top of page
May 2029 min read

An Exercise in Middle Woodland Geometry III


Part III - Geometry


"... in a useful physical theory the state of a system represents its method of preparation: two systems are in the same state if they have been prepared in the same way, or more precisely if our information about the method of preparation is the same in each case. This conclusion may be modified slightly: two states or methods of preparation need not be distinguished if they are equivalent in respect of any prediction which might be made - that is, if they correspond to the same assertion concerning the result of any experiment which might be performed on the system." [Giles 1964]


"I am interested in sites, the fixed places in the topography where man may periodically pause and carry out actions. I am concerned with site patterning both in the frequency with which occupations occur at different places, and in the processes which generate associations among archaeological materials at sites. Site patterning in both within-place and between-place contexts is a property of the archaeological record. The accuracy with which we are able to give meaning to the record is dependent upon our understanding of the processes which operated in the past to bring into being the observed patterning. Put another way, our accuracy depends upon our ability to correctly infer causes from observed effects." [Binford 1981]


"I find that archaeologists, like mathematicians, have a hard time thinking (or describing or imagining?), simply, of a team of very smart, creative people in the field with some long cords." [Hancock 2024a]


The preceding three quotes represent three views from a purely mathematical abstraction to what is called "mid-range theory" in archaeology, to a strictly empirical "embodied" comment on the utility of human groups. I will use these quotes as a template in this article. What exactly is Hopewell geometry and to what extent could it have been used to build their remarkable earthworks?


The word 'Geometry' comes from the ancient Greek words 'land' + 'measure.' [Wikipedia. Geometry] This fits well for a study of Hopewell earthworks. In the previous article, I talked about the problem of measurement. [Measurement] James Marshall categorized the earthworks he surveyed. Two of his categories were 'geometric" and 'crypto geometric.' [Marshall 1996] Within these categories he finds a Hopewell geometry.


An important concept is that of a 'construction.' This comes from old French, meaning 'together' + 'to pile up.' [Wikipedia Construction] Again this fits well with the earthworks. Construction in geometry deals with proving geometrical concepts using a minimum of physical tools. Construction in building deals with creating physical structures using multiple physical tools. For our purpose, it is important to look at the minimum physical tools needed for earthwork construction. The fashioning of these tools must also be within the culture's level of material technology. The connection between the two different meanings of construction is the creative human mind. It is this connection that I will use to show a path between these two meanings.


I have been thinking about bias and how it affects our thinking. I’ve run into several definite biases while trying to understand the Hopewell phenomenon:


  • Gender biases – Try as one might, it is incomprehensible not to see how deep gender biases exist in science. Archaeology and to some extent anthropology have erased women from existence. In biology, especially behavioral biology, there is even bias towards male animals! Things are rapidly changing but there is still a need for more female scientists and sometimes old ideas die hard. For instance, it has been considered a fact that women have inferior spatial skills to men. This is based on studies where male and female college students are given a spatial test where they look at a 3D drawing of an object and then try to match it to drawings of other objects, one of which is the original object rotated to a different view. It turns out that women did worse because the test was timed and if the women were given a few seconds more they were as successful as men. This suggests that women use different cognitive tools to solve the problem than men and take a few seconds more. Whether this shows a biological gender difference or a cultural difference is currently unknown, a good example of how picking apart bias can advance science. [Toth and Campbell 2019]

  • Material culture bias – If all perishable artifacts survived, they would overwhelm our current collections of nonperishable objects by about 20 to 1. [Jolie & McBrinn 2010] The fact that most of these perishables were made by women reinforces gender biases in archaeology.

  • Cognitive capacity bias – There is no biological evidence for any evolution of cognitive capacity in modern humans during the last 50,000 years. If anything, we might be losing some capacity through selective disuse. The idea that modern humans are somehow cognitively superior to older cultures is a misreading of evolutionary theory into ideas of cultural change. Smart creative humans have made smart creative things since there were modern humans.

  • Greek mathematics bias – The dominance of Greek deductive logic and the success of using this logic to discover proof has led to weaknesses and delays in developing other forms of thinking such as non-Euclidean geometries, an understanding of randomness, and complexity. This has also led to a misunderstanding and diminution of the mathematics of other cultures. If what Gerdes proposes is correct: [Gerdes 2003] that an understanding of geometry comes from material culture, particularly the perishable material culture of weaving and basketry. Perhaps women invented geometry in addition to agriculture.


We will never know how Hopewell culture created, transmitted, and used mathematics. It is useful to speculate on what mathematical concepts, at minimum, they needed to accomplish what they did. Looking at what is known about their material culture as a whole, and this includes the earthworks as artifacts, there are a few clues:


  • They liked high contrast and color. Earthworks are often capped with two contrasting colors of clay, their Great Houses are often paved with colored clay, gravel, or light or dark stone. Ohio has many amateur collections of artifacts going back to the first settlements because almost everywhere one looks, one can find something on the ground. Looking at just the flints one can tell what are Hopewell by the range of colors.

Hopewell Points [KET Artifacts]

  • They liked shiny surfaces. Mica can be used as a mirror and copper can be polished to a mirror surface. It was thought that all the ditches around and inside the earthworks were drainage ditches but there is ample evidence now that these were water features. [Riordan 2020] These formed both a barrier and a mirrored surface. Other art objects show they were comfortable with mirror symmetry and four-part symmetry.

  • Their geometric earthworks used squares and circles singularly and in combination. They also created ‘avenues’ of parallel lines. There are also ellipses but Marshall states that these are not true ellipses but draftman's ellipses, of which there are several types, and two half-circles connected by lines. As far as I can tell, an octagon appears only twice; in Newark and at High Banks in Chillicothe. Circles appear everywhere in their artwork, most spectacularly in the large copper disks they wore in their ears. [Ruhl 2005] Squares not so much although many of their finer ceramics had tetrapod ‘feet’ allowing them to stand up and split cane matting used to cover clay caps on tombs have a square pattern.

Copper Cut-out Ohio [Ruggerie, Adena and Hopewell Art]

Copper Frog or Salamander Effigy (Ohio) [Ruggerie, Adena and Hopewell Art]

Tetrapod Jar, Mound City, Chillicothe, Ohio [Ruggerie, Adena and Hopewell Art]

Split cane mats, Story Mound I, Hoecake Site, Mississippi County, Missouri. [Marshall 1984, p. 124, fig. 9.6]


  • I know of no artifacts that could represent a 'map' of an earthwork. Maps have their own biases as there is an assumption that they are a 'Western' invention. In a feminist critique of GIS, Donna Haraway calls maps "The God Trick." [Haraway 1990] She compares the simplicity of a map, stripping bare a region into its basic elements, to the male gaze. Although this critique of GIS is probably necessary the idea that women have inferior spatial skills has been challenged. The God Trick is for everyone. [Wheatley 2014] Maps do exist in indigenous America. There is what is believed to be a map carved in a rock at the confluence of the Ohio and Mississippi rivers. [Pauketat 2009] Rock art worldwide is believed to depict local directions Maps started showing up when the buffalo economy produced maps on buffalo hide. [The Illinois State Museum. Native American Mapping Tradition] This was after the Middle Woodland period of the Hopewell. In the final article of this series, I will talk in detail about spatial cognition, maps, and the skill of wayfinding. [Memory]

Shapes, squares, and circles have special properties, called invariance. All squares and circles, whatever size, are the same. Squares are defined by rules: a square has 4 equal-length sides and each side connects to the other forming a 90-degree angle. Only a shape that follows these rules can be called a square. No matter the length of the side, or how the square is moved or rotated, it is still a square. This is the invariant. A circle is defined by its construction, it is a shape constructed by following a set distance around a point. Like a square, this set distance doesn’t matter as long as one follows the construction rule, one gets a circle. A square has four sides but a circle only has one. A circle can also be thought to have infinite sides as a circle can also be constructed by repeatedly doubling the sides of a square. Notice that this construction is an estimate of a circle as one can never double the sides an infinite number of times.


Squares and circles have another invariant property in that both shapes contain a constant, a special number that never changes.

For a circle, this constant is called π and it is the ratio of the diameter, the distance from one side of a circle to the other going through the center to the circumference, the distance walking around the circle. π cannot be measured but if one measures the diameter and circumference and calculates π for many different-sized circles and then averages them, an approximation of π can be found. Another way to estimate π is to create a string the length of the circle's diameter and another string the length of the circumference. By laying the diameter string along the circumference it turns out that the diameter lays down three times plus a small piece of diameter. By folding this piece repeatedly it can be shown that it is about 1/7th of the diameter. Thus 3 + 1/7 or 22/7 is also an estimate for π. [Volker 2005]

Fig. 1) Constructing an estimate of π. [Author drawing]


I had not heard of this estimate before but it turns out that it was known to the Greeks from at least the 3rd century B.C.E. when Archimedes proved that 22/7th was greater than π. [Wikipedia. Proof That 22/7 Exceeds π.] Notice this was at least 200 years before the Hopewell phenomena. I am unconvinced that the Hopewell used this estimate in their construction. However, constructing an estimation of a value using ratios of numbers calculated with just cordage is important. Also important is using the folding of a cord to do integer division.



The above mud wall tablet dating 2000 - 1900 B.C.E. is the earliest recorded example of the Pythagorean Theorem. The tablet contains a series of exercises based on the problem of repairing a mud wall. The exercises use a type of triangle called a Pythagorean triple. The sides of the triangle form an integer ratio of (3:4:5) where 5 is the hypotenuse. Thus:


We are talking about different types of numbers here:


  • Integers, or counting numbers.

  • Rational Numbers can be written as a/b where b != 0. Integers are also rational numbers and are written as n/1.

  • Irrational numbers are numbers that cannot be written as rational numbers. There are two types: algebraic numbers and transcendental numbers. Algebraic numbers are solutions to polynomial equations consisting of one or more variables multiplied and added then set equal to zero.

  • Transcendental numbers are real numbers that are not algebraic. π is transcendental.

Fig. 2) Square with inscribed and circumscribed circles. [Author drawing]

Fig. 3) Approximately where equal area and perimeter circles would cross. [Author drawing]


Constructing a circle is easy. All one needs is cordage the length of the radius, Dc/2, and a center point. There are other ways to construct a circle. Gerdes suggests that the concept of a circle originally came from a spiral winding.

Creating a circular basket by winding a cord [Gerdes 2003 p 55-56]


Marshal claims that some Hopewell circles are spirals. He says that the large circle in Liberty Work is a spiral. [Marshall 1987]


One problem is the ability to scale up. Various measurements of the Great Circle in Newark give a diameter of between 360.9 and 365.5 meters [Volker 2005] Did the Hopewell have the technology to make cordage 180 meters long? What materials would they use? How heavy would it be? How would they prevent stretching in a wet Ohio climate? What is the time and energy required to make a cord this long? Fort Ancient is a large non-geometric hilltop earthwork in Southwestern Ohio. In 2005 gradiometer data within the earthwork uncovered a woodhenge, a set of post holes that form a circle and may have contained tall wooden posts, There is evidence that at one time the posts were removed and a circular earthwork was constructed. [Riordan 2020] Excavation at the Great Circle found no post holes but they could have missed them. [Lepper 1966] How this circular row of post holes was constructed, is an open question.


Like a circle, there are many ways to construct a square.



The above diagram is from the Baudhāyana-śulba-sūtra, a copy of an older Indian ritual Vedic text. The copy is from 800-500 B.C.E. The text describes the construction of a Vedic Fire Altar by first constructing a square using circles. Notice that the inscribed circle is used as a starting construction. The inscribed circle has a radius that is 1/2 the length of the square.

Fig. 4) 9x9 square constructed using 3:4:5 Pythagorean Triple. [Author drawing]


Figure 4 shows a second method of construction using a 3:4:5 Pythagorean Triple. This was drawn with a compass and a straight edge. The unit of measure is arbitrary. A 3x3 square is first constructed, then 8 more 3x3 squares resulting in a 9x9 gridded square. Notice that the center of the square is automatically created as long as a side is an odd number of units. This construction can be transferred to the field using a cord divided into 3:4:5 units.


John Hancock is Professor Emeritus of Architecture at the University of Cincinnati where he taught history, theory, and design. For the last 20 years, he has been interested in the Hopewell Earthworks, from creating 3D models of some sites and interactive media projects combined into a travel and tourism site called Ancient Ohio Trail. [Center for the Electronic Reconstruction of Historical and Archaeological Sites. Ancient Ohio Trail] I found this resource when planning a trip to Ohio in October of 2022. These articles are the latest expression of an obsessive interest that resulted from this trip. Hancock was the lead author of the UNESCO World Heritage nomination dossier for the Hopewell Ceremonial Earthworks, which were inscribed in September 2023.  [Hancock 2023] In the quote below, Hancock outlines a third way to construct a square.


"...the simpler principle, also known to modern carpenters, is that a square’s two diagonals are equal. Mark a cord into four equal lengths (A), walk those marked points out to a four-sided shape and then adjust it until its two diagonals (B) are equal.

...

These two cords contain the relation of the square’s side (its perimeter cord folded twice) to its diagonal. Any perfect square could be constructed with such a set of interrelated cords, whether the intent was a certain side-length, a certain perimeter, or a certain diagonal.

...

Diagonals were important, intentional features of Hopewell squares, as shown by the several cases where the diagonals mark a solstice alignment. This suggests that laying down the B-cord was the first step in a square’s positioning, rather than the sides, perimeter, or mid-point axes, and that construction teams walked out a four-part A-cord afterward." [Hancock 2024b]


The above quote makes an important point that at least one diagonal and pair of sides of a square earthwork is oriented to a landmark on the horizon or a celestial event. This idea needs to be tested on as many square earthworks as possible. Marshall gave evidence that some earthworks were not oriented to celestial events but I don't believe he looked at landmarks. [Marshall 1995] That was a discovery by Hively and Horn at Newark and in the Scioto Valley (called the Core.) [Hively and Horn 2013] [Hively and Horn 2020a} One takeaway from all this is that perhaps the square was the initial unit of construction and the large circles were built out of a square template. Stephen Luecking discusses a method used by the ancient Egyptians, called "rope pulling" to suggest several possible constructions. For instance, a large circle can be constructed using chords and right triangles. [Lueking 2004]

Creating a large circle using chords [Lueking p357]


The rising and setting points of the sun or moon on the horizon form a straight line across a landscape. If the distance is great enough roughly parallel lines can be drawn using the same points on the landscape. What is useful is that there is a built-in error factor involved that could be a signal of the use of this technique. Also, this suggests that "sticks and strings" [Hancock 2024a] can be reduced even further to "sticks and steps." [Rogers and Naumenko, 2016]


All these construction techniques are just speculation without a clear research approach to test these constructions against the actual data from the earthworks themselves. This is at one level an exercise in experimental archaeology. [Outran and Hurcombe 2024] First, the various constructions must be carried out on our equivalent of a sand floor or birch bark. Then this has to be taken into the field with "a team of very smart, creative people" [Hancock 2024b] to construct an outline of an earthwork. Then an error model can be generated for this type of construction. By error model, I am taking a cue from a paper by Isaac Ullah, a computational archaeologist at San Diego State. [Ullah 2015] The paper deals with errors created during surveys of Neolithic pastoralist sites in central Jordan at Wadi al-Hasa. Out of 16 sites listed, only 8 were found. It was later confirmed that only one was an original site. This is an outline of methods for correcting and measuring the accuracy of legacy survey data, meaning any surveys done before 2000 and unscrambled GPS. This includes correctable errors, rounding of coordinates, and directional biases. These also include intrinsic errors, by looking at common methods surveyors use; the act of 'constructing' a survey, the paper outlines an error model, a GIS surface based on the cognitive difficulties of reading maps and understanding a landscape. Each of these cognitive difficulties is represented by a set of models of the survey workflow, a spatial computational representation.


Pinson Mounds is a large ceremonial site in Tennessee. In 1922 William E. Myer published a map of the site originally created in 1917. Myer hired a local surveyor, Gail Buck, and interviewed several collectors in the region. In addition to 34 mounds two roughly circular embankments enclosed two regions he called the Eastern Citadel and the Inner Citadel. A third larger embankment enclosed the whole site. From Myer's notes, he added mounds that he had heard about but that neither he nor Buck had ever seen. Today only the embankment around the Eastern Citadel exists and only 13 mounds. In 2011 Mainfort et al. published a critique of the map and the survey done to create the map. Using a 1947 aerial map and a modern topo map, the authors were unable to find any of the outside earthworks or the earthworks around the Inner Citadel. They found that the map of the Eastern Citadel was topologically correct but the distances between structures were off. They created what is called a distortion map, and found that locations of mounds, entranceways, and embankments in the Eastern Citadel were between 90 and 115 ft off. [Mainfort et al. 2011] In 2005 a study of supposed astronomical relationships had been published [McNutt 2005] so the author had been questioning preliminary results. [Kwas & Mainfort 2007] [McNutt 2007] In 2020, a LIDAR and geophysical survey of the site by Edward R. Henry confirmed most of Mainfort's findings and created a baseline for any future studies of the site. [Henry 2020]


A mathematical construction is a mechanical physical act that is directly related to the physical act of constructing a structure. These are workflows but in this case, the workflow must be inferred from how the human mind abstracts space and form to create objects and the resulting objects themselves. This is a theme I wish to return to again.


I have deliberately left out any mention of a Hopewell unit of measurement. This is because I believe that a unit of measurement is a separate problem [Measure] from geometry. A circle and a square are two abstract mathematical forms. Any drawing or construction of a circle or square is a representation or an estimate of this form. I think an important question is: 'How circular is a circle?' or 'How square is a square?' For circles, there is a modification of the construction for finding the center of a circle:


Take any two random unique points on the circle and join them with a line, called a cord. The length of the cord can be set to a minimum size. Bisect the cord and draw a line perpendicular to the cord which intersects the circle. This is the diameter of the circle constructed by these two points. Do this until all the points of the circle are used up. If the circle is perfect the diameter lines will all intersect at a single point.

Finding the center of a circle using chords [bunglesmate, How to Find the Center of a Circle.]


In reality, one would never get a single point but a cloud of points within which resides the 'true' center. The circle has been reduced to its imperfections. I would think that this method would be very useful but although I have found mention of this concerning Archaeology [Larson and Boswell 2013] I have not yet found any references.


An image of the circle needs to be processed so that just the circle in black exists on a white background. Here are some modified steps:


  • Randomly pick a pair of points P1 and P2. The points should be greater than some minimum distance apart.

  • Find the midpoint M1 of the line L1 between the points P1 and P2.

  • Construct another line R1 at right angles to L1 passing through M1.

  • Repeat steps 1 through 3 for points P2 and P3 to give you a second line R2.

  • Find the point of intersection of R1 and R2. That is the candidate center C1 of the circle.

  • Do this n times. There will be a cloud of points around the actual center of the circle.

  • Calculate a possible center as the centroid of the convex hull of the point cloud.


For a square, a similar construction is carried out, except the two points picked can't be on the same side.


The relationship between a square and a circle was important to the Hopewell. A circle can have the diameter of the side of a square or the diameter of the diagonal of a square. The circumference of a circle can be equal to the perimeter of a square. Finally, the area of a square can be the same as the area of a circle. All four relationships have been claimed to exist between circles and squares in Hopewell geometric earthworks. [Volker 2005] [Marshall 1987]


One way to measure the area of a shape is to lay a grid of squares over the shape and then count the squares inside the shape plus the squares that the shape crosses. The intersecting squares are counted because of the Archimedean Principle [Measure] Marshall believed that the Hopewell used a grid to lay out their earthworks.

Post hole array in Harness Mound  [Marshall 1987 p 41]


Gerdes thinks that the conceptual idea of using a grid to measure the area of a shape like a circle comes from building cylindrical baskets with square bottoms.


Each slat of the mat is either 1, 2, or 3 squares wide so it can be divided into a grid of squares. Notice that it doesn't matter what size the square is, smaller squares will give a more accurate estimate of the area of a shape. The mathematical name for the abstraction of a grid is called a lattice. This is abstracted even further to a set of points that are the edges of the squares. A book on geometry by the mathematician David Hilbert I found this exercise by Fredrick Gauss.


Gauss placed a lattice over a circle with the following rules:


  • the center of the circle is a point on a square lattice

  • the radius of the circle contains an integer number of these points

If r is the radius, the circle inscribes a square with a side of 2r. Gauss counted the points inside the circle and the points just outside the circle belonging to squares crossed by the circle. Then he made the circle larger and counted again. Using this method he was able to calculate closer and closer estimates of π. Thus he used a lattice to find a rational approximation of an irrational number.


The claim that the builders knew how to square a circle has led to all sorts of fantastic claims about them being Vikings or Greeks or space aliens. For one thing, they are not squaring the circle, an impossible task using Euclidean construction, but a similar method called circling the square. A Euclidean squaring of the circle was proven impossible in 1882 by C.L.F. Lindemann [Wikipedia Squaring the Circle]. The reason is that π is a transcendental number and Euclidean constructions can only construct algebraic numbers. Circling the square to create a circle and square of equal area involves an estimate of the point along the diagonal of a square where the diameter of an equal-area circle lies. John Volker provided an answer. [Volker 2005] Luecking references Robert Lawlor's Sacred Geometry: Philosophy and Practice as a source for circle and square construction. [Lawlor 1982] This does give constructions for circle to square but no simple fractional estimate. As far as I know Volker's solution is unique.

Volker's Calculation [Volker Fig 11 & 12, equation]


This relates the radius of the equal area circle to the diameter of the square. A little algebra will show that the two are equivalent. Volker derives this ratio from measurements of the so-called Observatory Circle and the Newark Square at the Newark Earthworks. I will go into detail about this in the next article. He uses Marshall's postulated Hopewell dual measurement system consisting of a basic unit and the diagonal of a square constructed out of this basic unit. These two units of measurement constitute Marshall's Hopewell grid. That Volker used these units to find this ratio in Newark does give credence to Marshall's ideas.


One difficulty I see here is how one can show empirically that a circle and square with this relationship have equal areas. I would say from the drawing above that it looks likely that they are equal but that doesn't mean much. Maybe it is the ratio of small integers that is important. Taking a square and its inscribed and circumscribed circle it is clear that the equal-area circle fits somewhere between these two forms. I had the idea that maybe 4:5 is the first ratio that fits between these two circles. 1:2 doesn't and 2:3 doesn't but 3:4 does as well as 4:5. Volker's construction is more empirical as one needs an odd number in the denominator to have the construction work out, a construction that could have been discovered through basket making.


Fig. 5) Construction of equal area (4/5) and equal perimeter (9/10). [Author drawing]


Volker also mentions circles and squares having equal circumferences or perimeters but does not give a simple ratio. Hively and Horn specifically state that they do not believe the Hopewell created such earthworks. I calculated that the circumference of an equal-area circle is less than an equal perimeter one and the circumference of the circumscribed circle is greater so I decided to split the difference along the diagonal. To my surprise, this worked!


Fig. 6a,b) Graphs of equal area and equal perimeter. [Author]


Using a unit square I calculated the area and circumference of a circle starting with the diameter of the inscribed circle and stepping to the diameter of the circumscribed circle. I then subtracted the area and perimeter of the square and took the absolute value. The first graph shows the full diagonal while the second is zoomed in. Notice that because the area is a squared number the values converge to equality much faster than the perimeter values.


Approximating irrational numbers, both algebraic and transcendental by simple ratios of integers has been an important strand of mathematics since the beginning. Any irrational number can be approximated this way with not just one ratio but a set of ratios that converge to the initial number. A simple representation of the set is called a continued fraction.

Generalized continued fraction

Continued fraction where the numerator is always 1 and the rest are positive integers. Any irrational number can be written as this form of continued fraction. The notation for this is: [a0:a1,a2,a3,...]


The Golden Section also has the simplest continued fraction representation which is: Φ = [1;1,1,1,1,1,1,1,1,1,1,1,...]. These approximations converge toward the actual number at different rates. Φ converges at a certain rate, while π converges quicker. This can be shown as the slope of a convergence plot.


As an exercise for the reader, find the continued fraction representation for the approximations of equal areas and equal perimeters discussed in this article.




Circleville Earthwork [Marshall 1987, p. 43, fig. 7]


Circleville earthworks no longer exist The drawing above is by John Marshall. It is from a plat map from before 1810 that he found in the Pickaway County Courthouse. [Marshall 1987]


The relationship of a 5x5 grid to a circle gives square diameter values in the form of a low number ratio that gives circles of equal area and equal circumference to the square. using a 9x9 grid, Marshall showed that for the two concentric circles at Circleville, the ratio of the radius of the inner circle to the outer circle is 4:4.5 grid units or 8:9, the ratio of the sides of the circumscribed squares.



I took his measurements and did the following back-of-the-envelope calculations.

Fig. 7) Circleville Calculations [Author]


The diameter of the outer circle and the square's diagonal are within .17% so it looks like the circle circumscribes the square. The circumference of the inner circle is within 1.4% of the perimeter of the square. It is still less than the 9/10 approximation: within .035% of the calculated value. The issue here is that any inner circle close to the circumscribing circle will be close to the perimeter of the square. Hence, there is no way to tell whether the equality of circumference and perimeter is on purpose. So the relationship at first pass, looks like the outer circle circumscribes the square. This is extraordinary and merits a closer look.


Another extraordinary finding is that the diameter of the inner circle is exactly 8/9ths the diameter of the outer one. This result is too good to be true but if further measures confirm this it would support Marshall's idea of a 187' 9x9 grid.


On a corkboard, I constructed a 5x5 grid square and a circumscribing circle in an attempt to match Circleville's layout. The actual site had a second concentric circle inside and constructing this would require a 9x9 grid and a much smaller unit of measure.


Fig 8a) Author's thumb as a unit of measure [Author]



Fig. 8b) Construction of a square and circumscribed circle [Author]


  • As a unit of measurement, I used the distance from the tip of my thumb to the second knuckle. This unit I will call one Bt (Bill's thumb.)

  • I laid out one unit on e the string and folded it over eleven times to create a string twelve units long and divided into 3:4:5 units.

  • With this, I measured out a 5-unit line.

  • I constructed two right angles, extended the sides, and constructed the last two right angles to construct the square.

  • I used the measuring string to lay out a diagonal and then created a second string the length of the diagonal.

  • Using the measuring string I divided the diagonal string into 5 new units of measurement, which I will call one dBt (diagonal of Bill's thumb.)

  • Using the measuring string I found the center of the circle and using the diagonal string I constructed the circle.


This construction is a first attempt and pretty crude. However, I did learn a few things:


  • The string I used is a little too loose, flexible, and stretchy for such a small construction. I need to use a thinner stiffer string, maybe even wire.

  • I didn't start the square with respect to my construction space so the circle doesn't fit. This relationship between available space and placement is important as the layout of the geometrical figures could be affected by the physical space they had to build on. How is the positioning of the circles and squares; some connected by avenues, some touching, some overlapping, how are these related to the physical constraints of the space?

  • Once I have this method down I need to be able to take well-lit ortho digital images of each finished construction. Then I can build an error profile for each construction.


In the next article, I will detail John Volker's calculations at Newark. Whether the equal area circle and square found in Newark is a singular construction is an open question. A close reading of Marshall hints at others. Modern methods such as LIDAR, magnetometer, and ground-penetrating radar are now being used. The best results can be used to judge the rigor and biases of older methods.


Marshall claims that the builders knew and used the Fibonacci ratio estimates of the Golden Section in their construction. This has been a claim about the proportions of buildings through antiquity. The basic question is: 'Do we visually recognize a proportion that is the Golden Section?' This is a question of aesthetics, a psychological and perhaps cognitive question. Research results have been confused and contradictory with possible biases. I am skipping any further discussion, perhaps until a future article. [Elliott et al. 2015] [IOSA et al. 2018]


Geomagnetic data has shown that many of the small enclosures that accompany both mound groups and many of the geometric enclosures are not circular or square like the old maps have shown, but a rounded form part way between a square and a circle. These have been found in Ohio [Burks & Cook 2011], Indiana [Davis & Burks 2020], Kentucky [Henry et al. 2021], and North Carolina [Wright 2019]. A common method of rounding a square is to create a small square on each corner, draw an inscribed circle in each, and then erase the upper right quadrant of the small square. This process called 'rounding' a square is not the correct shape. Called 'squircles' by Jarrod Burks, these refer to a special form of a mathematical object called a 'Super Ellipse.' This is defined as follows:


Notice that this is a mathematical object drawn by a computer.


The question becomes: How can a complex mathematical object that can only be drawn by a computer show up as the shape of these enclosures? Barring alien intervention these had to be constructed. I tried to research a way to construct these by hand. The only mention was using templates like French Curves. All others relied on drafting programs or code. [ndesmic 2021] There are several methods for drawing an ellipse and a host of mechanical devices for drawing one. [Garcia 2024] This is an area for further research.


How does a culture communicate mathematics and a measurement system without a written language? This is a major question that I will discuss in [Memory].


Concluding Remarks


Every culture creates a system of knowledge that represents the very human process of manipulating and understanding the World. While perhaps not a 'true' science these systems are at least equivalent to a science in importance. That Hopewell culture encoded their mathematical knowledge into the landscape is a tantalizing speculation that merits rigorous confirmation. Modern mathematics, including geometry, rational approximation theory, and measurement theory can point the way.






Part VI - Landscape


Part VII - Memory



 

  • Binford, Lewis C. “The Archaeology of Place.” Journal of Anthropological Archaeology 1 (1981): 5–31.

  • bunglesmate. “How to Find the Center of a Circle.” Instructables. Accessed March 15, 2024. https://www.instructables.com/How-to-find-the-center-of-a-circle/.

  • Burks, Jarrod, and Robert A. Cook. “Beyond Squier and Davis: Rediscovering Ohio’s Earthworks Using Geophysical Remote Sensing.” American Antiquity 76, no. 4 (2011): 667–89.

  • C, Stephen. “Answer to ‘Determining Whether an Array of Points Roughly Make a Circle.’” Stack Overflow, May 4, 2018. https://stackoverflow.com/a/50175345.

  • Carr, Christopher. “Being Scioto Hopewell Appendices.” Christopher Carr Archaeology. Accessed April 29, 2024. https://www.christophercarrarchaeology.com/being-scioto-hopewell-appendices.html.

  • Center for the Electronic Reconstruction of Historical and Archaeological Sites. “Ancient Ohio Trail.” Accessed May 18, 2024. https://ancientohiotrail.org/.

  • Davis, Jamie, and Jarrod Burks. “Vol. 1 Ch. 1 Indiana Earthwork Sites New Insights from LiDAR DEMs and Aerial Photographs.” In Encountering Hopewell in the Twenty-First Century, Ohio and Beyond, Vol. 1. University of Akron, 2020.

  • Elliott, Mark A., Joy Kelly, Jonas Friedel, Jennifer Brodsky, and Paul Mulcahy. “The Golden Section as Optical Limitation.” PLOS ONE 10, no. 7 (July 8, 2015): e0131045. https://doi.org/10.1371/journal.pone.0131045.

  • Garcia, Pablo. “Drawing Machines.” Accessed September 20, 2024. https://drawingmachines.org/category.php?id=32.

  • Gerdes, Paulus. Awakening of Geometrical Thought in Early Culture. Minneapolis: MEP Publications, 2003.

  • Giles, R. Mathematical Foundations of Thermodynamics, Nacnillon, 1964.

  • Hancock, John. personal correspondence 2024a




  • Houston, George W. “M. J. T. Lewis: Surveying Instruments of Greece and Rome.” Isis 93, no. 2 (June 2002): 298–99. https://doi.org/10.1086/344987.

  • Iosa, Marco, Giovanni Morone, and Stefano Paolucci. “Phi in Physiology, Psychology and Biomechanics: The Golden Ratio between Myth and Science.” Biosystems 165 (March 1, 2018): 31–39. https://doi.org/10.1016/j.biosystems.2018.01.001.

  • Jolie, Edward A., and Maxine E. McBrinn. “Retrieving the Perishable Past: Experimentation in Fiber Artifact Studies.” In Designing Experimental Research in Archaeology, 153–93. University Press of Colorado, 2010. http://www.scopus.com/inward/record.url?scp=84872417511&partnerID=8YFLogxK.

  • KET Artifacts. “Hopewell Projectile Point.” Accessed April 30, 2024. https://www.projectilepoints.net/Points/Hopewell.html.

  • Kwas, Mary L., and Robert C. Mainfort. “Antiquarians’ Perspectives on Pinson Mounds Revisited: A Response to Mcnutt.” Southeastern Archaeology 26, no. 1 (2007): 145–50.

  • Lawlor, Robert. Sacred Geometry: Philosophy and Practice. First Edition. Crossroad, 1982.

  • Larson, Ron, and Laurie Boswell. Big Ideas Math Geometry. Brainly, 2013. https://brainly.com/textbook-solutions/q-17-problem-solving-archaeologist-finds-circular-plate.

  • Lee, CMG. Best Rational Approximants for Irrational Numbers. Own work. Accessed April 26, 2024. https://commons.wikimedia.org/wiki/File:Diophantine_approximation_graph.svg.

  • Lepper, Bradley T. “The Newark Earthworks and the Geometric Enclosures of the Scioto Valley: Connections and Conjectures.” In A View From the Core - A Synthesis of Ohio Hopewell Archaeology, edited by Paul J. Pacheco. The Ohio Archaeological Council, Inc., 1996.

  • Luecking, Stephen. “Pulling Ropes and Plumbing Lines: Geometry for the Neolithic Engineer.” BRIDGES Mathematical Connections in Art, Music, and Science, 2004.

  • Mainfort, Robert C., Mary L. Kwas, and Andrew M. Mickelson. “Mapping Never-Never Land: An Examination of Pinson Mounds Cartography.” Southeastern Archaeology 30, no. 1 (2011): 148–65.

  • Marshall, James A. “An Atlas of American Indian Geometry.” Ohio Archaeologist 37–49, no. 2 (1987).

  • Marshall, James A. “Astronomical Alignments Claimed to Exist on the Eastern North American Prehistoric Earthworks and the Evidence and Arguments Against Them” 45, no. 1 (1995).

  • Marshall, James A. “Towards a Definition of the Ohio Hopewell Core and Periphery Utilizing the Geometric Earthworks.” In A View From the Core - A Synthesis of Ohio Hopewell Archaeology, edited by Paul J. Pacheco. The Ohio Archaeological Council, Inc., 1996.

  • Marshall, Richard A. “Burial Pattern and Tomb Construction Story Mound I, Hoecake Site (23-Mi-8) Mississippi County, Missouri.” In Middle Woodland Settlement and Ceremonialism in the Mid-South and Lower Mississippi Valley, by Robert C. Jr. Mainfort. Mid-South Archaeological Conference, 1984.

  • McNutt, Charles H. “The Pinson Observatory.” Southeastern Archaeology 24, no. 2 (2005): 142–64.

  • McNutt, Charles H. “A Reply to Kwas’s and Mainfort’s Response to the Pinson Observatory.” Southeastern Archaeology 26, no. 1 (2007): 151–54.

  • ndesmic. “How to Draw Squircles and Superellipses.” DEV Community, March 9, 2021. https://dev.to/ndesmic/how-to-draw-squircles-and-superellipses-3d14.

  • Outran, Alan, and Linda Hurcombe. “Experimental Archaeology.” Oxford Bibliographies, 2024. https://www.oxfordbibliographies.com/display/document/obo-9780199766567/obo-9780199766567-0271.xml.

  • Pauketat, Timothy R. Cahokia Ancient America’s Greta City on the Mississippi. Penguin, 2009. https://www.penguinrandomhouse.com/books/296469/cahokia-by-timothy-r-pauketat/.

  • Riordan, Robert V. “Vol. 1 Ch. 2 At the Heart of Fort Ancient,” Vol. 1, 2020.

  • Rogers, Brian, and Olga Naumenko. “Perception of Straightness and Parallelism with Minimal Distance Information.” Attention, Perception, & Psychophysics 78, no. 5 (July 1, 2016): 1381–91. https://doi.org/10.3758/s13414-016-1083-x.

  • Ruhl, Katharine C. “Hopewellian Copper Earspools from Eastern North America: The Social, Ritual, and Symbolic Significance of Their Contexts and Distribution.” In Gathering Hopewell, edited by Christopher Carr and D. Troy Case, 696–713. Interdisciplinary Contributions to Archaeology. Boston, MA: Springer US, 2005. https://doi.org/10.1007/0-387-27327-1_19.

  • Ruggerie, Mike. “Adena and Hopewell Art.” Tumblr. Tumblr (blog), November 2016. https://adenaandhopewellart.tumblr.com/post/153834163439/adena-tablets.

  • The Illinois State Museum. “Native American Mapping Tradition.” Accessed May 1, 2024. https://www.museum.state.il.us/exhibits/lewis_clark_il/htmls/resources/mapping/native_american_mapping.html.

  • Toth, Adam J., and Mark J. Campbell. “Investigating Sex Differences, Cognitive Effort, Strategy, and Performance on a Computerised Version of the Mental Rotations Test via Eye Tracking.” Scientific Reports 9, no. 1 (December 19, 2019): 19430. https://doi.org/10.1038/s41598-019-56041-6.

  • Volker, John J. “The Geometry of the Newark Earthworks.” Unpublished from Bradley Lepper, 2005. Access is made available with permission from the author.


Comentários

bottom of page