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Journal of Archaeological Science: Reports journal homepage: www.elsevier.com/locate/jasrep

Astronomical signiﬁcance of architectural orientations in the Maya Lowlands: A statistical approach A. César González-García a,⁎, Ivan Šprajc b a b

Instituto de Ciencias del Patrimonio, Incipit, Consejo Superior de Investigaciones Cientíﬁcas, CSIC, Avda. de Vigo s/n, 15705 Santiago de Compostrela, Spain Research Center of the Slovenian Academy of Sciences and Arts (ZRC SAZU), Novi trg 2, 1000 Ljubljana, Slovenia

a r t i c l e

i n f o

Article history: Received 7 March 2016 Received in revised form 14 July 2016 Accepted 18 July 2016 Available online xxxx Keywords: Maya architecture Astronomical orientations Statistical analysis Archaeoastronomy Maya calendar

a b s t r a c t A recently accomplished systematic study of 271 architectural orientations measured at 87 archaeological sites in the Maya Lowlands led to several conclusions about the underlying astronomical motives. In order to test these proposals, we have performed a number of statistical analyses, employing kernel density estimations (KDEs) and cluster analyses without preconceived ideas on the directionality of the different buildings. Since most buildings have roughly rectangular ground plans, both north-south and east-west orientation axes were analyzed. Our KDE analyses, in which the errors assigned to alignment data were considered, are based on the assumption that the direction in which a particular orientation group was astronomically functional is indicated in histograms by narrower maxima with larger amplitudes. The distribution of azimuths and declinations suggests that the orientations were functional predominantly in the east-west direction, largely referring to the Sun, but the existence of alignments to the major extremes of Venus and the Moon on the horizon is also highly likely. The analyses of the distribution of sunrise and sunset dates corresponding to solar orientations, as well as of the intervening intervals, support the idea that the orientations recorded the dates separated by multiples of 13 and 20 days, allowing the use of easily manageable observational calendars intended to facilitate a proper scheduling of agricultural activities and the associated rituals. As revealed by cluster analyses, the East Coast and the Usumacinta basin share similar orientation trends, which are, however, notably different from those in the rest of the Maya Lowlands, probably reﬂecting regional variations observed also in other cultural elements. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Fray Toribio de Motolinía (1971: 51), a Spanish friar who arrived in Mexico soon after the Conquest, writes in his main work Memoriales that the Aztec calendrical feast of Tlacaxipehualiztli “took place when the sun stood in the middle of [the Temple of] Huitzilopochtli, which was at the equinox, and because it was a little out of line, [King] Moctezuma wished to pull it down and set it right” (Aveni 2001: 236ff). The complementary information can be found in a map of Tenochtitlan attributed to Cortés, where the face of the Sun is shown between the twin sanctuaries of the Templo Mayor (Aveni 2001: 236ff, Fig. 84; Šprajc, 2000a). Even if, aside from some drawings in prehispanic and Conquest-period codices, these seem to be the only documentary sources alluding to the astronomical orientation of a prehispanic temple, it can now be afﬁrmed that the practice of orienting ceremonial buildings on astronomical grounds was common in Mesoamerica. Archaeoastronomical research carried out during the last few decades has revealed that the distribution of orientations in ⁎ Corresponding author. E-mail addresses: [email protected] (A.C. González-García), [email protected] (I. Šprajc).

http://dx.doi.org/10.1016/j.jasrep.2016.07.020 2352-409X/© 2016 Elsevier Ltd. All rights reserved.

Mesoamerican civic and ceremonial architecture is clearly non-uniform, exhibiting concentrations around certain azimuthal values. The presence of such orientation groups at a number of sites spread far apart in space and time can only be explained with the use of astronomical objects at the horizon as reference objects (cf. Aveni and Hartung 1986: 7f). In view of the prevailing orientation patterns it is rather clear that most buildings were aligned upon sunrises and sunsets on particular dates, which have been interpreted in terms of their relevance in the agricultural cycle and in computations related to the calendrical system (Aveni, 2001; Aveni and Hartung, 1986; Aveni et al., 2003; Tichy, 1991). These and other hypotheses have also been forwarded for the orientations in the Maya Lowlands. However, due to the deﬁciencies and low precision of most of the available alignment data, many of these hypotheses remained unconvincing and impossible to test, disclosing the need for further ﬁeldwork with different approaches. In order to obtain a reliable set of data on architectural orientations in the Maya Lowlands, a systematic study was carried out in 2010 and 2011: at 87 archaeological sites in southeastern Mexico and northern Guatemala, 271 architectural orientations were measured, applying a much more rigorous methodology in data collection than the one employed in former studies. Our ﬁeld methods and techniques can be summarized as follows:

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1. The orientations were measured in ﬁeld, employing established archaeoastronomical measurement procedures (Aveni 2001: 119ff; Ruggles 1999: 164ff) (formerly published data were often derived from site maps, many of which are notoriously inaccurate). 2. In order to obtain a coherent data set, only structures with evidently civic or ceremonial functions (temples, elite residences, administrative buildings) were selected for measurements. 3. At every building both north-south and east-west azimuths were determined (in former publications the azimuth of only one line measured at a building was given as deﬁning its orientation in both directions; since many Maya buildings have patently rhomboidal ground plans, the perpendicular to a north-south line cannot be considered as representative of the building's east-west orientation, and vice versa). 4. At every building as many lines as possible were measured (walls, alignments of columns, etc.), and mean azimuths of east-west and north-south lines were determined, including their possible errors estimated on the basis of divergences among individual lines and uncertainties due to the building's present state of conservation. Wherever preserved, the alignments most likely to have been observationally functional were given greater weight (e.g. axes of symmetry of upper parts of buildings). 5. Horizon altitudes, regularly missing in formerly published data, but necessary for calculating declinations corresponding to the azimuths, were always determined, with either ﬁeld measurements or calculations based on cartography. The results of this study are comprehensively presented in a monograph (Sánchez Nava and Šprajc, 2015), which includes the methodology employed, all the relevant data on orientations, the analyses that led to the recognition of their most likely celestial referents, and interpretations of their cultural signiﬁcance based on contextual evidence. The most important conclusions, relevant for the purposes of the present article, are the following: 1. Most orientations included in the study were motivated by astronomical considerations and were functional predominantly in the east-west direction. 2. Orientations to sunrises or sunsets on certain dates prevail in the sample; the dates recorded by a building on the eastern or western horizon tend to be separated by multiples of 13 or 20 days, i.e. elementary periods of the Mesoamerican calendrical system. 3. Some orientations were probably related to the extremes of Venus and the Moon. 4. The whole study area manifests a high degree of uniformity and longevity of orientation patterns, with relatively few regional and temporal variations. The objective of this article is to establish to which degree the above summarized conclusions can be supported with statistical analyses of the alignment data. 2. Methodology While Aveni (2006: 58) afﬁrms, in relation with the use of statistics in archaeoastronomical research, that “the standard procedure for testing for intentionality in astronomical orientations has consisted of selecting putative astronomical targets, matching them with archaeological alignments, and then conducting some sort of statistical test of the likelihood of coincidence”, we have adopted a more objective approach, in which no astronomical target is given a preconceived significance. A similar approach was followed by Higginbottom and Clay (1999), who examined the suitability of several statistical tests to pinpoint the existence of non-uniform distributions of orientation data. In order to test the propositions listed above, we have employed two techniques, which are summarized below.

2.1. Kernel density estimators Histograms give the number of occurrences of a given characteristic within a population, e.g. of an orientation with a particular azimuth value. Such value is often given in terms of relative frequency or percentage. An appropriate smoothing of the azimuth histogram by a function called the kernel produces the azimuth kernel density estimate (KDE hereinafter) and its representation is a curvigram: each azimuth (or other) value in our sample is multiplied by the kernel function with a given passband or width, and all resulting kernels are summed up to produce the ﬁnal KDE, or distribution function. The key choices then are the type of function we use and the passband or the width of this function. For the ﬁrst choice, there are several possible functions, although the Gaussian and the Epanechnikov kernels are the most commonly used. For the bandwidth, although there are theoretical prescriptions in the literature as to the best choice (Rosenblatt, 1956; Parzen, 1962; Poluektov, 2014), these are not practical. A practical prescription for the optimal choice of the bandwidth is h ≈ 1.06σn−1/5, where n is the number of values we have and σ is the standard deviation of the sample (Silverman, 1998); however, this is only usable if we use a Gaussian kernel and the data behaves as a Gaussian distribution. It is important to note that for other kernels this is based more on a trial-and-error method; however, depending on the number of structures, the spread in values and the errors implied in the data acquisition process, one may use larger or smaller values for the passband. In the following we will consider the estimated error assigned to each value by Sánchez Nava and Šprajc (2015: Table 1) as the passband. As explained below, we have analyzed the distribution of both azimuths and the corresponding declinations and, in the case of solar orientations, also of the corresponding sunrise and sunset dates and the intervening intervals. For each alignment the possible error in azimuth was estimated on the basis of uncertainties regarding the originally intended value and depending on the present state of the building, and the corresponding errors of declinations, dates and intervals were calculated (Sánchez Nava and Šprajc, 2015). To be able to ascertain whether a concentration of values is signiﬁcant we have to compare the distribution of our measured data against the result expected if the null hypothesis were true, and see if the result is signiﬁcant. In our case, in order to test if the orientation is not at random or homogeneously distributed to each point in the horizon, the comparison should be against a random sample of the same size as our database. However this may pose a problem: a random sample with a small volume of data could give a rugged histogram to compare with, while a large sized random sample is not much different from a homogeneous distribution. This is why, given the relatively large size of our sample, we propose to do this comparison with a homogeneous distribution in azimuth of the same size as that of our sample. We normally use a normalized relative frequency to scale our KDEs or histograms. To do so we divide the number of occurrences of a given Table 1 Declination values and statistical signiﬁcances for the most prominent pairs of maxima (each pertaining to the same orientation group) in Fig. 3 (δE: east declination; δW: west declination). The data for each pair are presented in columns numbered consecutively in the ﬁrst row. The values and signiﬁcances of the eastern maxima (Fig. 3a) are given in the second and third rows; the corresponding data for the western maxima (Fig. 3b) are listed in the ﬁfth and sixth rows. In the fourth row the signiﬁcances of the azimuth (A) maxima corresponding to each group are given for comparison, whereas the last row provides a measure of the error estimated by the calculation of the signiﬁcance using different kernels. The mean error is of 0.3. Maximum

1

2

3

4

5

6

7

δE (°) Signiﬁcance A sign. δW (°) Signiﬁcance Error

−28.11 3.93 1.83 28.23 3.73 0.06

−23.52 3.61 1.76 23.32 3.62 0.12

−20.58 6.97 2.85 20.05 8.08 0.27

−18.62 7.63 3.78 18.42 8.77 0.31

−14.03 21.13 7.22 13.51 21.69 0.64

−10.43 19.42 6.76 10.23 20.05 0.36

−0.61 4.92 1.97 0.74 6.93 0.30

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azimuth value by the mean number of occurrences for that sample; this is equivalent to dividing or comparing the results of a uniform distribution of the same size as our data sample and with a value equal to the mean of our data. The quantity (f(obs) − f(unif)) / σ(unif) will be the expression of our comparison for declinations and dates, where f(obs) is the frequency of the observed event, f(unif) is the frequency of the uniform event, and σ(unif) is the standard deviation of the uniform distribution. What is the meaning of this? Obtaining a curvigram scaled with respect to the uniform distribution, we can see whether our data is signiﬁcant in comparison with that distribution. The scale is given by the standard deviation of such uniform distribution; if our data has a maximum that rises up to a value of 3 this means that it is 3 times larger than that standard deviation, or 3σ. This is a standard criterion in natural sciences to indicate that a given ‘signal’ is signiﬁcant. We will use these concepts along the paper and try to estimate, based on the above criteria, whether certain orientations in the Maya Lowlands are more signiﬁcant than others and what possible implications these results may have. The hypothesis we want to test is that the buildings composing our sample (and selected on the basis of the above speciﬁed criteria) were oriented on astronomical grounds, speciﬁcally to the rising and setting positions of certain celestial bodies. Second, we will evaluate to what extent the results of analyses suggest the directionality of orientation groups. Since most buildings have roughly rectangular ground plans, the orientation of each of them can be described with two azimuths, which correspond to four points on the horizon. It is reasonable to assume that, if several structures aimed at a particular target in the same direction, the curvigram corresponding to that direction will indicate a clear maximum with a sharp distribution, i.e. we expect a large amplitude and small spread or dispersion on both sides of the maximum.

2.2. Multivariate techniques Multivariate techniques are commonly used in archaeology and have been used in the recent past for archaeoastronomical data (see e.g. González-García and Belmonte, 2010, González-García, 2013). These techniques produce some kind of graphical output and statistics that are descriptive, because they simplify the data and present possible patterns within them (Fletcher and Lock 2005: 139). The technique we will apply here is Cluster Analysis, which tries to search for regularities in data distribution that may disclose groups or clusters among them. Using a given distance-measuring algorithm, cluster analysis ﬁnds groups among the data and then ﬁnds distances among the clusters, building a hierarchy of distances. A plotting procedure of such a hierarchy is known as a dendrogram. In a ﬁrst step, we have used the values of azimuth and declination to ﬁnd the groups in orientation. Later, we have grouped the data according to customary cultural areas. In this second case, we use the following seven (7) numbers to characterize the distribution of the data in each group, following González-García and Belmonte (2010; see Table 1): the mean declination, the median declination, the standard deviation of the declination distribution, the maximum declination of the distribution, the minimum declination of the distribution, the declination of the ﬁrst and second (if existent) maximum in the declination kernel density diagrams of the group. In case the second maximum is nonexistent this is taken to be equal to the ﬁrst one. We have used IDL software to produce ﬁrst the cluster analysis data and then the distances among groups. In the present work, we have used a weighted pairwise average where the distance between two clusters is deﬁned as the average distance for all pairs of objects between each cluster, weighted by the number of objects in each cluster. The relative distance is given on the left side of the diagrams. This quantity will be used to ﬁnd correlations between the different groups.

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3. Results To assess the differences, we initially analyzed the azimuth distribution applying both Gaussian and Epanechnikov kernels. Fig. 1 presents the results obtained with both kernels applied to the azimuths of north-south and east-west building axes azimuths, to make the comparison easier and the differences more clearly discernible. The greyshaded area is the result using an Epanechnikov kernel, while the solid black curve is the one obtained with a Gaussian kernel. Indeed, the results do not change signiﬁcantly, but the variations in the amplitude of the maxima provide a way to compute the effect of using different kernels, and thus a handle on the errors introduced by that variable, which we use later to assign possible signiﬁcances to the different maxima in the curvigrams. From now on all curvigrams and results are done using the Gaussian kernel, both because it is a more commonly used shape and because after several tests, examining the shape of the maxima and its sharpness, it was deﬁned as the best option. It should be stressed here that we use two senses for signiﬁcance: ﬁrstly, if a given maximum in one of our histograms has a value above 3 we shall consider it as being signiﬁcant. However, we will later compare two related maxima, for example for structures that have orientations towards east or west. Both east and west maxima might be signiﬁcant in the above sense, but with different values. In order to ascertain if such difference indicates a preference in directionality (east vs. west for instance) we will compare the values of those maxima. If the difference is 3 times larger than the error introduce by considering the different kernels, then we shall consider the larger maximum as more signiﬁcant than the other(s). Fig. 1 shows the distribution of azimuths of the north-south and east-west axes (only northward and eastward values are considered) of the 271 structures measured in the Maya Lowlands. The values of both distributions are normalized by the mean relative frequency, to provide a handle on their signiﬁcance. In a sense, this is like comparing both distributions with a uniformly distributed set with the same number of elements. The distribution of northward azimuths displays a larger spread and presents two maxima at values close to azimuths 11° and 14° and with signiﬁcance values close to 5 above the mean. The distribution of eastward azimuths presents more pronounced concentrations with two maxima at values close to 100° and 104° and signiﬁcances of 7 and 7.5 respectively. The differences between the two pairs of maxima in the two distributions are of 1.5 and 2 respectively. Considering the mean difference of 0.3 resulting from the use of the two kernels

Fig. 1. Azimuth distribution of 271 structures in the Maya area. The long dashed vertical lines indicate the cardinal directions, short solid lines indicate the solstice sunrise and sunset for a latitude of 20.5° and horizon altitude of 0°, while the short dashed lines give the moonrise and moonset for the major lunar standstills for that latitude. There are two curvigrams showing the result of employing an Epanechnikov kernel (grey-shaded area) or a Gaussian one (black solid line). The results are normalized by the mean value of the relative frequency, thus a value of 2 means that such azimuth is two times above the mean. The differences in using both kernels are minimal, but in some cases they affect the amplitude of the maxima.

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explored in Fig. 1, the differences are signiﬁcant, suggesting that the buildings were deliberately oriented along their east-west axes and that the concentrations of north azimuths are but a consequence of the buildings' roughly rectangular ground plans. Beside the two maxima mentioned above, other signiﬁcant peaks can be observed in Fig. 1. However, the celestial coordinate that allows identiﬁcation of the astronomical phenomenon(-a) possibly related with an alignment is the declination (δ), whose value, expressing angular distance from celestial equator to the north and south and ranging from 0° to ±90°, depends on the azimuth of the alignment, geographic latitude and horizon altitude corrected for atmospheric refraction. Therefore, in order to assess the astronomical potential of orientations, the distribution of declinations corresponding to the azimuths (Sánchez Nava and Šprajc, 2015: Table 1) must be analyzed. As mentioned above, the orientation of each building has been deﬁned with the azimuths of its north-south and east-west axes, and each axis corresponds to two declinations on the opposite horizons. For the sake of simplicity, the declinations corresponding to the azimuths within the northern, southern, eastern and western quadrants of the horizon (centred on cardinal directions and spanning 90° each) will be designated as north, south, east and west declinations, respectively. Fig. 2 shows the declination histogram for all orientations (dark grey) compared with a homogeneous distribution of the same number of elements (light grey). The latter distribution is the one we should expect if the orientations were distributed uniformly (or randomly) along the compass directions considering a ﬂat horizon without atmospheric refraction and a ﬁxed latitude of 20.5°; in other words, this is the expected distribution of the null hypothesis. If the latter were true, we should expect a larger concentration of north and south declinations. This is, indeed, seen in the actual data, but the unexpected parts are the concentrations around particular east and west declinations inside the lunisolar range, reinforcing the results based on the azimuth distribution and exposed above. It is clear that the two distributions are dissimilar and that the null hypothesis can be rejected with a great conﬁdence, especially concerning the eastern and western directions. Since it is, therefore, unlikely that the orientations were functional to the north or south, we will now concentrate on the east and west declinations. In the attempt to determine the directionality of orientations, we have compared the signiﬁcance of east and west declinations with that of a uniformly distributed sample of the same number of elements in azimuth and then translated into declination. The result of this comparison is given in Fig. 3. In each of the two diagrams there are at least 7 clear maxima above the 3σ limit. These are given in Table 1, together with the values of their

Fig. 2. Distribution of declinations corresponding to the north-south and east-west axes of all structures in the study area (dark grey) compared to the distribution of declinations calculated for the same number of homogeneously distributed azimuths at the latitude of 20.5° and on a ﬂat horizon without refraction (light grey). Solid and dashed vertical lines mark solstitial declinations of the Sun and the major standstills of the Moon, respectively; the dotted line corresponds to the equinoctial declination of the Sun.

signiﬁcance; the data are arranged in pairs, each corresponding to the same orientation group. Note that the signiﬁcance is given in terms of adimensional units, and therefore hereinafter all tables including such measure do not include a unit. In this sense a value of 3.93 means that such peak is 3.93 times larger than the dispersion of the base distribution. The error in these tables also refers to this measurement of the signiﬁcance and thus it is also expressed in these adimensional units. According to the above exposed hypothesis, if one of the peaks in a pair is signiﬁcantly larger, it suggests the direction in which the corresponding orientation group was intended to be astronomically functional. Table 1 also includes the signiﬁcances of the corresponding azimuth maxima, for comparison. The fact that the latter are notably smaller than those of the declination maxima supports the opinion that the orientations targeted astronomical phenomena on the horizon, rather than having been distributed in accordance with some kind of geometrical template, such as the one proposed by Tichy (1991). The last row in Table 1 provides the average error due to the use of the different kernels in the declination data. We will consider that the difference between peaks is signiﬁcant if it is larger than 3 times the average error due to using the different kernels. The differences are clear, especially for some peaks. Considering these errors, we could estimate that there is a signiﬁcant difference between the peaks in columns (1), (3), (4) and (7). If, however, we consider the mean error introduced by the use of different kernels (0.3), only columns (3), (4) and (7) present signiﬁcant differences (above the 3-σ level), pointing to the western directionality of the corresponding orientation groups. From now on we use this last mean error to estimate the signiﬁcances. Among the signiﬁcant maxima in Table 1, only pair 1 is beyond the span of solar declinations. While the latter will be discussed below, in relation with the corresponding sunrise and sunset dates, the eastern and western maxima in column 1 are at declinations of − 28.11° and 28.23°, with signiﬁcances of 3.93 and 3.73, respectively. The error introduced by using a different kernel for this particular maximum is nearly 0.06, so the difference in signiﬁcance indicates that the prominence is larger towards east with a 99% conﬁdence level. These two maxima may be related with major lunar standstills, which occur at 18.6-year intervals, when the Moon reaches extreme declinations of about ±28.5° (cf. Morrison, 1980; Ruggles, 1999: 36f, 60f; González-García, 2015). However, for calculating lunar declinations corresponding to alignments (Sánchez Nava and Šprajc, 2015: Table 1), the lunar parallax must be considered, resulting in that these maxima appear at values of − 27.71° towards east and 28.61° towards west, with signiﬁcances of 3.95 and 3.75. The larger signiﬁcances might indicate a marginal preference (although not statistically signiﬁcant) for lunar rather than non-lunar declinations, perhaps more focused towards the east. It is interesting to note, however, that the western peak exhibits a better agreement with the lunar standstill declination. Possibly the eastern peak reﬂects observation of lunar extremes at full Moon, because in that case the absolute values of declinations recorded by alignments tend to be smaller (Sánchez Nava and Šprajc, 2015; note that these declinations are valid for the center of lunar disc along the alignment; if calculated for the upper or lower limb, the variations would be about ±6 arc minutes). We must note here that some peaks might be masking the blending of several maxima due to the bin size used for our kernel. If we consider a constant bin of 1° for the histograms, we ﬁnd that the signiﬁcance is larger towards west, although in that case the maxima are lower than the 3-sigma and the differences between east and west are not signiﬁcant beyond the imposed 3-sigma level. It is interesting to indicate the possible link with the Venus extremes of two maxima lying below our signiﬁcance criterion; as already noted by Sánchez Nava and Šprajc (2015), the peak at west declination of 26.92° (Fig. 3b, with a signiﬁcance of 2.81) agrees well with the maximum northerly declination of Venus. The east declination peak corresponding to this orientation group (value of − 27.12°, with a signiﬁcance of 2.69: Fig. 3a) cannot refer to the maximum southerly

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Fig. 3. Curvigrams of east (a) and west (b) declinations for all structures in the study area, normalized by comparing with a sample of uniformly distributed orientations.

declination of the planet, considering the asymmetry of its maximum extremes visible on the eastern and western horizon: the extreme declinations of Venus when it is visible as morning star hardly exceed those of the Sun at the solstices (Šprajc 1993: 19ff; Šprajc, 2015). It may be mentioned that Group 4 in Table 1 possibly involves orientations to minor standstill positions of the Moon (when its extreme declinations are about ±18.5°), but there is no sound evidence supporting this hypothesis, since another possible astronomical referent of these orientations is the Sun. Contrary to what happens with the major lunar standstills, the minor lunar standstill occur in areas of the sky where the Sun would also be observable. Indeed, the minor standstill has properties that make it interesting in itself (see e.g. GonzálezGarcía, 2016), but the lack of independent evidence supporting the interest in minor standstill prompts us to give preference to the Sun in these cases. In the absence of other signiﬁcant peaks beyond the range of solar declinations, we now want to test if the orientations refer to the rising or setting sun on particular dates along the year. For these purposes, however, the analyses of declinations do not necessarily provide reliable results, considering that, due to precessional variations in the obliquity of the ecliptic and in the heliocentric longitude of the perihelion of the Earth's orbit (the latter element determining the length of astronomical seasons), on the one hand, and to the intercalation system used in the Gregorian calendar, on the other, one and the same solar declination does not necessarily correspond in any time span to exactly the same Gregorian date. Consequently, if we attempt to determine the directionality of a group of solar orientations on the basis of differing statistical signiﬁcances of the maxima of the corresponding east and west declinations, the results obtained may well be spurious. The following analyses aimed at establishing the most likely directionality of prominent groups of potentially solar orientations are, therefore, based on the corresponding sunrise and sunset dates, which were determined by Sánchez Nava and Šprajc (2015) (Table 1) and are valid for the most likely period of the building in question,1 as well as on the intervening intervals, which may also have been relevant (see below).

1 The dates corresponding to the declinations in relevant periods and given in the proleptic Gregorian calendar (Sánchez Nava and Šprajc, 2015: Table 1) – because the latter preserves a stable relationship with the tropical year during relatively long periods – were determined with the aid of Horizons system elaborated by the Solar System Dynamics Group, Jet Propulsion Laboratory, NASA (EE.UU.) and available online (http://ssd.jpl.nasa. gov/horizons.cgi). For the purposes of analyses of the distribution of both dates and the intervening intervals (v. infra), the dates were determined with precision (i.e. with the fraction of day), assuming that vernal equinox fell invariably on March 21.0 (i.e. March 21, Gregorian, at 0:00 h of Universal Time). Since the moments of the tropical year in which the Sun. reaches a certain declination vary relatively slowly through time, the uncertainties as to the exact dating of particular structures have an insigniﬁcant effect on the validity of date calculations. For details see Sánchez Nava and Šprajc (2015).

Since our null hypothesis is that there was no particular interest in speciﬁc dates, the distribution of sunrise and sunset dates corresponding to orientations compatible with the Sun's positions along the horizon is compared, in Fig. 4, with the distribution of dates that would result from a homogeneous distribution of orientations. It should be noted that, when the latter is converted into sunrise or sunset dates, it is more probable to have more dates close to the equinoxes than around the solstices. However, our distribution of both sunrise and sunset dates is not consistent with such distribution, suggesting a preference in orientation practices for particular dates along the year. Again the difference in signiﬁcances could give us a hint to break the dichotomy in the dates. This is shown in Fig. 5.

Fig. 4. Curvigram of dates (day-of-year numbers) of sunrise (top) and sunset (bottom) for the orientations corresponding to declinations within the solar span. The light grey area presents the distribution of dates derived from a homogeneous distribution of orientations.

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Fig. 5. Statistical signiﬁcance of dates corresponding to orientations as compared with a homogeneous distribution of dates along the year. There are a number of both sunrise and sunset dates that appear signiﬁcant above the 3-sigma level.

Fig. 6. Statistical signiﬁcance of intervals separating the pairs of sunrise (top) and sunset (bottom) dates.

The sunrise dates concentrate largely in February and October, while most of the sunset dates fall in April and August. The largest maxima correspond to sunrise dates February 12 and October 30 (day-of-year numbers 42 and 302), with very similar signiﬁcances. It may be noted that the orientation group contributing to these maxima is identical to Group 5 in Table 1. If we use the difference due to the use of different kernels as a proxy for the error on the maximum, we may estimate this error as 0.11 for the normalized frequency presented in Figs. 5 and 6. For several Mesoamerican regions it has been argued that the dates recorded by solar orientations tend to be separated by multiples of 13 and 20 days, elementary periods in the Mesoamerican calendrical system (Šprajc, 2001; Sánchez Nava and Šprajc, 2015; Šprajc and Sánchez Nava, 2015). To assess the validity of this hypothesis for the Maya Lowlands in statistical terms, we will now analyze the distribution of intervals delimited by sunrise and sunset dates, with the attempt to determine the directionalities of the most prominent groups of solar orientations. Let us recall that any solar orientation (except a solstitial one) matches two sunrise and two sunset dates, and each pair of dates divides the year into two complementary intervals whose sum is equal to the length of the tropical year, but not necessarily both pairs of dates and intervals were achieved on purpose. If the dates to be recorded by a group of orientations were chosen on the basis of certain intervals, we can expect that a higher signiﬁcance of intervals separating the dates marked on one horizon will suggest the direction in which the corresponding orientation group was observationally functional. Fig. 6, top panel, presents the distribution of intervals in days delimited by sunrise dates. There are three signiﬁcant pairs of maxima above the 3σ level, corresponding to intervals of 104.58, 124.72 and 130.59 days and their complements of 260.66, 240.52 and 234.65, with values of the local maxima of 4.2, 3.55 and 3.2, respectively. If we

ﬁt a Gaussian distribution to the maximum we get a handle on the dispersion of each maximum, obtaining values of 2, 3 and 1.5 for these three peaks. To do this we deﬁne the region of interest, the interval of values to be ﬁtted, and then perform a least square ﬁtting to a Gaussian with a given maximum. As a by-product of such ﬁtting we obtain the Gaussian dispersion. Fig. 6, bottom panel, presents the distribution of intervals delimited by the sunset dates, with maxima at complementary intervals of 112.77/252.47, 133.55/231.69 and 139.38/225.86 days and with amplitudes of 3.8, 3.65 and 3.14, their Gaussian dispersions being of 2.2, 2.7 and 1.6 days, respectively. Now, since each orientation group corresponds to one concentration of intervals on the eastern horizon (east interval) and another one on the western horizon (west interval), the east interval peak of 104.58 should be compared with the western peak at 112.77 days and, likewise, the eastern maxima at 124.72 and 130.59 days with the western peaks at 133.55 and 139.38 days, respectively. This information is summarized in Table 2. For Group 1, the east maximum at 104.58/260.66 days has a larger statistical signiﬁcance (4.2) than the west maximum (3.8) and the difference (0.4) is larger than three times the estimated error of 0.11 introduced by the use of different kernels, while the Gaussian dispersion of both is similar (2 vs. 2.2) although marginally larger for the west maximum. These indicators suggest that this orientation group was functional to the east, recording sunrises on February 12 and October 30, whose signiﬁcance has resulted already from the analysis of dates (above). In the case of Group 2 (sunrises around February 22 and October 20 and sunsets around April 16 and August 28), the signiﬁcances of east and west maxima - considering the estimated error - are similar, however the dispersion is larger for the east than for the west maximum, perhaps pointing to a western directionality of this group, although nothing conclusive can be said at this stage. In Group 3 the difference in signiﬁcances is minimal and the astronomically functional direction cannot be proposed.

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Table 2 Values and signiﬁcances of the peaks found after computing the interval of days for three prominent groups of solar orientations. Group 1 East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersión (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

104.58 260.66 Group 2

4.2

2

112.77 252.47

3.8

2.2

East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersión (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

124.72 240.52 Group 3

3.55

3

133.55 231.69

3.65

2.7

East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

130.59 234.65

3.2

1.5

139.38 225.86

3.14

1.6

It should be noted that, in some of the maxima found in Fig. 6, different (but similar) alignment groups might be blended into a single group, giving perhaps spurious results. In order to investigate this effect, we have performed a new comparison between east and west intervals. To avoid merging of close neighboring concentrations we decided to use a constant error for each interval, considering a constant bandwidth for the Gaussian curvigram of 1.4 days, as a proxy to the mean error in days. Admittedly, by introducing a uniform and arbitrary error, we do not consider the fact that some buildings are much better preserved than others and that, therefore, some azimuths determined by ﬁeld measurements are closer to the originally intended values than others. These variations are reﬂected in different errors assigned to individual azimuths by Sánchez Nava and Šprajc (2015) (Table 1) and considered hitherto in our analyses. However, it is quite possible that the errors they estimated are often conservatively large. Thus, our experiment involving a uniform error may allow discrimination of some neighboring alignment groups. Indeed, a few more maxima above the 3-sigma level have been obtained with this method; the results are shown in Fig. 7 and Table 3. Groups 1 and 2 in Table 3 are identical to those in Table 2, but the values are slightly different. Eastern directionality is now suggested not only for Group 1 but also for Group 2, which corresponds to sunrises on February 22 and October 20. In fact, this group is only one of two slightly different orientation groups, which are merged in Group 2 in Table 2; the second one is now (Fig. 7) represented by the east maximum of 120.52/244.72 days, which is slightly below the 3-sigma level, but the fact that there is no corresponding pair of maxima among the west intervals (expected to be around 130/235 days) indicates their greater dispersion and, therefore, suggests eastern directionality of this group. Group 3 in Table 2 has no counterpart in Table 3 (no maxima above the 3-sigma level have been obtained), but three others appear: Group 4 corresponds to the solstitial dates; Group 5 involves orientations to sunrises around March 1 and October 12 and to sunsets around April 8 and September 5; and Group 6 includes intervals delimited by the so-called quarter-days (March 23 and September 21, ±1 day), i.e. the dates which, together with the solstices, divide the year into four approximately equal parts. These results allow us to identify the most prominent orientation groups and to suggest their directionalities. Given the statistical signiﬁcances discussed above, the orientation groups with azimuths of about 100°/280°, 104°/284° and 115°/295° were functional to the east (declinations around −13°, −10° and −23.5°; Groups 5, 6 and 2 in Table 1). However, western directionality is indicated for orientations with azimuths around 97°/277° and 91°/271° (declinations near 7° and 1°); the latter group corresponds to Group 6 in Table 1, also with clearly more signiﬁcant western maximum.

4. Regional patterns and cluster analysis We have used the orientation data as input for the Cluster Analysis in order to see how different structures group according to this criterion. As input data for the ﬁrst analysis of this kind, we used the east azimuth and the east and west declination for each structure, with a coding for its construction period. These data are given in Sánchez Nava and Šprajc (2015). The dendrogram ﬁgure including the 271 structures renders unreadable and thus we provide in Fig. 8, top left panel, a graphical representation of such grouping extracted from the distances obtained from the Cluster Analysis and with a color coding. The top-left inset shows a legend for each of the colours indicating the cluster they belong to and

Fig. 7. Statistical signiﬁcance of intervals separating the pairs of sunrise (top) and sunset (bottom) dates, based on the uniform bandwidth of 1.4 days. For details see text.

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198

Table 3 Values and signiﬁcances of interval peaks for the most prominent orientation groups derived from a uniform bandwidth of 1.4 days. Group 1 East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersión (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

105.10 260.15 Group 2

7.65

2.0

113.30 251.93

7.02

2.2

East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersión (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

125.30 239.93 Group 4

6.79

2.5

134.20 231.05

6.24

2.5

East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

0.025 365.22 Group 5

3.01

0.5

0.025 365.22

2.48

1

East intervals

West intervals

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

140.48 224.75 Group 6

1.61

2

150.05 215.20

3.59

0.8

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

Peak (days)

Signiﬁcance

Gaussian dispersion (days)

172.53 192.73

−0.3

1

182.62

3.9

0.5

East intervals

West intervals

which can be closely related to the east and west declination values indicated in the parenthesis. This ﬁgure indicates that there is no clear pattern with geographic areas, however some exceptions should be noted. The area of the Northeast Coast, including Cozumel Island, presents a fair number of orientations corresponding to declinations beyond the solar extremes, a similar situation to that found in one part of Southern Lowlands. Conversely, the central area displays a pre-eminence of orientations towards declinations of (− 13°, 13°), being the clearest regional pattern for the Maya area. Fig. 8, top right and bottom panels, displays the data divided in three temporal intervals, roughly consistent with the customary divisions in Preclassic (top right), Classic (bottom left) and Postclassic (bottom right). In Preclassic times the vast majority of orientations are inside the solar range, to within declinations (− 1°, 1°) to (− 13°, 13°), with only a handful of orientations outside these ranges. The orientation is highly variable in Classical times in all regions. A greater coherence appears for the East Coast and especially the Usumacinta area, with a fair number of orientations possibly towards the extreme positions of the moon and Venus. Finally, in Postclassic times such internal coherence seems to be reduced to the East Coast (signiﬁcantly Cozumel), while the rest of the area present orientations within the solar range. In order to test and reﬁne these results we built seven regional groups corresponding to some cultural zones characterized by certain archaeological features. We have now used the results exposed above concerning the directionality of orientations; for orientations that do not belong to the groups that have been deﬁned, the higher declination peak value has been taken as relevant for determining a unique directionality, either to the east or west. Based on these data, we have built a curvigram for each of the regions. Then we extracted a number of ‘genetic’ markers, following those tested and used by González-García and Belmonte (2010). The approach followed here is similar to that developed for Sardinia using similar techniques (González-García et al., 2014). These markers were mentioned in the Introduction and are presented in Table 4. They are based on our orientation data and include the ﬁve common statistical

quantities plus the principal and secondary maxima of the declination distribution for each region. The results were used to derive the dendrogram and map presented in Fig. 9. The dendrogram helps us identify three broad areas with differing orientation trends. A ﬁrst group is that of the Usumacinta basin (U) and the East Coast (EC). As indicated above, both regions present a considerable number of orientations outside the solar range and a number of them compatible with the extreme positions of the Moon and Venus. This could be hinted upon by the differences in the overall values with respect to the other groups and the existence in both groups of a secondary curvigram maximum (DM2) centred at declination −28°. The other two areas are composed of the Northern Lowlands (NL) and Puuc region (P), on the one hand, and the Central Lowlands and the Chenes (C) region, each of the two areas being characterized by a wide internal coherence in orientations, while the Río Bec (RB) region exhibits a slightly different trend. It must be stressed that these results depend upon the results encountered in the ﬁrst part of this paper. We have carried out a number of tests to verify the coherence of these results, considering a number of possible directionalities. Basically, we have considered the different possibilities for those cases where in the previous section we were not sure if the directionality was east or west. There are a number of points to be mentioned here: The association between the orientations of the East Coast and the Usimacinta groups seems robust, as in all cases the two appear close together in the dendrogram analysis. Similarly, the NL + P and CL + C clusters also seem close together, although given the low number of structures in the Chenes group, the variability is larger and in certain cases the association to other groups appears. The RB group also presents a larger degree of variability due to low number statistics that render the results uncertain in all cases. 5. Discussion Several conclusions concerning the intentionality and signiﬁcance of architectural orientations in the Maya Lowlands can be derived from the results of statistical analyses exposed above.

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Fig. 8. Distribution maps for 271 structures measured in the Maya area, showing the orientation ranges expressed by east and west declinations in a color code (see labels in each frame). Top left panel includes all data, while the other three present them separately for the Preclassic, Classic and Postclassic periods.

As mentioned in the Introduction, the alignment data analyzed correspond to the buildings with evidently civic and ceremonial functions. However, since we have no contextual indications as to what type of buildings were, indeed, oriented astronomically, some “noise” in our data sample is highly likely, attributable to the structures that, in fact, were not aligned to celestial phenomena. Nonetheless, the distributions of azimuths and declinations, differing from those resulting from a uniform azimuthal distribution of orientations, as well as the fact that the distribution of declinations presents maxima with greater signiﬁcance than those in the distribution of azimuths (Figs. 1 and 2, Table 1), indicate a predominant use of astronomical targets on the horizon.

Table 4 ‘Genetic markers’ for the seven cultural areas (see González-García and Belmonte, 2010 for a deﬁnition). The ﬁrst and second columns indicate the group name, and the number of structures considered in each of them. The next seven columns indicate the seven markers. The ﬁrst ﬁve quantities are the standard statistical quantities deﬁning a population; the last two give the location of the principal and the second maxima of the histogram for each sub-sample. For details, see text. Median δ (°)

St. dev δ (°)

Max δ (°)

Min δ (°)

D M1 (°)

51 1.16

2.01

13.1

20.05

−28.4

−11.25 18

88 −3.17

−9.71

11.57

20.10

−25.80 −12.0

7.0

−23.08 −23.54 −9.29 −11.10 −9.44

26.17 18.41 8.41 13.83 16.93

38.42 23.84 8.32 16.54 20.53

−44.1 −41.0 −14.4 −26.37 −28.81

−28.0 −28.0 9.25 7.0 19.0

# Northern Lowlands Central Lowlands Usumacinta East Coast Rio Bec Chenes Puuc

26 61 16 7 22

Mean δ (°)

−10.53 −18.37 −4.24 −7.69 −3.83

−23.75 −36.0 −10.0 −15.0 −13.75

D M2 (°)

Furthermore, the concentrations in the distribution of east-west azimuths and the corresponding declinations are more pronounced and exhibit higher signiﬁcances than those observed in the distribution of north-south azimuths and the corresponding declinations, suggesting that the orientations were functional predominantly in the east-west direction. As shown elsewhere (Sánchez Nava and Šprajc, 2015: 51ff), the placement of the access or main façade does not necessarily indicate the astronomically functional direction of the orientation; this conclusion, valid also for prehispanic architecture in central Mexico (Šprajc 2001: 69ff), is supported by numerous structures facing north or south, but whose orientations belong to the widespread groups, evidently functional in the eastern or western direction. A few declination maxima listed in Table 1 are statistically signiﬁcant, suggesting probable astronomical targets. While the maxima in column 1 most likely refer to major lunar standstills (and possibly to major northerly Venus extremes), others are likely associated with the Sun. The directionality of the latter is suggested by differences in significance of east and west declination maxima; however, for reasons exposed in the text, more reliable results were obtained by analyzing the distribution of the corresponding sunrises and sunset dates. Indeed, in comparison with the declination maxima, the distribution of dates exhibits maxima with higher signiﬁcances, supporting the idea that speciﬁc dates were targeted by solar orientations. As contended elsewhere (Sánchez Nava and Šprajc, 2015), the concentrations of dates in four periods of the year can be explained in terms of their relation with four agriculturally important seasons, particularly in the maize cultivation cycle (preparation of cultivation plots around February, beginning of the rainy season and planting of maize in April–May, the ﬁrst fruits in August and the harvest season in October–November). It

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Fig. 9. Left, dendrogram for the seven regional groups described in the text placed according to a given statistical distance. Right, map showing the results of the dendrogram analysis with a color code following those results.

has also been argued that the solar orientations tend to mark dates separated by multiples of 13 and 20 days, i.e. of elementary periods of the Mesoamerican calendrical system, thereby allowing the use of easily manageable observational calendars intended to facilitate a proper scheduling of agricultural activities and the corresponding ceremonies. Particularly important for these purposes must have been the 260-day calendrical count, in which the cycles of 13 and 20 days were intermeshing, so that every date had a name composed of a number from 1 to 13 and a sign in the series of 20. Given the structure of this calendrical cycle, the sunrises and sunsets separated by 13-day intervals and their multiples occurred on the dates with the same numeral, while the events separated by periods of 20 days and their multiples fell on the dates having the same sign. If the orientations marked agriculturally important dates separated by such intervals, they not only allowed determination of the critical moments by means of direct observations; with the use of observational schemes composed of elementary periods of the formal calendrical system, it was relatively easy to anticipate the relevant dates (which was important because cloudy weather may have impeded direct observations on these dates), knowing the structure of a particular observational calendar and the mechanics of the formal one.2 It is worth pointing out that, according to a relatively common opinion, modern farmers know when to plant and harvest by observing different changes in the nature, and thus have no need to determine relevant dates by means of astronomical observations. In fact, multiple ethnographic data contradict such afﬁrmations, demonstrating that seasonal changes in natural environment are not sufﬁciently exact and reliable indicators of the most convenient moments for agricultural tasks and related ceremonies. Several present-day indigenous communities still rely on astronomical observations for determining canonical dates appropriate for carrying out agricultural rituals, whose objective is to ensure a proper sequence of seasonal changes and an abundant harvest. Signiﬁcantly, the most important of these dates exhibit a close correspondence with those most frequently recorded by orientations. While some of the dates that continue to be ritually important have no apparent relation to Christianity and are thus obviously of prehispanic origin, in most cases they correspond to Christian feasts, but the contents of ceremonies have evidently prehispanic roots; it is thus clear that the prehispanic rituals performed at key moments of the agricultural cycle were incorporated into the approximately coincident Catholic feasts. Moreover, not only is the signiﬁcance of computing time by calendrically signiﬁcant intervals – sometimes explicitly related 2 The need for astronomical observations is understandable, considering that the Mesoamerican calendrical year of 365 days, due to the lack of intercalations, did not maintain a perpetual concordance with the tropical year of 365.2422 days (for the whole argument and the corresponding bibliography, see Šprajc, 2000b, 2001: 135ff).

to agriculture – attested in prehispanic codices; the importance of multiples of 13 and 20 days in scheduling agricultural tasks and rituals has been ethnographically reported among several indigenous communities, which also preserve fragments of the ancient calendrical system. While the whole discussion and the corresponding bibliography has been presented by Sánchez Nava and Šprajc (2015: 89ff), their interpretations can now be additionally supported by the results of statistical analyses. The analysis of the distribution of intervals, taking into account errors in days calculated on the basis of estimated errors in azimuths and derived declinations, resulted in three statistically signiﬁcant maxima (Fig. 6, Table 2). The most persuasive is undoubtedly the maximum on the complementary east intervals of 104.58 and 260.66 days. Its signiﬁcance, notably larger than that of the west interval maximum corresponding to the same orientation group, suggests that this group was intended to record sunrises on February 12 and October 30, separated by 260 days. That this interval, corresponding to the largest orientations group, ﬁgures most prominently agrees with both its equivalence to the length of the 260-day calendrical count (which means that the phenomena separated by this interval, a multiple of both 13 and 20 days, fell on the same date of this cycle) and the fact that the two dates can be interpreted as canonical or ritually important dates delimiting the agricultural cycle. Similar dates continue to have such a function in a number of modern communities, marking the moments for performing rituals with notably agricultural signiﬁcance (see above). It thus seems reasonable to suppose that other prominent groups of solar orientations were also intended to record dates separated by multiples of 13 and 20 days, i.e. of constituent periods of the 260-day calendrical cycle, allowing the use of different versions of observational calendars. The existence of multiple orientation groups matching a number of different dates can be accounted for by the fact that, while some of the dates included in observational schemes marked canonically important moments of the agricultural cycle, others must have had an auxiliary function, facilitating the prediction of the dates most signiﬁcant in practical and ritual terms. Indeed, as shown by Sánchez Nava and Šprajc (2015), at least one of the east and west intervals corresponding to each of the orientation groups they identiﬁed closely corresponds to a multiple of either 20 or 13 days. The differing signiﬁcances of the east and west interval maxima of Group 2 in Table 2 (124.72/240.52 and 133.55/231.69 days) suggest the western directionality of this orientation group, although in this case it is the east maximum that closely corresponds to the signiﬁcant interval of 240 (=12 × 20) days, which delimits sunrises on February 22 and October 20. We have shown, however, that this maximum is actually produced by a mixture of two slightly different orientation groups, which we have been able to single out by analyzing the interval distribution with a uniform error arbitrarily reduced to 1.4 days. The

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maxima corresponding to the ﬁrst (and larger) group, marking the dates mentioned above, have now turned to be more signiﬁcant than those resulting from the former analysis, and the eastern directionality of this group is clearly indicated both by the higher signiﬁcance of the eastern pair of intervals (125.30/239.93) and by the fact that the presumably relevant one (239.93) exhibits a closer correspondence to the calendrically signiﬁcant interval (240) than the one (240.52) derived from the former analysis (cf. Fig. 7 and data for Group 2 in Tables 2 and 3). The other group, with a smaller number of orientations, tends to record sunrises on February 19 and October 22, separated by an interval of 120 (= 6 × 20) days; while the east maximum of this group (120.52/244.72 days) is below the 3-sigma level (Fig. 7: top), the fact that no corresponding maximum appears among the west intervals (expected to be around 130/235 days) indicates their greater dispersion and, therefore, the eastern directionality of this orientation group. For Group 3 in Table 2, the directionality cannot be ascertained on statistical grounds, probably because, again, two orientation groups targeting different events seem to be blended, one recording the interval of 130 (=10 × 13) days on the eastern horizon (from October 17 to February 24) and the other 140 (=7 × 20) days on the western horizon (from April 13 to August 31). Neither of the two groups appears in Table 3, resulting from the second analysis, apparently because, due to their different (but similar) astronomical referents and the reduced interval error, they are dissolved to a rather indistinguishable continuum. On the other hand, the second analysis suggests the existence of three additional statistically signiﬁcant groups (Table 3). Group 4 corresponds to solstitial orientations, whose signiﬁcance has been revealed already by the analysis of declinations (Table 1: column 2). To judge by the differing signiﬁcances of east and west interval maxima, as well as by the fact that the east declination peak is closer to the Sun's solstitial declination than the west one (Table 1), most orientations of this group recorded the winter solstice sunrise. For Group 5 the western directionality is indicated, although the corresponding orientations tend to record a signiﬁcant interval (140 days) on the eastern horizon. While the possibility cannot be discarded that the higher signiﬁcance of the west interval maximum reﬂects intentionality, meaning that, for some presently unknown reason, this group of orientations was functional to the west (recording dates around April 8 and September 5), it is also possible that this result is, again, a consequence of a fusion of different orientation groups, intended to mark intervals of either 140 or 220 days (cf. Sánchez Nava and Šprajc, 2015: Table 1). Finally, Group 6 was evidently functional to the west, recording sunsets on quarterdays of the year (March 23 and September 21), with an intervening interval of 182 (=14 × 13) days. The western directionality of this group has also been clearly indicated by the analysis of declinations (Table 1: column 7). As revealed by cluster analyses (Figs. 8 and 9), the orientation patterns in the study area exhibit rather few regional and temporal variations. While relatively close similarities can be observed among the Northern Lowlands, Puuc, Chenes, Central Lowlands and Río Bec regions, the latter seems to manifest a more notable difference in comparison with the remaining four regions, perhaps reﬂecting speciﬁc Late Classic developments materialized in a number of cultural features, including the Río Bec architectural style, which is a deﬁning characteristic of this region (cf. Potter, 1977; Gendrop, 1983). Probably the most interesting result of these analyses is the fact that the Usumacinta and East Coast regions share similar orientation trends, which are, however, notably different from those observed in the rest of the Maya Lowlands. The general similarities between the two areas must be somehow related with the coastal trade network, which became increasingly important with the decline of the Classic Maya culture in the central and southern Lowlands during the 9th and 10th centuries A.D., and in which the merchant groups originating from the Gulf Coast region had a paramount role. As a consequence, the Postclassic period on the East Coast is characterized by a mixture of cultural elements from different Mesoamerican regions (cf. Miller, 1982: 64ff).

201

However, since no close relations with the Usumacinta region, speciﬁcally, seem to be evident in the archaeological record, we have no clues as to the immediate causes underlying the resemblances in orientation practices in the two distant areas. While only further research may reveal the details and extent of these similarities, the two zones are evidently characterized by a relatively large number of orientations that cannot relate to the Sun, including those corresponding to the major lunar standstills. Signiﬁcantly, most of the latter are concentrated along the East Coast and are largely Postclassic; this is in agreement with independent evidence attesting to the importance of lunar cult in this area, which was particularly popular during the Postclassic period, but must have had earlier origins (cf. Freidel and Sabloff, 1984; Milbrath, 1999: 147f; Sánchez Nava and Šprajc, 2015). On the other hand, comparing artiﬁcial cephalic forms in different sites of the Maya Lowlands, Tiesler and Cucina (2012) conclude that the types of cranial deformation in the Usumacinta basin differ from those predominating in the Petén and Northern Lowlands, and add that these differences might express deeper cultural divergences, perhaps related to linguistic differences and distinctive ideological schemes; they mention other data supporting this idea (Tiesler and Cucina, 2012: 116). 6. Conclusions In the light of the results of statistical analyses discussed above, it is highly likely that a considerable number of important buildings in the Maya Lowlands was oriented to the major lunar standstill positions and/or Venus extremes on the horizon, and that most of the solar orientations, which predominate in the sample, recorded dates separated by multiples of 13 and 20 days. Such a generalization is plausible particularly because the two most prominent groups identiﬁed as statistically signiﬁcant by our analyses are widely distributed not only in the Maya area but also in central Mexico (Šprajc, 2001), and considering that the most widespread of the two records the most signiﬁcant interval, multiple of both 13 and 20 days and equivalent to the length of the sacred 260-day calendrical cycle. Another supporting element is that an interval close to a multiple of 13 or 20 days is found in each of the groups we have identiﬁed as statistically signiﬁcant. In view of the coincidence of the four major concentrations of dates with the agriculturally important seasons of the year, we can also conclude that the orientations enabled the use of easily manageable observational calendars intended to mark the time of the agricultural activities and the accompanying rituals. Notwithstanding the emphases of this discussion, it should not be overlooked that the buildings marking certain celestial events did not serve as observatories, in the modern sense of the word, or as devices serving practical needs only, because their functions were primarily religious, residential or administrative. An important aspect of orientations was their symbolic signiﬁcance. Since the astronomically relevant directions are most consistently incorporated in monumental architecture of civic and ceremonial urban cores, evidently commissioned by the governing class, it is clear that both practical uses of astronomical knowledge and broader cosmological beliefs formed a very important part of the ideology of power. While the cultural signiﬁcance of architectural orientations and their astronomical underpinnings is obviously of primary importance, it is worth stressing that, instead of exploring these aspects, extensively discussed by Sánchez Nava and Šprajc (2015), our objective has been to provide an independent statistical test, whose results, we believe, lends a considerable support to their interpretations. Acknowledgements ACGG is a Ramón y Cajal Fellow of the Spanish MINECO. The research project resulting in the alignment data analyzed in this contribution was ﬁnanced and authorized by the Instituto Nacional de Antropología e

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