10/01/2024.
Today I'm going to write about areas.
The Octagon/Circle (here oriented up/down instead of its usual 51.8° azimuth)
was clearly constructed using the OCD (Observatory Circle Diameter) as a basis of measurement. In the lower portion we see that OCD in the Observatory Circle and the square (blue) with a side of one OCD. In the Octagon portion we see the square again (blue again); it's sides were pushed out (symmetrically) the right amount to catch the moon's lunistices.
One can also envision a circle (red) that (almost) encompasses the Octagon using those points on the square.
Here's an amazing fact: the area of the red circle is twice the area of the Observatory Circle! How impressed ought we be?
Not at all. It is a simple mathematical consequence of the construction. If the radius of the Observatory Circle is r, that means its area is πr2. For the other circle, the radius comes from the diagonal of the square, R, which is (√2)r, so the area of the larger circle is πR2 = 2πr2. The doubled area emerges automatically from the construction.
This is a cautionary tale about looking for and finding areas (or other mathematical quantities) that are automatic consequences of the mathematics. (Note that I am NOT here denying incredible feats of the Indigenous Peoples of the Hopewell Culture. But let's credit them correctly.)
Let's try to be objective as we look at some features of the Newark Earthworks Complex.
Along the way, we will use a special mathematical tool called "Degrees of Freedom".
What we will do here is look at the relative sizes of the Observatory Circle and the Great Circle. There have been many proposals about how that ratio came about. Let's work our way through them.
If you go on an Octagon tour, you will probably hear about the fact that this ratio matches the relative apparent sizes of the Moon at apogee (when it is farthest away) and perigee (when it is closest and a "Super Moon"). You can see that in this picture, which is a false-color LiDAR picture showing elevations around the Earthworks.
Now, the size of the moon is an accident of history. Eclipses (either solar or lunar) are an accident of the fact that the moon just happens to be the right distance from the earth right now in history to allow these eclipses. In the distant past, and distant future, the total eclipses won't happen.
So then we have to ask, are these relative sizes for the two Circles an accident, or was this something that the Indigenous Peoples of the Hopewell Culture consciously designed into the Newark Earthworks Complex?
What other possibilities are there for determining these relative sizes?
The following was in Hively & Horn's original 1982 paper, Figure 5. (Hively/Horn 1982) On page S10, they say,
Thus, the Fairground Circle avenue axis is parallel displaced from the corresponding axis in the Observatory Circle along a mean azimuth of 129°·7.
Here's that what means:
You may recall that the line from the center of Observatory Circle to the Great Circle points to the southernmost moonrise (and is 6 OCDs long). Redrawing that portion of their picture very carefully, you find that if you draw a line parallel to that southern moonrise axis which is also tangent to the Observatory Circle, it intersects the Great Cirle at a very interesting spot: the exact center of the opening. Coincidence? Deliberate?
H&H state that this opening points towards the southernmost northern moonrise, but a careful look clearly shows that it doesn't. Instead, this gateway appears to point to the spring or fall cross-quarter (the Irish Beltane or Lughnasadh).
Either one would have required the possibility that the Indigenous Peoples deliberatly chose the diameter of the Great Circle to make that work. With a different Great Circle diameter, this would not work.
In the picture, γ is the angle (south of east) of the southernmost southern moonrise (so its azimuth is 90° + γ). φ is the angle (north of east) of the cross-quarter day. If r is the radius of the Observatory Circle and R is the radius of the Great Circle, then simple trigonometry says that r = R·sin(γ+φ).
If this method is how the Indigenous People's deliberately chose the size of the Great Circle, then the apogee/perigee correspondence is, just like eclipses, an accident or coincidence.
That said, it is certainly possible that the Indigenous People's noticed this coincidence. If so, I suspect they would have seen it as further evidence of the sacred nature of the relative sizes of these two circles.
However, there is another possibility for their relative sizes. Bill Romain has suggested that the radius of the Great Circle is √5/2 times the radius of Observatory Circle. (Romain 2015: 62) One gets that number by constructing a right triangle with sides of 1 and ½: then the hypotenuse is √5/2, and we know the Indigenous Peoples were masters of squares and triangles.
The picture shows the Great Circle (with aqua-like topographic colors showing its location and elevations) with the Observatory Circle inscribed (in black) inside of it. (Again, LiDAR data.) The red lines show two diameters of the Observatory Circle and the half-length sides required to make a right triangle. The slender green lines show the hypotenuses of those right triangles. That looks like a pretty good fit.
So, what does it mean if this was the conscious decision of the Indigenous Peoples for determining the size of the Great Circle? First, it does make one wonder whether there was a specific motivation beyond using a simple-length right triangle. But beyond that, if true, it means that the apogee/perigee suggestion is coincidence. Furthermore, it means that our second option (Beltane) regarding the opening of the Great Circle is also coincidence. (And again maybe the Indigenous Peoples knew about it, or maybe they didn't).
For any of these, a conscious decision to use one precludes a conscious decision to use one of the others.
Have you noticed that I haven't talked about areas between the geometric figures yet? Here goes.
If you go on an Octagon tour, sometimes you will hear that the area of Observatory Circle is the same as the area of Wright Square, that the perimeter of Wright Square is the same as the perimeter of the Great Circle, and that the area of the Great Circle is the same as the area of the square that defines the Octagon.
This is not wrong (within the limits of reasonable measurements). But we need to look at the degrees of freedom to get a better feel for what is going on: what we have here are 3 equations with only 2 variables.
We use r for the radius of Observatory Circle—it is the base length and not a variable. We also use R for the radius of the Great Circle and L for the length of the side of Wright Square. Then
- Area of Observatory Circle and Wright Square: πr2=L2.
- Area of Great Circle and Octagon Square: πR2=(2R)2.
- Perimeters of Great Circle and Wright Square: 2πR=4L.
You can derive the 3rd equation from the first two, so this is a consequence of them. (The other thing that could have happened is that equation 3 contradicted the first two—if that happens that tells you something is wrong with your assumptions.)
It is generally agreed that the perimeter relationship is of little importance.
In The Geometry of the Newark Earthworks (Volker 2005: 20), John Volker says
It should also be noted that if the proposed area relationships between the Newark enclosures were the primary intention of the builders, then the perimeter-circumference correspondence would occur incidentally—a sort of mathematical 'bonus'.
And Hively & Horn, in Hopewell Topography, Geometry, and Astronomy in the Hopewell Core (Hively/Horn 2020: 124), say
If the area duplication of circles and squares was a deliberate feature of the design of the Great Circle (with the area of a 1054 ft square) and the Wright Square (with the area of a 1054 ft diameter circle), we would expect as a geometrical consequence that the Great Circle and the Wright Square would have nearly equal perimeters.
The design principle motivating the construction of the Great Circle and the Wright Square does not appear, however, to be the equality of perimeters.
But what those equations, and their relative importance, tell us is that it is pretty clear that the Indigenous Peoples knew how to make circles and squares of "the same size".
Let's look at that some more.
Today, we think that having "the same size" means calculating an area, multiplying various lengths and adding them together. We think of specific formulas that apply to circles and squares. We think of Greek geometrical proofs. But are those necessary? How might we make a square and a circle the same size if we were unaware of all that Greek mathematical framework?
We are biased by knowing about π. But I am unaware of any evidence that the Indigenous Peoples worked with π. (That said, just because such evidence was not preserved does not mean it didn't exist.) But there are other, really clever, ways for them to have done such constructions.
Saying that they may not have used π does not demean them. "Human genius" does not necessarily mean "Mathematical genius". It can also mean "Practical engineering genius". Which is what I think they had in spades.
This picture provides a starting point.
We know that the "same size" square has to fit somewhere between the two extremes that are shown.
Let's look further just to test ourselves and our naive, unpracticed abilities.
Just look at those possibilities and see how close you can get to determining which one has the circle and the square "the same size".
As a hint (and you can be darn sure that the Indigenous Peoples, or at least their "Practical Engineers", figured this out), you don't need to gestalt the whole picture. You can look to see if the corner of the square outside the circle looks to be "the same size" as the circular arc outside the square.
So, now that you've picked, "you" have the circle and the square close to being "the same size". Are there "practical engineering" things you can do to really narrow it down, narrow it down enough to construct giant earthworks in an "engine of world renewal" with the sizes of the circles and squares, and what they represent, precisely in balance?
There are a couple of things that immediately come to mind. The first involves modeling in clay. If you make clay squares and circles and carefully put water in them up to the same level, you can made sure they hold the same amount of water. Or you could instead place sand or small peoples inside them to even them out.
Or. They could use leather cut-outs and a balance scale. The corners of the square beyond the circle should weigh the same as the circle arc beyond the square.
But what I rather prefer is the idea of making an unfired clay square with a specific thickness, then shaving the corners and then carefully redistributing that clay to make a perfect circle with that exact same thickness. That would have given them the relative dimensions of the square and circle. Practical engineering.
But there was something else going on.
Oh, by the way. The option where the areas match was option "E". How well did you do? But "A" was a close second.
"A" is Bill Romain's suggestion that I showed in my 4th picture, that the radius of the Great Circle is √5/2 times the radius of Observatory Circle.
[This illustrates another possibility regarding degrees of freedom. Generally, different options are mutually exclusive. But sometimes one can have near matches with logical explanations. Which we will now explore.]
This picture redraws the 4th picture to create a full square (in red).
Do you notice that dividing the sides of the square in fourths makes it all work? (Of course it does, since the sides of the right triangle are in the ratio 2:1.)
If you have a naturally curious scientific (and/or practical engineering) mind, though, you might ask, what happens if I divide such a square in thirds, or fifths, or sixths, etc.?
And that gives you this picture.
This is what happens if you go to fifths (or maybe tenths describes it better since we are also looking at the centers of the pink squares).
The blue square has the standard side of 1 OCD, which is why it frames the Observatory Circle. The red square is the size of Wright Square (which is supposed to have the same area as the Observatory Circle), based on the Middleton survey. You can see how that square has been divided into fifths (pink). The green lines run center-to-center inside the one-fifth size mini-squares. You can see just how good of a match that is.
For the Observatory Circle, this is an even better fit than using fourths.
There is another reason that this is very special, and quite convincing.
Here we see the (larger that Wright Square) 1 OCD square that defines the Octagon divided into fifths the same way.
The ends of the green lines closely approximate the size of the Great Circle. But even better, and more convincingly, we see that the length of the corridor is given by the diagonal of one of those fifth-sized boxes.
This makes it quite clear that the Indigenous Peoples used this method to make their circles and squares be the "same size", and found it extremely important to do so.
Also note that this really has not required a knowledge of π.
Even Hively & Horn implicitly recognize this in their 2020 book chapter, where they mention π not at all:
The geometry of the Octagon shows that the Hopewell were likely concerned with the relations between the dimensions of the sides and diagonals of squares together with the diameters of circles that could be inscribed within or circumscribed about a given square. It also seems plausible that they could compare areas in some quantitative fashion. Experimentation with these ideas could plausibly lead to the suggested algorithm.
This has been my exploration of such experimentation.
One final item, bringing things back to thinking about degrees of freedom.
This picture shows one of the standard handouts, still in use, which shows a supposed deliberate alignment for the maximum southern moonset.
This could not have been a deliberate part of the design—we've run out of degrees of freedom. The starting point is wholly determined by the tip of the Octagon square. That the line is tangent to the Observatory Circle is wholly dependent upon the length of the corridor, which, as we have already seen, was determined by the areal relationships between circles and squares.
This has to be a coincidence. (Though, again, such a coincidence could have been noticed by the Indigenous Peoples and seen as a sign of the sacredness of their work.)
[Note added 10/20/2025: It was pointed out to me that the Indigenous Peoples shortened that leg of the Octagon that the red line crosses, so that that line does not go over the leg. Thus, they in all likelihood must have recognized that coincidence, and enhanced it.]
References:
Hively/Horn 1982
Geometry and Astronomy in Prehistoric Ohio, by Ray Hively and Robert Horn. Archaeoastronomy, No. 4 (1982).
Hively/Horn 2020
Hopewell Topography, Geometry, and Astronomy in the Hopewell Core. by Ray Hively and Robert Horn. Encountering Hopewell in the Twenty-first Century, Ohio and Beyond (2020), Vol. 1 Ch. 5.
Romain 2015
An Archaeology of the Sacred: Adena-Hopewell Astronomy and Landscape Archaeology, by William F. Romain. The Ancient Earthworks Project, Olmsted Falls, OH (2015).
Volker 2003
The Geometry of the Newark Earthworks, by John Volker. Unpublished manuscript on file in the Department of Archaeology, Ohio History Connection, Columbus.