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Large Geometric Earthworks and the Concept of Multiplication

  • 6 days ago
  • 6 min read

Updated: 5 days ago

Similar Triangles [Mathspace 2026]
Similar Triangles [Mathspace 2026]

Around 100 CE in southern Ohio, a remarkable cultural florescence produced architecture, art, mathematics, and astronomy within an egalitarian, peaceful environment. The population voluntarily contributed thousands of hours building huge ceremonial centers; Newark covers over four square miles, including circular, square, and octagonal earthworks that relate to each other in different ways geometrically and may even have been built to a standard of measurement. The builders encoded not only their mathematical and astronomical knowledge into their earthen enclosures, but also their relationship to the world around them. Earthworks and mounds have a long history in eastern North America, but these earthworks were of a scale, geometric, and astronomical precision unique from anything seen before or after in the eastern woodlands. One important question is: How were these built? What skills did the builders need to construct such enclosures?


This article is strictly speculation, but speculation based on what I know and what I have discovered. I invite any comments. James Marshall. Robert Romain and many, many others have done important work on the types and forms discovered. Discoveries and new technologies require updates to correct errors and a continual ongoing reinterpretation.


When measuring the various shapes of the geometric earthworks, a unit of measure called the Observatory Circle Diameter (OCD) or its multiple appears several times. Coined by Hively & Horn [Hively & Horn 1982], James Marshall championed a different measure. [Marshall 1987] John Volker showed that the two units of measure were related, leading to his remarkable discovery that a circle having 4/5ths of the radius of a circle circumscribed on a square has the same area as this square. [Volker 2025] Robert Romain rightly assumes that the OCD or the Marshall measure were some multiples of a smaller measure based on the human form. [Romain, William 1996] Much speculation has focused on how a standard of measure could exist in a society without writing and be used by that society for hundreds of years. The claim that a standard measure exists is an assumption that may be true but has never been analytically tested. The theoretical background and even the code are available [Kasiński 2019], but the data have not been put in one place. [Kubicka & Kasiński 2020] [Brzozowska-Jawornicka, & Kubicka-Sowińska 2021]


Another feature of the geometric earthworks is just that: their geometries, the relationships between the circle and the square, and the use of a special class of triangle, those where two or three sides are counting numbers. But what did this knowledge of such triangles give these people? I want to propose the cognitive skill of understanding multiplication. Now, the people of this time had no more or less cognitive ability than the people today. Einstein may have found an original proof of the Pythagorean Theorem in third grade, but he didn't discover the theorem; he was taught it. Out societies advantage has been a long, stable use of the written word along with increasingly sofisticated means to store and distribute it. This didn't exist in the first century CE in Ohio. They accomplished what mathematician Paulus Gerdes calls 'organizing of spatial experiences,' [Gerdes 2003, p. 3] from knowledge derived through making ceramics, weaving, basketry, and wayfaring. Understanding multiplication allowed these people to create these immense earthworks, to physically embed their mathematics into the landscape.


So how do children learn multiplication? There are two competing ideas. One is that multiplication is repeated addition. One starts this way, then they memorize the multiplication tables. It is strictly procedural, a way that multiplication is calculated. Another, first proposed by Paget, is a way to understand multiplication using what is called a conceptual field. This consists of an invariant element (something that does not change) and a schema, an organized pattern of thought that resolves around this invariant. [Vergnaud 1983] Students first taught a conceptual method did better at a wider variety of problems than students who were first taught the procedural method. [Park & Nunes 2001]


Four simple triangles have as their sides two or more counting numbers. Notice that in two cases the third side creates a new number, a magic number specific to the particular triangle. The triangles can be defined as a specific relationship between the sides:


  • 1:1:1

    This is the only triangle that does not have a 90° angle [Maple Tech.2026] Romain claims that a triangle of this type with sides equal to the side of the Seal square would be circumscribed by the Seal circle. This works within 6.6%, which is weak but interesting, especially because the actual diameter of the circle has not been pinned down. [Romain 2000, p. 52]
    This is the only triangle that does not have a 90° angle [Maple Tech.2026] Romain claims that a triangle of this type with sides equal to the side of the Seal square would be circumscribed by the Seal circle. This works within 6.6%, which is weak but interesting, especially because the actual diameter of the circle has not been pinned down. [Romain 2000, p. 52]

  • Magic1:1:1

    This is the diagonal of a square. The magic number is √2. [Maple Tech.2026] This is also the diagonal of a circumscribed circle. If the map that Marshall found of Circleville is correct, the square is circumscribed by the large circle at this earthwork.
    This is the diagonal of a square. The magic number is √2. [Maple Tech.2026] This is also the diagonal of a circumscribed circle. If the map that Marshall found of Circleville is correct, the square is circumscribed by the large circle at this earthwork.

  • 2:Magic2:1

    The magic number is √3. [Maple Tech.2026] I have confirmed Romain's statement that this exists at Seal, although the explanation in his book is incorrect. The error here is 3.321%. Again, there is the issue of the accuracy of the diameter of the Seal circle [Romain 2015, p. 146]
    The magic number is √3. [Maple Tech.2026] I have confirmed Romain's statement that this exists at Seal, although the explanation in his book is incorrect. The error here is 3.321%. Again, there is the issue of the accuracy of the diameter of the Seal circle [Romain 2015, p. 146]

  • 5:3:4

    First Pythagorean Triple. [Maple Tech.2026] Although I don't believe that this actual relationship has been found, this is an optimal way to create a measuring device to create a right angle and to lay out a large square. I was able to reconstruct Circleville Earthworks using a piece of string divided into 5:3:4 parts using my thumb as a unit of measure. There are other ways. Notice that this does not necessarily mean that the builders had knowledge of the Pythagorean Theorem.
    First Pythagorean Triple. [Maple Tech.2026] Although I don't believe that this actual relationship has been found, this is an optimal way to create a measuring device to create a right angle and to lay out a large square. I was able to reconstruct Circleville Earthworks using a piece of string divided into 5:3:4 parts using my thumb as a unit of measure. There are other ways. Notice that this does not necessarily mean that the builders had knowledge of the Pythagorean Theorem.

These relationships don't just represent a single triangle of a certain size, but a whole set of triangles of different sizes that share the same relationship. These relationships are invariant; they do not change as the triangle gets bigger or smaller. These are called similar triangles.


Similar Triangles [Mathspace 2026]
Similar Triangles [Mathspace 2026]

So what does it mean to get bigger or smaller? We call it multiplication, and the schema is represented in modern symbols as:


f(ax) = af(x)


The letter a represents a multiplier, which can be any number, including any standard of measure. f(x) in this case represents the various relationships between the sides given by these four triangles.


Early and Middle Woodland earthworks can be roughly classified into general types. Temporal classification is much more local and complex but generally older to newer:


  • Small earthworks in solidary locations

  • Sites composed of various small earthworks. The Mann site has an interesting feature in which the small circles are arranged to form a larger one.

  • Hilltop enclosures of various sizes, including quite large.

  • Large non-geometric enclosures.

  • Large geometric enclosures


The discovery of multiplication not only allowed the builders to design and construct massive geometric earthworks, but also to encode their mathematical discoveries into the design. Mathematics can seem magical or sacred, especially its usefulness in the physical world. Many mathematicians, even today, hold a Platonic view of mathematics: that it resides somewhere outside reality, and that they are discovering mathematics rather than creating it. This is opposed to the view that mathematics is embodied, that we are wired this way. We create mathematics based on our wiring, and because we are part of the physical world, our mathematics reflects this.


I have confirmed a few sites; there are many more. Maybe some pattern will appear as the data becomes richer.



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