Statistical Analysis of Hively and Horn's
Archaeoastronomical Claims at the Newark
Octagon
Christopher S. Turner
Poster presented at the 2004 joint meeting of the Southeastern
Archaeological Conference and the Midwestern Archaeological Conference
October 21-23, 2004 St. Louis
Abstract
In
1982, Earlham College professors Ray Hively and Robert Horn published an
analysis of calendrical sightlines at a Hopewell geometrical earthwork. The
octagonal enclosure in Newark, Ohio, was shown to define specific lunar rise
and set points along the horizon surrounding this notable site. This poster
presents statistical analyses of these claims, using both chi-square and
resampling tests. Using these methods, the Hively and Horn results are found to
be statistically significant. Such results lend credence to their other,
"ad hoc" sightlines (as the authors called them), and to the overall
plausibility of other substantiated Hopewell archaeoastronomical claims.
Introduction
Research
published in 1982 by Ray Hively and Robert Horn detailed the presence of
astronomical sightlines at the octagon component of the Newark earthworks
(33Li10)(Figure 1). The Newark Octagon, believed to date to the late Middle
Woodland (c AD 250), is one of what was once a score of such sites scattered
throughout southern Ohio. These
geometric earthworks, with dimensions exceeding 300 meters, are endemic to the
heartland of the Ohio Hopewell.
Standard to the methodology
of archaeoastronomical studies are statistical analyses of proposed alignments.
Here I present such an analysis of the Hively and Horn Newark Octagon
sightlines.
Null Hypothesis
I
will restrict my analysis to sightlines specific to the octagon embankments and
the eight vertices formed where they converge. Between these eight points,
Hively and Horn claimed to find seven of the eight lunar extrema azimuths
marked. As with all such assertions in archaeoastronomy, the obvious question
is “What are the chances that these sightlines are the result of random
association?” How likely is it that the
alignments formed by these eight points would randomly indicate seven out of
eight specific sightlines?
The
null hypothesis is, then, this:
Events marked by the octagon number no
greater than what would be expected from random occurrence.
As is
conventional, if a statistical probability under 5% likelihood can be
demonstrated for the observed outcome, the null hypothesis is considered false.
Framing the Problem
What
are at issue here are visual sightlines. The inter-vertex lines at the octagon
define these. Specific directions are expressed in degrees of azimuth around
the horizon. Thus, statistically, the 360-degree horizon effectively becomes
the sample universe.
The number of total possible sightlines
between the eight vertices of the earthwork is simple to calculate: it is n
(n-1), where n is the number of vertices. Thus, there are
8 x 7 = 56
possible sightlines defined between the eight gateways of the octagon (Figure
2).
To calculate probability, one must define
a sampling interval. Here it is defined in degrees of azimuth. It would be
meaningless to define the sampling interval as smaller than the known accuracy
of the survey data upon which the archaeoastronomical analysis is based. A
value of one or one-half degree is commonly used. It is problematic to insist
on greater accuracy due to questionable archaeological context, such as erosion
of earthen embankments, or displacement of megalithic or architectural
features, for instance.
Lastly, as noted earlier, the celestial
events marked by the octagon, according to Hively and Horn, are the lunar
extrema, of which there are eight. It is these eight independent azimuths along
the horizon that are considered as “hits” in the probability analysis.
The Hively and
Horn Data and the Statistical Sampling Interval
As mentioned already, Hively and Horn
posited that seven of the eight lunar extrema are indexed by sightlines defined
by the octagon vertices. However, as shown in their data table, three of the
seven sightlines involve azimuth errors greater than one-half degree, putting
them outside of the range of the one-degree sampling interval used in this
statistical analysis. This leaves only four valid sightlines. Hence, when using
the one-degree sampling interval, the statistical likelihood of only four
“hits” will be tested.
Looking at the data in another way, I
expanded the sampling interval to twice the largest error value found amongst
the proposed sightlines. The largest error in the Hively and Horn data table is
1.3°,
so I set the sampling interval to 2.6°. In this way I am able to
test the likelihood of the occurrence of all seven of their proposed lines.
Given these conditions, I calculated the
expected frequencies of lunar extrema sightlines that would occur randomly.
With a sampling interval of one degree and 56 randomly generated sightlines, we
should expect 1.24 lunar extrema sightlines. With the sampling interval set to
2.6°,
we should expect 3.24 lunar extrema sightlines. These expected frequencies were
compared with the Hively and Horn results using chi-square (Guassian)
statistics, and with resampling (Bayesian) statistics.
Chi-Square
Analysis
The chi-square results are shown in
Figures 3 and 4. Because we are examining whether there are more sightlines than would be expected
randomly, the null hypothesis is said to be directional.
Also, because the test is of a simple binary choice (hit or miss of the lunar extrema azimuth), it is said to have one degree of freedom.
Both chi-square results negate the null
hypothesis. With the one-degree sampling interval, the probability of finding 4
lunar extrema sightlines is 1.3 %. When the sample interval is 2.6°, the
probability of finding 7 such sightlines is 3.1%.
Resampling
Results
The chi-square test is an approximation,
and it is said to overstate significance, especially when the number of degrees
of freedom are low, or when the value of the “expected” factor is low, as are
both the case in the above examples. For this reason I have also included
statistical resampling results. Resampling is done with computer software
(Simon 1999). The program repeatedly generates (100,000 times in these
examples) the probability model being tested, and counts the results. Here, 56
sightlines are repeatedly and randomly simulated, and the program counts the
number of times the target intervals (in this case, the eight lunar extrema)
are “hit”. The mean of this value should (and does) converge on the “expected”
frequency.
Additionally, the software program counts
the number of times that the frequency of lunar sightlines meets or exceeds
that found by Hively and Horn. This value is expressed as a percentage as with
probability. Again, because we are examining what percentage “meets or exceeds”
the critical value, the hypothesis is directional (or “one-tailed”).
As anticipated, the resampling
probabilities indicate somewhat less statistical significance than the
chi-square results, but they are still under the 5% critical value. With the
sampling interval at one-degree, the probability of finding 4 lunar sightlines
is 3.7% (Figure 5). With the 2.6° sampling interval, the probability of 7 sightlines is
4.3% (Figure 6).
Independence
of Sightlines
To meet theoretical constraints,
samples being tested with chi-square must be mutually independent. That is to
say, in this case, the random sightlines must be independent of each other. The
actual sightlines at the Newark earthwork, however, are obviously interrelated
because they are defined by a regular polygon. The key word here, though, is regular. It is arguable that a given
symmetry of sightlines defined by an octagon could favor the likelihood of
multiple lunar extrema azimuths being selected, more so than especially a
randomly generated set of such lines. It is important to note, then, that the
octagon is not strictly symmetrical. Hively and Horn demonstrated how given
angles formed by the embankments were distorted from regular symmetry in the
direction of the lunar extrema azimuths.
The twin earthwork to the Newark Octagon,
the High Bank Octagon, is even more distorted from a regular polygon. The High
Bank earthwork complex, in Chillicothe Ohio, was also analyzed by Hively and
Horn (1984). It was also found to index the lunar extrema and the four solstice
azimuths.
The Hopeton Works, also in Chillicothe,
exhibit minimal geometric regularity, the polygon anomalously somewhere between
an octagon and a square. Yet it too indexes the lunar and solar extrema, as well as the equinox and cross-quarter
dates (Turner 1982, 1983, 2004).
Geometric regularity was important to the
Hopewell, but was secondary in importance to the accuracy of the calendrical
sightlines. Surely the Newark Octagon was the apex of Hopewell geometric
earthwork construction, the octagon being the most complex figure they crafted.
It is not however, and never was intended to be, an equilateral octagon. As
noted by Hively and Horn, the angles at the vertices alternate, so we are left
with a flattened, or bilateral octagon. It seems evident that the builders
chose this particular bilateral octagon, out of the many such possible, to best
match the azimuths of the lunar extrema. The intent was not to adhere to a
perfect equilateral octagon.
Given the variation in geometric symmetry
between Hopeton, High Bank, and the Newark Octagon, but remembering that each
one indexes the corpus of lunar extrema, it is arguable that the overarching
intent of such earthwork construction was to mark the calendrical sightlines.
Any symbolism incorporated into the architecture by virtue of the geometric
design should be interpreted as being secondary to calendrical sightlines
themselves. Hence the non-independent nature of the sightlines, rather than
being a theoretical obstruction, is exactly that for which we are testing: that the interdependence of the sightlines
is a result of the calendrical functioning of the earthwork. The criticism
that the sightlines are non-independent by virtue of the octagonal construction
is answered by evidencing the primary influence of the calendrical intent, and
the intentional distortions from idealized geometric regularity as found in
these earthworks.
An Earthwork
Phylogeny
The above three cited earthworks, Newark
Octagon, High Bank, and Hopeton, share morphology: they fall under the rubric
of what Martin Byers calls the “C-R” groups, i.e. circle-rectilinear (Byers 1998:139). The majority of Hopewell geometric sites have a three-component,
or tripartite arrangement. The “C-R” groups by contrast consist of a circle, or
more properly an ovoid, conjoined to a polygon.
The two sites, Newark Octagon and High
Bank, are further distinguished by having each an octagonal polygon, and
because their near-perfect circular components share nearly the exact same
diameter. Hopeton, their awkward cousin, fails to trace an octagon or boast
precise circularity, but it joins the two in a group: C-R earthworks that mark lunar sightlines.
This apparent interrelation has been long
noticed, and was commented on by Maclean (1879:83) “It would appear that the
Hopeton and High Bank Works were either modeled after that at Newark, or else
the last was a combination of the other two”.
In this triad, we see a clinal variation
of both geometrical and calendrical precision. Also, though carbon dates are
not yet conclusive, I would assert that presently they suggest, or are
consistent with, a chronology of:
Hopeton, 1st century AD; High Bank second century; and Newark
Octagon third century AD. This temporal ordering may be evidenced in the degree
of geometrical complexity found at Newark Octagon relative to High Bank, and
Hopeton. Similarly, there appears to be increased specialization in the
selection of specific types of celestial sightlines marked. Hopeton, which is
the least geometrically regular of the three, and perhaps the oldest, notably
indexes the largest number and greatest variety of celestial alignments. Also,
Hopeton has many multiple or redundant sightlines.
High Bank, clearly intermediate in
geometric symmetry and complexity between the other two, also marks an
intermediate number of celestial events. And this is not some spurious parsing
of the types of calendrical sightlines. High Bank indexes all of the lunar
extrema events, and the four solstice azimuths, and nothing else. It trends toward an octagonal outline (and
accompanying angular constraints) at the expense of omitting Hopeton’s equinox
and cross-quarter sightlines.
Finally we have Newark Octagon, the most
morphologically symmetrical of the three, but what’s more, yielding the most
restricted set of celestial phenomena: its
sightlines mark only the lunar extrema. This statement can be restated as
such: the Newark Octagon has been designed, especially in comparison to High
Bank and Hopeton, to not mark any solar
phenomena. Arguably and demonstrably, calendrical accuracy of sites such as
Hopeton or High Bank was sufficient. No practical reason exists compelling the
avoidance or non-inclusion of solar sightlines at the Newark Octagon. We could
conjecture that a taboo as such, or perhaps a refinement of celestial mana, was
at play, restricting the embodiment of solar angles and the symbolism they carried,
magnifying or concentrating things lunar.
Hence it appears that the Hopewell
intentionally included the lunar lines and excluded the solar at the Newark
Octagon. The statistical question thus arises: what is the probability that
this octagon could index some of the lunar lines but still miss all of the
solar ones in question?
A Final
Statistic
Remembering the earlier discussion, we
have 4 lunar extrema sightlines marked with the one-degree sampling interval,
or we can include all seven Hively and Horn lines by using the larger sampling
interval (2.6°).
There are ten rising and setting solar
events that are not marked by the Newark Octagon that are marked at other
Hopewell sites: these are the solstices, equinoxes, and cross-quarter dates.
Using the one-degree sampling interval,
the likelihood of 56 random sightlines both:
§ marking
4 out of 8 lunar extrema azimuths, and
§ not-marking
any of 10 solar azimuths
is P = 0.74%.
When using the 2.6° sampling interval, i.e.
when including all seven of the Hively and Horn sightlines, the likelihood is
one in 1492 (Figure 6).
(Author’s note: In the original text of this
paper, which I distributed to accompany the conference poster presentation in
2004, the last probability figure of 1 in 1492 was stated as for a directional hypothesis.
Since then, specifically when I was editing my 2011 article in Time and Mind, I reasoned that because the above combined
probability result is for a combination of “hits” and “misses”, that the analysis would then be non-directional,
or “two-tailed”. Therefore, in the Time
and Mind article (page 308), I cited the combined resultant likelihood as
per a non-directional hypothesis as 1/746, half of the directional result of
1/1492 Now, I have been reminded that in
order to posit non-directionality in the statistical results, that the two
hypothesized events, though independently tested directionally and using both
“tails”, (the right tail in the lunar case and the left tail in the solar
case), must be mutually exclusive in order to warrant the “non-directional” interpretation. This is definitely not the case here. The
avoidance of solar sightlines and the
“hitting” of lunar sightlines are not mutually exclusive events which are being
tested simultaneously by the same null hypothesis. The mutual exclusivity presented
in each null hypothesis is, in the first case: do the inter-vertex octagon
sightlines “hit” azimuths that index the 8 lunar extrema more than the expected
number of times , or do they not? The second null hypothesis, concerning the
solar sightlines, asks: do the
sightlines “miss” the 10 azimuths that index solar events less often
than what would be expected, or do they
not. Therefore I would further adjudge my conclusion here and suggest that
my original combined probability figure of 1 in 1492 is sound. Note that in
each example above the terms “more than” or “less often than” denote these
hypotheses as bring directional.
Summary
Hively
and Horn revealed that the Newark Octagon, a Middle Woodland Ohio Hopewell
geometric enclosure, marks seven of eight lunar extrema azimuths, as defined by
inter-vertex sightlines. Using chi-square and resampling tests, I analyzed
these calendrical alignments for statistical significance. Using two different
sampling intervals, all results negated the proposed null hypothesis.
The
tests are not invalidated by the non-independence of the sample (sightlines).
The very interdependence that the sightlines encode is due to their
intentionally being aimed at specific calendrical azimuths, not to the
overarching geometric outline of the enclosures. This calendric intent is
exactly what the archaeoastronomical results suggest in the first place.
Geometric symmetry was not the primary goal in constructing these earthworks.
The degree of geometric regularity present in the earthworks varies greatly
from site-to-site. Specific embankments are skewed away from geometric symmetry
and toward calendrically significant azimuths.
Within
the triad of enclosures, Hopeton-High Bank-Newark Octagon, there are clinal
variations in geometric and calendrical complexity that may also reflect a
relative chronology between the three sites. Hopeton embodies the greatest
number of types of calendrical sightlines, High Bank less, and Newark Octagon the
least. The latter earthwork marks only lunar events and appears to be
constructed to specifically avoid solar azimuths.
I
tested this last point statistically, and found extremely low probabilities
that this particular combination of circumstances occurs randomly.
Statistical
analyses are fundamental to argument veracity in archaeoastronomy. However, the
methods are often questioned as suspect. The above paper is not intended as the
final word on Hopewell archaeoastronomy, but it can serve as a jumping off
point for further discussion, or more importantly as an adjunct to the range of
other forms of argument available to anthropologists.
Figures
Figure 6
References
Cited
Byers, A. Martin
1998
Is the Newark Circle-Octagon the Ohio Hopewell “Rosetta
Stone”? A Question of Archaeological Interpretation. In Ancient Earthen Enclosure of the Eastern Woodlands, edited by Robert C. Mainfort and Lynne P. Sullivan, pp. 135-153. University Press of
Florida, Gainesville.
Hively, Ray
and Robert Horn
1982
Geometry and Astronomy in Prehistoric Ohio. Archaeoastronomy
supplement to
the Journal for the History of Astronomy 13(4):S1-S20.
1984
Hopewellian Geometry and
Astronomy at High Bank. Archaeoastronomy
supplement
to the Journal for the History of
Astronomy 15(7):S85-S100.
MacLean, J.P.
1879
The Mound
Builders. Robert Clarke and Co., Cincinnati.
Simon, Julian L.
1999 Resampling Stats: the New Statistics.
Version 5.0.2. Resampling Stats, Inc.
Turner, Christopher S.
1982
Hopewell Archaeoastronomy. Archaeoastronomy 5(3):9.
Center for
Archaeoastronomy, College Park, Maryland.
1983
An Astronomical
Interpretation of the Hopeton Earthworks. Manuscript on file at the Hopewell
Culture National Historical Park, Chillicothe,
Ohio; and at the Ohio Historical Center, Columbus.
2004 Middle Woodland Archaeoastronomy in Ohio.
Paper presented at the 7th Oxford Conference on Archaeoastronomy,
Flagstaff, AZ.
No comments:
Post a Comment