense, not exactly, a circle with a diameter of 1 metre of 39.375" gives either a circumference of 6 Egyptian Royal Cubits or 5 units of pi metres.
Plato's dialogue continues:
THEAETETUS: We divided all numbers into two classes: those which are made up of equal factors multiplying into one another, which we compared to square figures and called square or equilateral numbers;—that was one class.
SOCRATES: Very good.
THEAETETUS: The intermediate numbers, such as three and five, and every other number which is made up of unequal factors, either of a greater multiplied by a less, or of a less multiplied by a greater, and when regarded as a figure, is contained in unequal sides;—all these we compared to oblong figures, and called them oblong numbers.
SOCRATES: Capital; and what followed?
THEAETETUS: The lines, or sides, which have for their squares the equilateral plane numbers, were called by us lengths or magnitudes; and the lines which are the roots of (or whose squares are equal to) the oblong numbers, were called powers or roots; the reason of this latter name being, that they are commensurable with the former [i.e., with the so-called lengths or magnitudes] not in linear measurement, but in the value of the superficial content of their squares; and the same about solids. (Italics my own)
SOCRATES: Excellent, my boys; I think that you fully justify the praises of Theodorus, and that he will not be found guilty of false witness. (3)
It is a reminder that while two lines measures in a shape may be related by an irrational ratio, one of these may be related in a rational way to the area of the shape. Plato reminds us that when a number is called power or root, it is not just incommensuarable with another linear measure, but is also commensurable with the area. Perhaps more focus could be put on area measures rather than linear measures to understand the relationship between the various ancient units, and why square roots or pi might define their ratios.
Notes
Bath, G.J., 2021, Stone Circle Design and Measurement: Standard Units and Complex Geometries: 2: Stylised Plans and Analysis of over 300 Rings
Plato, Theatetus, translated Benjamin Jowell, The Internet Classics Archive | Theaetetus by Plato (mit.edu)
Ibid
Petrie, W.M. Flinders, 1883, The pyramids and temples of Gizeh, London : Field & Tuer ; New York : Scribner & Welford
5. Schneider, Michael, 1995, A Beginers Guide to Constructing the Universe, Harper Collins New York
6. Unfortunately, neither Schneider nor Gleason provide a source for their claims. I have not been able to find any reference to the square root of three in relation to any Greek god, in any of the ancient sources that I've looked at. However, this book may well have inspired both researchers: Jesus Christ, Sun of God: Ancient Cosmology and Early Christian Symbolism Paperback – Illustrated, 1 Oct. 1993, by David Fideler