A few posts back, we took a first look at Hadrian’s Library based on the plans and data from Athanasios Angelopoulos, focusing on the Library itself and the adjacent reading rooms and auditoriums. The general assessment is that rather than consistent use of Greek Feet in the design, what we actually see is a variety of different ancient units being deliberately used, as we often see in so many examples of ancient architecture.
One always gets the sense that the architects expected and welcomed interpretive efforts, and frequently tried to put as many of the interpretive tools that we would need into our hands as expeditiously as possible. Part of that is proving to be the nature of the units themselves and their relationships to one another, and part of that is what we seem to see with Hadrian’s Library complex, where a deliberately diverse collection of units appear to be on display.
A good way of conceptualizing our basic metrological toolkit.
(I’ve deliberately made the corkscrew the mile to reflect the fact that there may be some tools that are best suited only for certain tasks).
I will be honest and admit that I’m having a few interpretive troubles with Angelopoulos’ diagram, as to what measures belong to which parts exactly, so I’m not certain how much data I can recover from the plan. I may also be having a bit of trouble getting some of the parts to add up correctly as they are described.
I’m particularly intrigued with the pool (cistern), but this is one of the areas I find somewhat confusing. In this diagram by Angelopoulos, we can see that the cistern has a substantial rim that is perhaps as wide as a foot or even more. This would easily afford it with both inner and outer length and width values, whereas I only have single figures for these attributes.
Diagram of Hadrian’s Library by Athanasios Angelopoulos, from http://athang1504.blogspot.com/2016/08/the-roman-agora-of-athens.html
However, we generally have reason to believe (based on experience) that the data is nonetheless valid and accurate until proven otherwise, and in that spirit I have attempted to look into the matter further.
Here I’ve adapted Angelopoulos’ diagram of the cistern and added several details along with conversions to “Imperial” Feet, our primary interpretive unit.
The semicircular sections at the ends are proving quite remarkable. Their radius value in feet would be 18.72047244, which hopefully readers are learning to recognize as 1/12 of the Venus Orbital Period in days, or as the value of the Petrie Stonehenge Unit in feet: 224.8373808 / 12 = 18.73644840, a value which can be reduced to Hashimi Cubits or Egyptian Royal Feet.
The diameter then would be 18.73644840 x 2 = 37.47289680. I sometimes comment that it seems curious that we don’t see this figure more, nor this value divided by Pi, because these figures have very significant and very powerful pedigree.
Readers have probably already guessed that the projected perimeter then is 37.47289680 ft x Pi = 117.7245771, so that the perimeter of the circle is measured in Harris-Stockdale Megalithic Feet (I am using the figure for the HSMF which I prefer of 1.177245771 ft rather than Harris and Stockdale’s stated figure of 10 x (sqrt 2 / 12) = 1.178511302 ft).
Let’s set that thought aside for just a moment so we can look at several other aspects. Assuming the cistern is a symmetrical construct, we can calculate that the widths excluding the diameter of the semicircles is at either side of the semicircle 0.539 m = 1.768372703 ft.
It’s possible that this figure represents 1.765868657 ft, which would be another expression in Megalithic Feet: 1.177245771 ft x 1.5 = 1.765868657.
Using these figures, the total projected width – I am presuming of necessity here the interior width – of the cistern would be 37.47289680 ft + 1.765868657 ft + 1.765868657 ft = 41.00463441 ft.
I find this too somewhat remarkable, because I’m fairly certain I recognize the figure. I’ve been seeing it since as long ago as my exchanges and collaborations with Michael L. Morton, but never before can I recall having seen it in a situation that is so suggestive that an ancient architect was aware of this unusual number and chose to commemorate it architecturally as directly as actual incorporation of the number as a physical measurement.
The number in question is, I presume, 41.00542110. While I cannot recall ever having found this number is ancient American architecture with any certainty, it entered the proceedings at very least as a useful abstraction. It did so because it is 1/2 of 82.01084221, which was part of an inquiry into whether whether the Mayan had used a larger number analogous to their known calendar number 819. 819 represents a formula that uses 364 days rather than 365: 819 = Venus Orbital Period 225 days x Solar Year 364 days = 81900; 225 x 365 = 82.125, and the reciprocal of the standard Remen is 1 / 1.216733603 = 82.18725919 / 10^n.
Thus 820.1084221 entered the list of potential candidates for the refined meaning of “225 x 365”. It boasted a Mayan pedigree, and even prompted the initial discovery of the “Best Lunar Month” value (giving rise also to rumors about a possible “Lunar Remen”).
The frequently seen diagram of Tikal Temple I by Carl Munck, labelled with Teobert Maler’s data.
82.01084221 = (1 / 1.219350970) x 10^n; 1.219350970 = (360 / 29.52390320) x 10; 29.52390320 = “Best Lunar Month”.
The “Mayan Pedigree” part is that the interpreted value of the length of Tikal Temple I’s temple platform = 38.81314681 exactly, and 38.81314681 x Pi = 12.19350969
The next thing I would like to call attention to regarding the diagram of Hadrian’s courtyard cistern is the length value 57.82 meters, which according to Angelopouos is also the width of the courtyard. 57.82 m = 189.6981627
Hopefully there are at least a few who are now equipped to recognize this number., or numbers like it. They belong to a class of numbers that we can call “Half Venus Cycle Builders” or “Calendar Round Builders”. They are too low to be valid figures for the ~18980 day Half Venus Cycle aka Calendar Round, but but we can divide 360 x 10^n by these “Builder” numbers to “build” valid figures for the HVC/Calendar Round.
The “Builder figure” for HVC A for example is 189.6334628: (360 x 10^n) / 189.6334628 = HVC A 18983.99126, and that may well be be what we are seeing here.
We find “Builder figures” along with actual HVC/Calendar Round values in Great Pyramid equations with relative ease.
It was probably beyond the scope of my blog posts to share this piece of history with readers, but it took me a long time to warm to the idea of these “Builder figures”. Originally I was against the very idea and thought of them as being little more than generally useless inferior versions of actual calendar Round approximations.
The first time I was finally forced to accept them, I was provided at the same time with the opportunity to understand why we are sometimes forced to write the HVC/Calendar Round “backward” as the “Builder figures” and why it can be very much worth the trouble to do so. This was early on in my Mesoamerican studies when I encountered the indisputable mathematical brilliance of the Aztec Sun Stone, from which I received tremendous encouragement that I was finally on the right track to understanding what all this ancient architectural math was for).
The Aztec Sun Stone, an ancient calendar formula calculator that uses measurements in “modern” “Imperial” Feet.
If we don’t deliberately invert the thickness projections for the Sun Stone into their reciprocals before performing the calculations, we will generate the “Builder figures” A and B rather than their “forward” forms, HVC/Calendar Round A and B, from the projected dimensions. Of course, all that we require to “build” the desired figures are these “builder” numbers and a 360* circle, which we seem to have been provided by the very shape of the artifact.
For what it’s worth, I can think of at least one example where the Maya seem to have built one of these things with the very same radius, diameter and circumference. The site is Yaxchilan and the location is structure S39 A4 where the artifact (“altar”) has the distinction of having been located indoors. The particular data point comes from older sources via Carolyn Tate’s book on Yaxchilan, but the subject was also covered by George Andrews.
Raw data and diagram for the Mayan “altar” at Yaxchilan and its location, from George F. Andrews, “Architectural survey Yaxchilan and Bonampak Chiapas, Mexico“. Diameter is 1.14 m = 3.744015748 ft = Circumference 11.75005126 / Pi
Two different independent academic data sources provide the data from which the interpretation of the Aztec Sun Stone was based on.
Allow me to go ahead and cut to the “show-stopper” here – the presence of a HVC/Calendar Round “Builder number” isn’t all that Hadrian’s cistern has in common with the Aztec Sun Stone – the projected proportions from the semi-circles at the ends of the cistern, and the proportions determined for radius, diameter and circumference of the Aztec Sun Stone (and Yaxchilan “altar”) are identical except for the location of the decimal point.
I am greatly devoted to the proposition that until proven otherwise, these mathematical constructions are part of an universally well-distributed ancient mathematical language, but even then I am still rather surprised to see such a remarkable set of parallels as this.
–Luke Piwalker
Luke Piwalker 