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    • Mar 26
    • 11 min read

An Exercise in Hopewell Geometry


GIS Model of High Banks Earthworks on a Modern Landscape [Chaney EarthWorks]


Part II - Measurement


"In analyzing the foundations of measurement, one of the main concerns is formalization: the choice of an empirical relational structure as an abstraction from the available data, the choice of an appropriate numerical relational structure, the discovery of suitable axioms, and the construction of numerical homomorphism (scales) ,,, However, this formalization process does not exhaust the problem of formalization by any means. The most important omission is an analysis of error of measurement. This involves difficult conceptual problems concerning the relation between detailed, inconsistent data and the abstraction derived from them, the empirical relational structure." [Krantz et al. 1971 p. 13]


What does it mean to measure something and what is meant by a unit of measure? It is best to start out using weight rather than length. Pan balances for weighing objects have been found since 2600 BC in Egypt. [Wikipedia History of Measurement]


Minoan Weighing Dishes 2000 - 1500 BC [Nagel 2012]


Two lumps of stuff, A & B, are placed on a pan balance, A on the left side and B on the right. There are three possible relationships between A & B:


  1. The left pan is lower, thus A > B in weight

  2. The right pan is lower, thus B > A in weight

  3. The pans balance, thus A = B in weight


So one thing measurement does is impose an order on a group of objects.


Another important property of measurement is the Archimedean Property, called thus by the Austrian mathematician Otto Stolz in the 1880s. Archimedes himself attributed the property to the Greek mathematician Eudoxus of Cnidus. [Wikipedia Archimedean Property]


If A< B, there is an integer multiple of A, called nA, where nA >= B. Thus we say that nA is a measure of B where A is a unit of measure. Notice a couple of things:


  • If B <= A then n = 1.

  • The size of A does not matter as long as it is not infinitely large or small. The value given A is arbitrary.

  • Unless one is extremely lucky the measure of B by A is an estimate of B. Thus any measurement has a built-in error that must be accounted for.

  • Any unit of measure A can be scaled up or down to another unit of measure by multiplying or dividing by some integer. This creates a system of measurement based on the original unit. For instance, the American Standard for length is the foot, 1/12th of a foot is an inch and 3 feet make a yard.

Archimedean Property [Arkadius 2009]


Any measurement system is a linear ordering of a unit of measure allowing for a measured estimation of an object and the ability to scale up or down to larger or smaller units. Historically a unit of measure has been human body-based. This is true in most cultures and persists today. Human-based measures are empirical and can create materials designed to fit individual differences. Standardization of measures was considered the sole result of a complex culture with a written language. Modern scientific standards of measure are no longer even material but instead based on quantal constants. [Kaaronen et al, 2023] [Price 2023] [Wikipedia List of Human-Based Units of Measurement] [Wikipedia History of Measurement] Both the Maya and the Aztecs had standardized measurement systems, the Aztecs even had a method of finding the area of a complex polygon as they taxed parcels of land much like modern U.S. counties do today. [O’Brien et al. 1986] [Williams and Jorge y Jorge 2008]


Archaeological studies of measurement systems usually start with written records and try to match commentary with actual measurements. [Morely 2010] This technique is not useful for cultures that have no written language. Sir Flinders Petrie (1853 - 1942) was a British Egyptologist who pioneered a more rigorous scientific approach to archaeology. He was one of the first to suggest that ceramics, including pieces of ceramics, could be arranged in a time series, now called seriation. [Poole 1998] In 1877 he published Inductive Metrology: Or, The Recovery of Ancient Measures from the Monuments [Petrie 1877] in which he suggested that measuring architectural artifacts alone could be sufficient to show a standard of measure for a culture. Alexander Thom (1894 - 1985) was a Scottish engineer. In 1955 he became interested in the stone circular monuments in England and France. After he retired from Oxford in 1961, he spent the rest of his life measuring megalithic stone circles. he was the first to use statistics to prove his theories that the circles were astronomical calendars and that this culture had a unit of measure he called the mesolithic yard. [Thom 1967] His life parallels James Marshall's in the midwestern United States, except Marshall didn't use statistics or accept the astronomy connection. [Marshall 1987] [Marshall 1995] Marshall also didn't reach the fame that Thom achieved. This had a downside for Thom as his ideas were the subject of several pseudo-science books. Several attempts have been made to replicate his findings using his data set. [Kendall 1974] [Freeman 1976] Both Kendall and Freeman gave the results as a weak "maybe" with Freeman noticing that the data supporting a unit of measure came mainly from Scotland.


Dee Travis Hudson (1941 - 1985) was curator of the Santa Barbara Natural History Museum from 1973 until his untimely death in 1985. His research into the Chumash peoples helped revive their culture. [Timbrooke 2024] In 1972, Hudson published a study of two Chaco Great Houses, Pueblo Bonito and Pueblo del Arroyo. [Hudson 1972] He got his data from excavations conducted by Niel Judd {Judd 1959] [Judd 1964] He used the same statistical method that Thom used and further divided the concept of a standard into two parts:


"Once personal units acquire a common meaning within a social unit in which case the method of measuring becomes culturally or socially shared the measurement behavior may take one of two forms. If the method of performing the measurement is shared, but the dimensional value of the unit varies, then this type of measurement behavior will be called measures hy custom in this paper. On the other hand, if both method and value for the unit are shared and do not vary, then this type of measurement behavior will be called measures by standard." [Hudson 1972 p28]


Pueblo Bonito has been found to have different styles of architecture that suggest four different building periods. Hudson found no signal for the first building period but clear signals for the other three. In addition, building period three showed a double signal. By dividing this period into two parts, split into the east and west wings of the structure, this resolved into two separate values. So he got four values for three building periods. Strangely these formed two separate clusters with Pueblo del Arroyo forming a much larger outlier. Hudson also found a signal for the use of a measure of standard as opposed to a measure of custom. I am a bit unclear about the meaning of this as the current research suggests that almost all cultures started first using human-based measures. Judd's original data is still available.


Sherry Towers is a statistician and data scientist now working in Germany. She is best known for her work on the contagion effect of mass shootings. In 2017 she published a study of the sun temple at Mesa Verde. She found evidence of the Golden Section, 3-4-5 Pythagorean right triangles, equilateral triangles, 45-degree right triangles, and a standard measurement system. She used aerial photography from Google Earth. One important thing she did was ground truth some of the measurements taken from the imagery to establish the accuracy. [Towers 2017] It is unclear in the paper what statistical methods she used to get her results.


This all reminds me of Marshall, and like Hopewell culture, Pueblo and Chaco culture now have several candidates for a measurement standard. To my knowledge, Hudson and Towers have published the only statistical-based research on measurement systems of a culture without written language in the Southwest. I've found an interesting recent study of Machu Pichu. [Kubicka and Kasiński 2020] Whether the Inca had a written language or not is still a matter of speculation. Some consider Khipu, a system of knots on separate strings tied to a long cord the Inca's written language. If so, it has not been translated. [Demaine and Demaine 2024] The official language is Quechua but the Inca were an empire with many different speakers. Two Quechua to Spanish dictionaries exist from the 16th century. They both contain references to several units of measure. Inca architecture had a standard over their huge empire with some local deviations. Kubicka and Kasiński looked at LIDAR data from a scan they took of a set of standard wall elements, called niches. They concluded that an imperial measurement system could exist. The values found in various structures differed depending on the rank of the people using the structure. Multiples of these values, however, aligned with known Inca measures. There is an R software package currently available for this type of analysis. [Kasiński 2020]


Another paper including one of the same authors looks at a 'Hellenistic' 1st-century AD Cyprus. Cyprus was first Greek, then after Alexander the Great, a Ptolemaic kingdom ruled for several hundred years. This brought in late Egyptian influences. At the time the house was built Cyprus was under Roman rule. Greece contributed three measurement systems while Roman and Ptolemaic one each. [Brzozowska-Jawornicka and Kubicka-Sowinska 2021] The authors concluded that the house was designed using a Ptolemaic measurement system. They make a good point that whatever value the analysis gives it only has worth in the power of the interpretation. In this case, there are no questions about if there is a measurement system or its parameters, just which was used. This does, however, give more trust to the analysis.

There seems to be more research on measurement systems in Europe, but, as I've mentioned, any results are controversial. One problem is that in many instances all of these measurements are lumped together when they each need a statistical model unique to the problem. This includes The Golden Section which has its own strange story to tell, the geometric measurements, what is called structural orientation, and related to this, archaeoastronomy. Here, I want to concentrate on a statistical model for a measurement system.


The statistical model for measurement standards is called a quantum or quantal model. I prefer quantal to quantum to distinguish it from quantum theory. This model was first proposed concerning chemistry, regularities in atomic weights could be explained by integral numbers of protons and neutrons. The first statistical attempt at this problem was in 1954. [Hammersley and Morton 1954] They were the first to call this a quantum hypothesis and offered a Monte Carlo approach. This was taken up by S. R. Broadbent [Broadbent 1955] [Broadbent 1956] whose method was used by both Thom and Hudson in their research. Broadbent is also believed to be the second author of a foundational paper on percolation theory. [Keeler 2024] Kendall, who tried to replicate Thom's data, further refined the model which he called cosine quantogram analysis. [Kendall 1974] Kubicka and Kasiński's analysis and current European studies use Kandall's approach.



In Europe, support for a standardized measurement system also shows support for a diffusion model for an assumed persistence of this system through the ages. [Rottländer 1996] This reminds me of a similar theory of diffusion for geometric knowledge. In the first article, I introduced the ideas of Paulus Gerdes who rejected the diffusion theory in favor of empirical mathematics derived directly from the workings of a material culture. [Gerdes 2003] With the propensity of human-based measurements and the act of construction including sacred architecture still a manifestation of material culture, I wonder if this standardization comes from a convergence of methods rather than some diffusion of an ideal. Perhaps this is what Hudson meant by measure by custom rather than measure by standard.


Hopewell culture did not have a written language although the culture was expansive over a large part of eastern North America. This wasn't a conquest but a spread of ideas and practices, perhaps through proselytizing or pilgrimage. Whether this was the result of migration out of the Scioto Core is unknown. A study of a Hopewell cultural group in Western Michigan shows that the people living there were locals. There is no evidence that this group built earthworks. [Chivis 2020] Meaning had to be transmitted from place to place. [Giles 2020] [Keith 2020] No one knows their language or what modern peoples still may speak a form of it and research into this is just beginning. [Yerkes 2020] Hopewell culture has a distinctive aesthetic and their earthworks also use a visible geometric aesthetic. Whether this aesthetic included a common measure is still an open question.



Part III - Geometry


Part IV - Newark, the Core, and Beyond


Part V - Landscape


Part VI - Time


Part VII - Memory


 

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