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An Exercise in Middle Woodland Geometry VII

Updated: 5 days ago


3D model of the Observatory Circle, Newark, Ohio [Hancock 2014, with permission https://earthworks.uc.edu]
3D model of the Observatory Circle, Newark, Ohio [Hancock 2014, with permission https://earthworks.uc.edu]

Sky


Now Talking God

With your feet I walk.

I walk with your limbs

I carry forth your body

For me your mind thinks

Your voice speaks for me

Beauty is before me

And beauty is behind me

Above and below me hovers the beautiful

I am surrounded by it

I am immersed in it

In my youth I am aware of it

And in old age I shall walk quietly

The beautiful trail.


The Dineh Night Chant [Vallejo 2024]


"The objective of research in cultural astronomy is not to study the sky in itself, but to study the practical construction of the heavens by, as well as the use of the heavens for, human life on earth. The heavenly environment is a research field through which we can study human societies rather than the mysteries of the universe."


Modern humans are losing the sky, especially at night and during a good part of the day, and many, maybe a majority, seldom see the moon and stars. [Bakich 2014] To older societies, the sky was an important part of the landscape and was treated as such. The sky is dynamic but also repetitive. This repetition seemed static over human lifetimes. Thus, the sky gave humans a way to measure time, but it also seems timeless. Knowledge of time gives a culture legitimacy and control over the future. The timing of the sun and moon's motions, planets and stars, and even more obscure sky events like comets, eclipses. or meteor showers, these can all be understood and even predicted with careful observation. [Ruggles 2015c] These recurrences don't align exactly with whole numbers of days or with each other, thus the use of ever more complex numerical calculations, which expanded human mathematics. Up until recently, the sky was not a part of a landscape that could be modified. Man-made objects in the sky and brilliantly lit cities hiding the night have changed the importance of the sky. What was modified in the past was the immediate landscape and sometimes the horizon itself. What part of the sky was important and why varied greatly and can only be understood in the way a culture used the sky and time. [Iwaniszewski 2015b]


Orientation

All structures have some orientation. Even a circle has an orientation to its center or an entrance or entrances. Anywhere one can trace a straight line distance, no matter how small, and extend it outward to the horizon. This includes the scattering of objects or grave goods, the direction in which a body is laid in a grave, fire pits and platforms, and the various alignments of a structure or between parts of the structure itself. In addition, structures that are far apart can be aligned with each other or have similar alignments. For example, early Mayan cities were aligned East to West, but later cities were aligned North to South. [Šprajc 2015b] There are only two known octagon and circle combinations in Ohio, at Newark and High Bank Works in Chillicothe. These are aligned at right angles to each other, yet they are 90 kilometers apart. [H & H 1984]


For alignments to be considered astronomical, they have to be oriented to some event in the sky. This often means some appearance of a celestial object at the horizon, but not necessarily. At the horizon, there are various distortion effects and, of course, weather. [Schaefer 1993] When a celestial object first appears, when it is half exposed, or just when it stops touching, these positions are important when calculating alignments. The Earth spins like a top with a slight wobble, and stars wander off alignment after about 100 years, the planets 1,000 years, and the sun and moon are the most stable. [Ruggles 2015d] Latitude and altitude also affect the position of a celestial rising or setting. [Burnett 2000] The night sky for any date, time, and place on the Earth can be modeled with programs like Stellarium. [Chéreau et al. 2023] A local horizon can also be included. GIS and 3D modelling can be helpful in both analysis and visualization. [Zotti 2015]


Positional Astronomy

The study of the rising, trajectory, and setting of a celestial object at a location on the Earth and a surrounding horizon is called 'Positional Astronomy.' [Ruggles 2015a] This is perhaps the oldest means of viewing and studying the sky, a precursor to all of modern astronomy. Three measurements are needed:


  • Azimuth - the clockwise angle of degrees from due north.

  • Altitude - the height difference from a point on the ground to the horizon.

  • Latitude

In addition, the position of a celestial object must be calibrated to a specific date. This allows for calculating a 'line of constant declination,' the path of a celestial object through the sky.

Declination of a horizon point [Ruggles 2015a, p. 461, fig. 30.1]
Declination of a horizon point [Ruggles 2015a, p. 461, fig. 30.1]

Intent

For an astronomical alignment to be considered archaeologically real, the alignment has to show intent, the builders aligned for a reason, it is not just random. This is usually done using a Monte Carlo method. [Wikipedia - Monte Carlo method] Using this method, repeated draws of alignments from a random distribution are tested against an orientation to find the probability of such an alignment happening.


Rejecting the Null Hypothesis using a Monte Carlo method only shows a possible intent. Stellar observations can be so numerous that there is no way to tell them apart from a random event. The only final proof is intent. Intent is shown through ethnographic study. How does a culture that is perhaps related to an extant culture under study view the sky?


The Hopi are a Pueblo people, meaning they live in adobe or stone multi-room and multi-story structures. Pueblo architecture started around 1,000 C.E. Individuals called 'sun watchers' kept track of the sunrise from a certain vantage point. The sunrise forms a calendar with the horizon. This calendar can be used for planting crops, but can also herald a ritual. This shows that the solstices were important, as was a proper place to view them. In this case, the orientation of the wall follows an eastern mountain chain rather than any celestial event. [Williamson 2015a]

Hopi Sun Watcher Horizon Calendar, Walpi, AZ [Williamson 2015a, p. 644, fig. 45.1]
Hopi Sun Watcher Horizon Calendar, Walpi, AZ [Williamson 2015a, p. 644, fig. 45.1]

Intent is as diverse as the means to use and display it. Shadows in passageways and across surfaces, human-made and natural, can be intentional or not. [Murray 2015a] [Murray 2015b] It depends on the culture, what they thought was important, and how they internalized this importance. [McCluskey 2015a] The correspondence between a woman's mensuration and the phases of the moon means that sometimes the moon was thought of as a goddess, but not always. Different phases could be male or female. [Milbrath 2015] The 260-day round of the Mayan ritual calendar is thought to be related to the human gestation period. It is also the time it takes corn to grow and ripen from seed. [Davies 2024] The archaeological site of Boca de Potrerillos in NE Mexico contains one of the largest concentrations of rock art in the country. This represents a tradition of hunter-gatherers much older than the civilizations of central Mexico. Among celestial alignments and rock symbols, the bars and dots of a common Mesoamerican counting system are pecked into the rocks representing a calendric round of 207 days, the gestation period for deer. [Murray 2015c] Evidence of day counts based on gestation periods of bison, wild horses, ibex, and reindeer has been found. [Rappenglück 2015]


History

Archaeoastronomy as a science is not very old. It all started in the 1950s with Alexander Thom. Thom was Chair of Engineering Science at Oxford. Throughout his life, he was fascinated by the construction of Megalithic sites in the British Isles, especially in Scotland. He was the first to use measurements from hundreds of sites and the first to use statistics. I have talked about him before in [Measurement] for his idea of a Megalithic standard of measure. In 1963, Gerald Hawkins published a paper in Nature, Stonehenge Decoded. Hawkins claimed that Stonehenge acted as a gigantic astronomical computer. This claim was quite popular, especially after Hawkins turned the paper into a best-selling book. The book generated much controversy in archaeological circles, even after the astronomer Fred Hoyle agreed with Hawkins's conclusions. Thom later argued that certain points at Stonehenge created a backsight where astronomical events could be viewed on a foresight, a point on the horizon. These foresights were notches, low places between hills. [Salt 2015] Salt sees a major difference between American and British archaeoastronomy in that British archaeologists come from a historical background while American archaeologists come from an anthropological one. This made American archaeology more prone to ethnological and sociological explanations, in other words, intent.. This may be so, but in both America and Britain during the 1990s and 2000s, there was a flurry of alignment collecting, an old saying came true: "If all you have is a hammer, then everything is a nail." Most of these turned out to be false positives as the analysis matured. To some archaeologists, intent is not even enough, they will only pay attention to Archaeoastronomy when it starts answering actual archaeological questions. Alignments are artifacts; what can be said about the culture that creates them? [Kintigh 1992]


One example of clear false positives is at Hovenweep National Monument. From about 900 AD to the mid-13th century, a Pueblo culture built a series of towers around the tops of several canyons in a forsaken part of what is now southeastern Utah. Some are square, some are round, some come in pairs or triplets, and no plan seems to be involved in their placement. The ruins have never been completely excavated. Alignments have been found at rock art sites near the towers. Ray Williamson and Craig Benson examined astronomical alignments at the towers. In a book published in 1987 [Williamson 1987], he mentions several possibilities. A tower in a structure called Hovenweep Castle had three 'portals' that were thought to have astronomical alignments with the lintel of doorways within the interior. He fails to provide any ethnographic evidence that these portal and doorway features were built intentionally, plus he failed to mention maybe five more portals on the building. Thus, he committed the statistical fallacy of "counting the hits and ignoring the misses." He also studied a structure called Unit-Type House. This was supposed to have four portals that had alignments. Unfortunately, a photograph of the Unit-Type House from 1917 showed the tower being devoid of portals. [Bates 2015]

Unit-Type House 1917 [Bates 2015, p. 616, fig. 42.6]
Unit-Type House 1917 [Bates 2015, p. 616, fig. 42.6]

For archaeoastronomy to be taken seriously, especially by archaeologists, it must lose the sensationalism of pop culture and become either an adjunct of archaeology or a science in itself. This means a set of 'best practices' [Ruggles 2015b], but also specific practices for looking at solar alignments, the moon, and planets and stars. [Ruggles 2015e] [Belmonte 2015b] [Ivaniszewski 2015b] [Šprajc 2015a] This includes best practices for cultural interpretations as well.. [Ivaniszewski 2015d] [Lopez 2015]


Most of the references I've used come from Clive L. N. Ruggles's monumental book, Handbook of Archaeoastronomy and Ethnoastronomy, 2015. [Ruggles 2015f] It contains hundreds of short articles on all aspects of archaeoastronomy. Although work in the Americas is extensively studied, there is nothing about the eastern Woodlands, and H & H's work in Ohio is not mentioned except once in the references. Whether this was because the work was obscure or whether this was a deliberate slight, I don't know.

28 Phases of the Moon
28 Phases of the Moon: Engraving [Miotte 1646]
Hively & Horn

Ray Hively and Bob Horn (henceforth H & H) are professors at Earlham College in Richmond, Indiana. Hively is an astrophysicist, and Horn is a philosopher. In 1974, they were asked to co-teach a year-long History of the Cosmos course. At the time, it was just speculation that Stonehenge in Britain and other sites in Europe used the rising and setting points of the sun and moon as a calendar. Finding no published maps of Stonehenge, they first looked at nearby Fort Ancient and then the Octagon at Newark. [H & H 2020a]


I have tried to separate the questions of a system of measure and the geometric knowledge behind these structures and have mentioned these researchers several times. [Measure] [Geometry] [Landscape] In this section, both measure, geometry, and landscape play an important role and are part of a complete astronomical package of discoveries.


H & H published their first paper in 1982. Here is their conclusion:


"Our data and analysis show that the Newark earthworks were carefully conceived and constructed to exhibit a remarkable degree of symmetry, precision, and geometrical harmony, apparently based on a single length (the OCD)."



Of the many features of the Newark Earthworks, only three have survived pretty much intact. These are the Observatory Circle, the Octagon, and the Great Circle. The Octagon, Observatory Circle, and another small circle form one group, and Wright Square and the Great Circle form another. A restoration of these enclosures was done during 1893-96 and again in 1933-36. These 'fixed' some features and raised some of the worn-down walls. This meant that any modern survey needs to take this into account. There are several maps of Newark, starting with Caleb Atwater's in 1820. [Atwater 1833] Squire and Davis's 1847 map is the most famous map of Newark. [Squier and Davis 1847] Neither of these maps is known for its accuracy. In 1892, W. H. Holmes published topographic drawings of the Octagon and Circle in Newark and High Banks. [Holmes 1892] James Middleton did a formal survey which was published by Cyrus Thomas. [Thomas 1894] This is considered the first accurate survey. James Marshall did survey work at Newark in 1960 and 1980 [Marshall 1987], but H & H do not mention it. H & H surveyed Newark, plus they used the Middleton survey data.


A map was drawn by the Salisbury Brothers in 1862. It was never published. In 1988, Brad Lepper was giving his first presentation about Newark earthworks. The archaeologist N'omi Greber was attending and told him about a card catalog entry about a map of Newark earthworks at the American Antiquity Society in Amherst, MA. Lepper obtained the map in 1989. [Lepper personal communication] The Salisbury Brothers map included structures never recorded, including what is now called 'Salisbury Square', a square attached to a very small circle on a flat ridge southeast across the Licking River from the main earthworks.


Different drawings of the Observatory Mound [Holmes 1892, p. 368, fig. 1 & 2]
Different drawings of the Observatory Mound [Holmes 1892, p. 368, fig. 1 & 2]
Drawing of the Observatory Circle and the Octagon [H & H 1982, p. S6, fig. 3]
Drawing of the Observatory Circle and the Octagon [H & H 1982, p. S6, fig. 3]

One important finding was how the diameter of the near-perfect Observatory Circle or OCD (321.3 m) is related to other features. Middleton first noticed the relationship below. In addition, a small circle below the Octagon is 1/7th OCD. The distances from the center of the Observatory Circle to the center of the Great Circle and from the center of the Octagon to the center of the Wright Square are 6 OCD. This suggests deliberate planning on the part of the builders.


In addition, point F is 6 meters closer to the center of the Octagon while the other points are almost exact. This was discovered when the GF and FE were surveyed, and the lengths differed from Middleton's measurements. All the other lengths were close. H & H suggest that this is because the Octagon sides were restored to a perfect length. This difference is believed also to be deliberate, as this modification allows for alignment to a lunar maximum.

A square with a side equal to the Observatory Circle Diameter (OCD) fits inside the Octagon. Circles drawn from points A G E C, having the length of the square's diagonal, intersect to create the other 4 points of the Octagon. [H & H 1982, p. S8, fig. 4]
A square with a side equal to the Observatory Circle Diameter (OCD) fits inside the Octagon. Circles drawn from points A G E C, having the length of the square's diagonal, intersect to create the other 4 points of the Octagon. [H & H 1982, p. S8, fig. 4]

They found 17 possible astronomical alignments:

17 possible alignments found at Newark. [H & H 1982, p. S16, table II]
17 possible alignments found at Newark. [H & H 1982, p. S16, table II]

Also in 1982, Christopher Turner started investigating the Ohio earthworks. Much of his work is unpublished but released as a series of blog posts in 2015. I could not find out for certain who he is; a latest published piece in 2021 labels him as an 'independent researcher.'

It also says that he has a doctorate in anthropology from the University of Louisiana. In 1983, he wrote a study on Hopeton Earthworks, which was never published, but he claims a copy resides at the National Park in Chillicothe and with the Ohio History Connection. [Turner 1983] He uses the Middleton survey for Hopeton plus aerial photographs taken in the 1930s by Dache Reeves. [Reeves 1934] He found 17 rise alignments and 11 set alignments. He also investigated foresights along the hills east and west of Hopeton, particularly for fire-cracked rock.


Hopeton rise alignments [Turner 1983, fig. 8]
Hopeton rise alignments [Turner 1983, fig. 8]

In 1984, H & H published a paper on High Banks Works. [H & H 1984] It is almost the same size as the Observatory Circle - Octagon in Newark, with the same OCD relationships. A shorter connecting avenue means that the circle-octagon pair at High Banks is exactly 2 ODC long. It is also rotated ninety degrees from the pair in Newark, a distance of 113 km.

High Bank Circle and Octagon relationships [H & H 1984, p. S93, fig. 6]
High Bank Circle and Octagon relationships [H & H 1984, p. S93, fig. 6]

In the figure below, gap 9 forms an anomaly in that it should be where point 12 is. This anomaly allows for three possible alignments.

Possible alignments due to the gap 9 anomaly. [H & H 1984, p. S95, fig. 8]
Possible alignments due to the gap 9 anomaly. [H & H 1984, p. S95, fig. 8]

They found 13 possible alignments at High Bank.

Table of 13 possible alignments. [H & H 1984, p. S98, table 3]
Table of 13 possible alignments. [H & H 1984, p. S98, table 3]

During the 1990s, astronomical alignments were found everywhere. Keith Kintigh called it 'celestial butterfly collecting.' [Kintigh 1992] James Marshall was an Illinois surveyor who investigated and surveyed hundreds of earthworks on his own. In the spring 1995 issue of the Ohio Journal of Archaeology, both he and William Romain had written articles. [Marshall 1995] [Romain 1995] Romain expands on the two dozen additional earthworks he believes to have alignments. Marshall, who appears first, does not support any alignments at all. He claims that during a meeting of the Historical Astronomy Division of the American Astronomical Society in 1992, he was heckled by the crowd during a presentation. He spends much of his paper grousing and presenting his theories, and doesn't name Romain or H & H by name. He supplies his survey drawings and makes a major point: How can these earthworks be constructed when many times trees obscured the horizon, and even if most of the nearby trees had been felled, the density of the forest prevented the use of a proper foresight horizon? It has been determined that the area around the Newark Earthworks was a mezoic prairie created by 1,000s of years of human burning. [Wymer 1997] The problem of using a foresight horizon still exists.


In his 2015 blog rewrite of his original 1983 article, Turner claims that Romain [Romain 1991] used his Hopeton measurements without referencing his article.


"Romain in his subsequent articles uses the same azimuths as generated from my article and applies them to maps of other mound sites using an inscribed plastic overlay. This method ignores intersite variation in horizon heights and does not yield valid archaeoastronomical results."


This might have been one source of Marshall's ire.


Anthony Aveni holds professorships in Astronomy, Anthropology, and Native American Studies at Colgate University, Hamilton, NY. From his unique position, he has been able to combine the European emphasis with statistics and measure with the American emphasis on cultural astronomy. [Aveni 2003a] He emphasizes the importance of having these alignments, first, be statistically relevant, and secondly, answer archaeological questions. He contends that there are two types of archaeoastronomy: "brown," studies that have found links with the underlying culture through ethnographic or written evidence, or "green," cultures with weak ethnographic links and no written record. [Aveni 2016] He considers Hopewell studies to be in a "green" state and has expressed this in critiques of both Romain's [Aveni 2003b] and H & H's work. [Aveni 2004] Aveni has also published useful astronomical tables of calculated dates of past events going back in time. [Aveni 1972]


Geometric relationships between elements of the Newark Earthworks [Volker 2003, fig. 3]
Geometric relationships between elements of the Newark Earthworks [Volker 2003, fig. 3]

In 2003, a student of Brad Lepper, John Volker, wrote a remarkable paper. [Volker 2003] In 2022, John Hancock asked me to look at this paper and determine whether it was accurate. My attempt to understand this paper resulted in a series of articles, in which this is the seventh. In [Geometry], I went over Volker's paper and came to the following conclusions:


  • An estimate for Phi based on the ratio 22/7, which has been known since Greek times.

  • An estimate for finding a square and circle of equal area. By using a square and its circumscribed circle, a circle with a radius of 4/5ths of the radius of the circumscribed circle will have the same area as the square.

  • An estimate for finding a square and a circle of equal perimeter. Using the same square, a circle of equal perimeter would have a radius 9/10ths of the radius of the circumscribed circle.

  • Notice that this method uses a square and its circumscribed circle to find an equal area and an equal perimeter circle using simple low integer value ratios. The term for this is more correctly 'circling the square' rather than 'squaring the circle.'

  • These ratios are called Diophantine approximations of irrational numbers. 22/7ths, 4/5ths, and 9/10ths are the first approximations of a set of integer ratios that converge on an irrational number. The sqr2 is the ratio of the sides of a Pythagorean Triple to the hypotenuse. The first ratio is (3+4)/5. The ratio of the alternate numbers of the Fibonacci series converges to the Golden Section. The lowest ratio is usually termed 2/3rds, although the actual lowest ratio is 1/1.

  • The converging series of integer ratios creates a line ending at the irrational number. Since the Golden Section is also one result of a quadratic equation, the series forms two lines. It also has a very shallow slope. The slope of the line is how fast the series converges to the irrational number. I show that the slope of the equal perimeter estimation is less than the slope of the equal area estimation. Any statistical tests of the geometry of Middle Woodland earthworks must keep this in mind.

  • Marshall postulated a 57-meter 'Hopewell grid' based on the grid layout of Teotihuacán. Volker found that the diameter of the 57-meter square is 57 * sqr2, very close to 1/4th (within .14%) of the OCD. Using this relationship between the grid side and diagonal, he was able to find the equal area relationship. This answered one of Marshal's critiques of Ohio archaeoastronomy.

Volker's discovery of the equal area ratio using Marshal's Hopewell grid. There are 4 half diameters from the center to five half diameters to the corner. This generalizes the relationship. [Volker 2003, fig. 12]
Volker's discovery of the equal area ratio using Marshal's Hopewell grid. There are 4 half diameters from the center to five half diameters to the corner. This generalizes the relationship. [Volker 2003, fig. 12]

In 2004, Turner presented a poster at the joint meeting of the Southeastern Archaeological Conference and the Midwestern Archaeological Conference containing the first statistical analysis of H & H's results from the Newark Octagon. Using a simple model of the Octagon gaps, and Moon maximum and minimum rise and set points along a 360-degree azimuth, he found an expected random alignment of all 8 alignments to be 1.24%. Using Chi-Squares and resampling, he finds that both tests reject the null hypothesis that the alignments are random. [Turner 2004]


The Observatory Circle is close to a perfect circle, just as the Wright Square is close to a perfect square. The Octagon, however, deviates slightly. The eight sides should form 4 pairs of parallel lines. Two of the pairs are off, they are not quite parallel. The other two pairs are very close to parallel. Why is this so when the builders could build so exactly? The two pairs of lines form the structure of five lunar alignments. These five are the most accurate alignments found at Newark. Did the builders deviate from a 'perfect' because they intended to mark the lunar alignments? Or is this just a random error?

The five most accurate maximum and minimum Moon rise and set alignments. [H & H 2006, p. 285, fig. 4]
The five most accurate maximum and minimum Moon rise and set alignments. [H & H 2006, p. 285, fig. 4]

In 2006, H & H published a statistical analysis testing this question. [H & H 2006] They created a Monte Carlo simulation of random octagons. Octagons usually have only one degree of freedom, the angle between the sides, marked beta on the figure below. This octagon has two degrees of freedom, the second one, alpha, being the angle between the center of the circle and the line through the center of the passage joining the two figures.

Octagon has two degrees of freedom [H & H 2006, p. 286, fig. 5]
Octagon has two degrees of freedom [H & H 2006, p. 286, fig. 5]

The calculation of the probability of intentional alignment uses the following procedure:


  • "... define a primary population of equilateral octagons (with four-fold rotational symmetry and an attached avenue axis) which could have been plausibly built independently of any astronomical considerations;"

  • "... define a target population of such octagons which have an astronomical significance comparable to or greater than that found for the actual Newark Octagon;"

  • "... choose a random sample from the primary population and determine what fraction of those octagons also belong to the target population (the fraction being the probability of chance alignment)."


They created 72 different models for defining the primary and target populations. These were based on various criteria for generating the octagons and picking alignments. They used a table of lunar and solar rise and set points calculated by Aveni [Aveni 1972] and a date of 250 C.E. They found that the optimal octagon for just lunar alignments vs lunar and solar alignments differed in angle by over 10 degrees. The angles for the Newark Octagon are within the Lunar-only range. In addition, the angles fall within a medium between a perfect geometric form and an optimal Lunar alignment. H & H believe the builders intended to encode both into the Newark Octagon. They also note that the angles of the High Bank Octagon fall within a solar-Lunar set of angles. In their conclusion, they mention the uniqueness of the Newark landscape for making astronomical observations and consider a solution to the foresight problem; the observations for the construction of the earthworks were done from prominent high points.


"... the earthworks in the valley were never intended to be the primary observatories at which the lunar cycles could be precisely monitored. Rather the geometrical structure and orientation of the geometrical figures were integrated with the lunar sightlines which had already been established by observations made from surrounding overlooks."


This is an important paper in that it establishes a statistical test for intent outside of any cultural knowledge. It, of course, doesn't prove intent, but it supports looking for intent in the archaeological record. The five alignments found by H & H record three maximum and two minimum Lunar standstills. There are important differences between the minimum and maximum standstills. The moon moves from north to south to north again during each month. With the maximum, the moon slows down in its procession, and the total visible maximum occurs for 18 months. The minimum standstill is less. Also, the moon passes through the minimum twice a month regardless of where it is in its procession. This makes the discovery of the minimum a difficult achievement. [Fisher and Sims 2018] [González-García 2015] [The Archaeoastronomy Database 2025] The phase of the minimum during the bi-monthly passage and possible lunar eclipses adds another level of complication. [McClure 2016] This also makes the Octagon-Observatory Circle complex a lunar calendar recording both a bi-monthly cycle of time and a longer 18.6-year cycle.


H & H's paper in 2010 introduces what they call the 'zero-altitude hypothesis.' [H&H 2010] They used the criteria that since the builders had very accurate control of what they were trying to accomplish, any deviation from that control must have been done on purpose. To find a proper alignment point, the backsight needs to be on a promontory with a view of the surrounding hills and the earthwork.


"The question of how the lunar extremes might have been discovered and marked with an accuracy of about 0°.5 is also addressed by the zero-altitude hypothesis. Such a feat would have required determined observation over several generations. Consistent and precise observations would have been difficult to make from valley floors, where the directions to rise and set points appear to vary significantly due to obscuration by local topography and seasonal vegetation. The most logical place to make and record repeatable observations of lunar rise/set points would indeed be from high places with unobstructed views of distant horizons. Such high places could be dedicated to long-term observations, as they would not have been prime areas for other activities."


They suggest two possible locations. One they call H1 southwest of the earthworks and another, Coffman's Knob to the southeast. The zero-altitude hypothesis is a solution to Marshall's critique of the Newark alignments.

Alignments from Coffman's Knob. The second foresight, H1, is west and slightly south. [H & H 2010, p. 135, fig. 5.5]
Alignments from Coffman's Knob. The second foresight, H1, is west and slightly south. [H & H 2010, p. 135, fig. 5.5]

They next turn to Chillicothe. Here, the Scioto River meanders through a broad valley formed millions of years ago by a north-flowing river, the Teays. They show that Liberty, High Bank, and Works East not only line up, but that line is parallel to the Teays Valley. In the Paint Creek watershed, they find two sets of three earthworks that line up and run parallel to that valley.

Teays River Valley alignment of Liberty, High Bank and Works East [H & H 2010, p. 140, fig. 5.8]
Teays River Valley alignment of Liberty, High Bank and Works East [H & H 2010, p. 140, fig. 5.8]

To the east side of the valley is a range of low mountains that afford extensive views with Mt. Logan to the south and Sugarloaf to the north. On the west side of the valley is Mt. Prospect, the highest land closest to the river. This is on the estate of John Worthington, the sixth governor of Ohio. He named the estate "Adena," and it contained the mound that is considered an early expression of these mound-building cultures, although it has been found to have been built slightly before the start of Mound City. [Lepper et al. 2014] Mt. Pleasant could be a common backsight of zero-altitude lunar rise and set points for Shriver and Hopeton, Works East, High Bank and Liberty, Seip, and Anderson and Frankfort.

Looking east at the Appalachian Range. Sugarloaf is on the left, and Mt. Logan is on the right. [H & H 2010, p. 145, fig. 5.12]
Looking east at the Appalachian Range. Sugarloaf is on the left, and Mt. Logan is on the right. [H & H 2010, p. 145, fig. 5.12]

The Great Seal of the State of Ohio is a view of sunrise to the east from Mt. Prospect.

The Great Seal of the State of Ohio [State of Ohio 1803]
The Great Seal of the State of Ohio [State of Ohio 1803]

The results here suggest a mainly lunar with some solar interest. The earthworks in Chillicothe and Newark were planned from observations that would have taken many generations. In addition, Newark is much more accurate. This suggests to Brad Lepper that Newark was built as a single project. [Lepper 2004]


I've already covered Christopher Turner's unpublished work in 2011 on H & H's zero-altitude H1 alignment in [Landscape]. [Turner 2011a] He used a more accurate GIS to find a better location for the point. He concluded that the second sightline through the Great Circle was not nearly as accurate as the sightline through the Octagon/Observatory Circle. This could mean that the Great Circle was older than the others. This is quite different from H & H's view that these 'errors' show some sort of intent on the part of the builders. Turner believes that this inaccuracy means that the Great Circle was built first. The Great Circe/Wright Square combination has things in common with Circleville, while the Octagon/Observatory Circle combination is related to High Bank and Hopeton. The Great Circle has a Great House that contains no burials, while University Circle has a platform mound that extrudes outside the wall. Wright Square has two sides out of true, only five entranceways and seven gateway mounds, two of which are out of alignment. Again, is this sloppiness or intent? Turner cites only two radiocarbon dates, around 300 CE for the Observatory Circle and 100 CE for the Great Circle. Brad Lepper, who did the excavation at the Great Circle, says that this is a misunderstanding of the dates; the date was taken from the soil at the base of the embankment because no charcoal was found. Soil is notoriously bad for radiocarbon dating. [Hajdas et al. 2021] This gives a minimum date; the Great Circle could not have been built before this date but could have been built anytime after. [Lepper personal communication]


Also in 2011, Turner published for the first time. [Turner 2011b] In the paper, he proposes another axis of alignment between earthworks around Chillicothe. He looks at Hopeton, Mound City, Anderson, Hopewell, and a possible location in the Logan Range near Bald Hill. Turner found a great quantity of cracked rock at this location in a 30-meter-wide depression, a possible signal fire. He uses the valley locations rather than Bald Hill for backsights so his analysis doesn't use zero-altitude alignment.. He doesn't specifically state but suggests that the builders used signal fires to align the earthworks rather than looking down from the higher elevation. I am a little unclear about what calendrical rise event is encoded by the Bald Hill location.


In 2013, H & H published an update to their work at Newark. [H & H 2013] They expand the number of super-accurate alignments from seven to eight, plus the additional semi-accurate alignments. They expand the number of zero-altitude locations adding H2 through H5 plus H1 and Coffman's Knob. They add a small circle south of the Octagon, Wright Square, the newly rediscovered Salisbury Square, the Great Circle, and an oval earthwork in the Raccoon Creek Valley to the alignments, plus four more hilltop locations H6-H9. They do not use a modern GIS and a viewshed, and they do not mention Turner's new location for H1. They do say that H1 could be moved around 160 to 200 ft without causing much error. Turner's new zero-altitude backsight is 406 m southwest, keeping the low error through Observatory Circle / Avenue / Octagon axis, but is off for the Great Circle. Remember that Turner claims that from the original H1, the Great Circle is not even visible.

Observation Points H1 - H5 and Coffman's Knob and their foresight locations. [H & H 2013, p. 109, tbl. 2]
Observation Points H1 - H5 and Coffman's Knob and their foresight locations. [H & H 2013, p. 109, tbl. 2]
Observation Points H6-H9 and their foresight locations.  [H & H 2013, p. 110, tbl. 3]
Observation Points H6-H9 and their foresight locations.  [H & H 2013, p. 110, tbl. 3]
Alignment of the small circle. [H & H 2013, p. 112, fig. 12]
Alignment of the small circle. [H & H 2013, p. 112, fig. 12]
Observation points H1 - H4 [H & H 2013, p. 113, fig. 17]
Observation points H1 - H4 [H & H 2013, p. 113, fig. 17]
Alignments to the corners of Wright Square. [H & H 2013, p. 114, tbl. 4]
Alignments to the corners of Wright Square. [H & H 2013, p. 114, tbl. 4]

This paper expanded the number of alignments to a large degree without much speculation about intent. The precision, as shown in the errors from azimuth, speaks for itself. As for the timeline for building Newark, the sheer scale and complexity of the project versus the interlocking geometric and astronomical intricacy present something of a paradox.


Also in 2013, Jarrod Burks published a magnetic gradient survey of High Bank Works. [Burks 2013] The images below show first, the Squire and Davis map, and second, the survey map. Notice that the small circle doesn't show up, replaced by two smaller crescents and an oval anomaly. Near the centers of both the Great Circle and Octagon are anomalies that could be structures. Also, there are long linear anomalies and many possible postholes. Another important difference between the new data and the Squier and Davis map is that a second gap in the Great Circle wall is missing. It is possible that there was a low point in the wall instead of a gap, and by Squier and Davis' time, this low spot had worn down to look like a gap.

Squier and Davis' map of High Bank with survey area marked in red. [Burks 2013, p. 10, fig. 2]
Squier and Davis' map of High Bank with survey area marked in red. [Burks 2013, p. 10, fig. 2]
Features and anomalies found from the magnetic gradient survey. [Burks 2013, p. 29, fig. 15]
Features and anomalies found from the magnetic gradient survey. [Burks 2013, p. 29, fig. 15]

The new data differs slightly from the 1984 Hively and Horn data. The main difference is the missing gap in the Great Circle. Also, the avenue between the Great Circle and the Octagon is longer. The six walls that were completely surveyed are slightly longer. On the other hand, many anomalies were found that intersect the alignment lines.

H & H's 1984 map of High Bank with the magnetic survey data in red. [Burks 2013, p. 47, fig. 24]
H & H's 1984 map of High Bank with the magnetic survey data in red. [Burks 2013, p. 47, fig. 24]
Possible large post holes and other anomalies near the 1984 H & H alignment lines. [Burks 2013, p. 50, fig. 26]
Possible large post holes and other anomalies near the 1984 H & H alignment lines. [Burks 2013, p. 50, fig. 26]

The magnetic survey of High Bank left most of H & H's conjectures intact, although I know of no recalculation of their findings based on this new data..


In 2016, H & H published a chapter in the book: The Newark Earthworks - Enduring Monuments, Contested Meanings. [H & H 2016] As the book is about meanings, their article is a rewrite of their major findings, answers to their critics, and some musings on the nature of the builders.


"With respect to the difficulty of measuring the standstill cycle, it is undoubtedly the case that marking the extreme points with a typical accuracy of 0.5 degrees would require strong motivation. It would demand the persistence to make regular and disciplined observations and to average the results of observations extending over many human generations. It would also require some way of transmitting the knowledge from one generation to the next. This would require a certain minimum of social, political, and ritual stability. However, these requirements of curiosity, intelligence, persistence, and the ability to transmit information from one generation to the next are precisely the traits suggested by the construction of the earthworks themselves."


Their additional discovery is a winter solstice alignment between points H3, H2, and a small circular earthwork, and a summer solstice alignment between H1 and H4. Observatory Circle in Newark is named after the platform mound attached to it, Observatory Mound. I think that Observatory Circle has been misnamed, as these hilltop sites are the true observatories. Belmonte talks about how many sites have been called observatories, but their purpose is different; they were built based on observations that had already taken place. [Belmonte 2015a] All of Newark is built as an encoding of knowledge, both astronomical and geometric. This encoding was used to sustain and project a set of rituals, and a culture. Of course it is almost impossible to change a name that has been in use for so long. [Lepper presonal communication]


William Romain has always contended that the Milky Way was in part a major element in Woodland religion, a "Path of the Souls," an often perilous journey in the afterlife to a final resting place. This belief is part of modern native beliefs and may go back into deep time, at least until the Early Woodland. In a paper published in 2017, Romain is looking for astronomical alignments and the use of geometry at Cahokia, the first North American city. [Romain 2017a] [Romain 2017b] The region has no topographic features at all; the massive earthworks and mounds are the topographic features. This period of Mississippian polities was very different from Ohio over 500 years earlier, with the domination of Three Sisters' agriculture, the widespread use of the bow and arrow, [Nassaney 1999] and a very different political and social system. He shows support that the city was laid out in a symmetrical grid oriented to the Milky Way and having alignments to both solar and lunar standstills. Also, a grid based on squares and a figure he calls a "root-two rectangle," a rectangle made out of a square with one side extended by its diagonal. This rectangle can contain both lunar and solar standstill in its diagonals.

Root-two rectangle with celestial alignments. [Romain 2017, p 14, fig. 12a]
Root-two rectangle with celestial alignments. [Romain 2017, p 14, fig. 12a]

He also states that the city is laid out in squares that are multiples of a standard measure, that of 47.5 meters. This measure is the same as the layout found in a mound group in Arkansas, Toltec Mounds (no relation to the actual Toltecs). Different from Marshall's 57-meter standard for Ohio. This is interesting as root-two has a first approximation of 7/5ths, the sum of the sides of a 3-4-5 right triangle divided by the hypotenuse. If his research is confirmed, it would put grid layout, geometric relationships, and solar and lunar alignments (especially lunar) as continuing themes in the Woodlands.


In 2020, H & H published a chapter in Encountering Hopewell in the Twenty-first Century, Ohio and Beyond. [H & H 2020] They propose a new hypothesis called TAG for Topography, Geometry, and Astronomy. These concepts informed the builders' placement, paterning, and alignment of the earthworks. This hypothesis is based on three principles.


  • "geometric designs based on exploration of circles and squares related by circumscription and inscription;"

  • "the discovery and implementation of a simple but accurate algorithm for constructing circles and squares with nearly equal areas;"

  • "deliberate siting and alignment of the structures along lines indicating the rise and set points of the sun at the solstices and the moon at the lunar standstills as viewed from prominent points in the local topography."


To show how these principles apply to both Newark and various earthworks in Chillicothe, they list a bewildering list of circumscriptions, inscriptions, equal areas, and multiples of equal areas, alignments to the solar solstices and lunar standstills, and multiple possible locations for observation. Here is a list of matching areas. I am paraphrasing H & H's list to save space. Notice that 1054 ft is the OCD and 1490 ft is the diagonal of a square with a side the OCD:


  • Observatory Circle = Wright Square

  • Great Circle = a square of side 1054 ft formed by Octagon vertices ACEG

  • Octagon = the area of a 1490 ft circle that circumscribes a square of side 1054 ft

  • The interior-mound octagon = a 1316 ft circle which would circumscribe the Wright Square

  • A 1316 ft circle circumscribing the Wright Square = a 1185 ft square circumscribing the Great Circle

  • The sum of the areas of the Observatory Circle and Wright Square = the area of the 1490 ft circle which passes through the mid-walls of the Octagon

  • The sum of the areas of the Great Circle and the 1054 ft square formed by the vertices ACEG = a 1490 ft square which circumscribes the Octagon by passing through the vertices ACEG

  • The sum of the areas of the Observatory Circle and the interior-mound octagon = the area of a 1490 ft square

  • The sum of the areas of the Great Circle and the walled Octagon = twice the area of the 1213 ft square formed by the vertices BDFH

  • The Cherry Valley Oval and its westward elliptical extension = the area of a 1490 ft square or twice the area of the Great Circle


To inform their geometric sense, they reference the work of Paulus Gerdes, a Dutch mathematician living in Africa. [Gerdes 2003] Gerdes believes that geometric thought arises from everyday practices of weaving and basketry; an empirical or embodied mathematics. He figures prominently in my [Geometry] article. H & H list the skills related to empirical geometry:


  • "the ability to design and build earthworks in the form of circles, squares, and octagons involving units of length derived from a standard square with a side of 1054 ft and a diagonal of 1490 ft;"

  • "the ability to compare the areas of geometrical figures such as squares, circles, and octagons;"

  • "the ability to construct large-scale circles, squares, octagons, and ovals with nearly equivalent areas."


They also make the following statement:


"We have sought to avoid any geometrical rationale that introduces hypothetical dimensions, grids, or geometries not immediately evident from the structure of the works themselves."


There are important problems related to both lists and the statement above. By algorithm, they mean the construction by John Volker that equated the area of a square to a circle by a simple fraction of 4/5ths. I was able to devise another method in [Geometry], but only in hindsight and with some modern mathematical knowledge. Volker's construction is truly empirical. There is no reason to assume that the builders had a way to test whether this construction provided equal areas. They might have known the approximation of pi, but they are all about relating circles and squares using the diameter and the side of the square. This requires grids. Volker's discovery of his ratio used the diameter of Marshall's grid. He found a relationship between Marshall's grid and H & H's OCD. With this relationship, he found a way to construct an equal area circle of any size. I believe that the reluctance of H & H to recognize this is that Marshall's grid comes from Mesoamerica. This doesn't mean that the builders originated in Mexico, but that some message of a universal layout made it to Ohio to inspire the brilliant minds of the builders. I believe that one can't have an empirical geometry without a grid, and the grid in this case belongs to Marshall.


At the end of their chapter, H & H list the challenges and objections of the TAG hypothesis:


  • "there is no established precedent for a similar achievement of this magnitude in the prehistoric world;"

  • "the only ethnographic evidence for knowledge of the lunar standstill cycle by Native Americans is very tentative and confined to the Mississippian and Chacoan cultures;"

  • "when topographic features are interpreted as astronomical backsights with no archaeological evidence that such backsights were actually noticed or used, it is easy to underestimate the likelihood of random chance alignments;"

  • "the adjustable parameters associated with defining possible alignments are so numerous that the theory can be fit to virtually any earthworks structure."


I would say that H & H have done a good job in answering 1, 2, and 4. Their data is both strange and compelling. Number 3, the ethnographic evidence, is a hard sell. Romain has done some work with Cakokia [Romain 2021], but I would say that following the non-Mississippian, the hunter-gatherers, is the way to go. So far, no relationship with the Northern Plains Medicine Wheels and celestial events has been found. [Vogt 2015] Earthwork construction and use continued in northern Michigan, Wisconsin, and western Ontario up until contact. [Howey 2012] Slowly, modern tribes are being linked to a deep history.


In 2021, Turner published a second paper. [Turner 2021] Turner is examining two sites in the Paint Creek Valley, north of Chillicothe, Baum and Seip, and a third non-geometric site in the valley, a stone work called Spruce Hill. He is using ESRI GIS and a 2008 DEM, probably 10 m x 10 m. He does not use zero altitude backsights from hilltop locations but uses positions within the earthworks as backsites and hills, including Spruce Hill as foresights. He finds solar alignments at Baum and both solar and lunar alignments at Seip. He also finds a quarter solar alignment (between the solstices). These two sites are part of the 'Tripartite Group' and do not include any circle or square that is fully realized. It is the square part, particularly the corners of the entrances to the square, where he finds the alignments. He notes that all of the alignments that he has found so far are at the corners of the entrances. He includes a viewshed of eight sites in the region.

Viewsheds of 8 sites in the Scioto-Paint Creek region. The viewshed is taken from a location at each site. Notice that the sites are not all visible to each other. Also, notice that the Seip-Baum-Spice Hill axis is almost perpendicular to the Teays Valley. [Turner 2021 p. 37, fig. 12]
Viewsheds of 8 sites in the Scioto-Paint Creek region. The viewshed is taken from a location at each site. Notice that the sites are not all visible to each other. Also, notice that the Seip-Baum-Spice Hill axis is almost perpendicular to the Teays Valley. [Turner 2021 p. 37, fig. 12]

The latest publication on the archaeoastronomy of Ohio earthworks is by H & H in 2023. [H & H 2023] They return to Newark to include many secondary features, circles, and parallel walled avenues, some rediscovered from aerial photographs by Dache Reeves in the 1930s. [Reeves 1934] With these structures, they find a set of solar alignments and three more observation points in the hills. They also use a 10 x 10-meter DEM for the first time. They also include the Milky Way for the first time and show alignments at this time with both the Milky Way and the solar and lunar results. They also discuss the use of parallel walls in the Octagon as shadows at right angles to the alignments. They also give new coordinates for some of their older observation points, including H1. They do not move it to the location Turner suggests or mentions in his critique.

Main elements at Newark. A circle attached to the Great Hopewell Road is below the map.[H & H 2023, p. 2, fig. 1]
Main elements at Newark. A circle attached to the Great Hopewell Road is below the map.[H & H 2023, p. 2, fig. 1]

Conclusion

In over forty years of finding alignments, notice that the polygonal earthworks encode most of them. It is tempting to see the earthworks themselves as a representation of the earth and the sky, the circles being the sun and the moon, and the polygons the earth. Many earthworks have a large and a smaller circle, plus small circles and crescents, along with a touching or connecting polygon..The so-called tripartite earthworks have all three combined, perhaps referencing an interaction of the sun, moon, and the horizon. One has a third even smaller circle, perhaps Venus or a star. It is a weird fact that during the time of humans, the moon is at a point where it is the same size as the sun as seen from anywhere on Earth. Millions of years ago, the moon was closer; millions of years from now, it will be further away. [Astronomy Staff 2000] The sun looks bigger because of the corona. Maybe Circleville, with its double circle, represents a solar eclipse. There are separate polygons at Newark, the Wright Square for the Great Circle and the Octagon for Observatory Circle. Salisbury Square has a small circle embedded on one side, perhaps a star or a planet. There has been much speculation as to the meaning of the circle and the square. [Byers 2005] [Romain 2000] Robert Riordan has spent years excavating Polluck Works in southwestern Ohio. [Riordan 2010] Polluck Works is a hilltop earthwork that has been faced with limestone. There are three entrances with three gateway mounds and three small crescents. Lower down in the valley is another earthwork called Bull Works. This consists of a small rectangle with some crescents and a few mounds. Riordan considers these two as part of the same system, the start of the square and circle geometries of central Ohio, all these being different expressions of a global belief system. At Newark, an additional element is at play, the equivalence to the circle and the square, of sky and landscape. And the transformation of one to the other. The watery world reflects the sky as the sky reflects the human transformations, from birth to maturity to old age and death. And the journey beyond.


The usual chain of events with astronomical alignments is that many are speculated on, but only a few stand the test of time. An example of this is Chaco Canyon and the work of Alice Sofaer. [Sofaer et al. 1982] Many alignments were found, but only a few survived further study. This is especially true of lunar minimums and maximums. [Williamson 2015b] In Ohio, this has not been true; the alignments keep on coming. Either this will change when a new generation of researchers start re-evaluating the present data, or something very unique has been found. Computational structures such as GIS and spatial statistical models will lead the way. Computational models must be properly archived and stored for verification and future use. Just as parts of a site might be left alone or reburied after a dig for future generations and better techniques, computational results need the same care.


 







Part VIII - Memory


Part IX - Quidnunc


 

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