arxiv:2110.12329v1 [cs.si] 19 oct 2021

THE NETWORK SIGNATURE OF CONSTELLATION LINE FIGURES

A PREPRINT

Doina [email protected]

University of Twente, The Netherlands

ABSTRACT

In traditional astronomies across the world, groups of stars in the night sky were linked into constellations—symbolic representations on the celestial sphere, rich in meaning and with practical roles. In cultures whereline or connect-the-dot figures were documented, these visual representations are constrained to the fixed back-ground of stars, but are free in their choice of stars and lines to draw. Over a dataset of 1591 constellationline figures from 50 astronomical cultures, we define metrics to measure the visual signature (or complexity)of a constellation, and answer two questions: (1) does the type of culture associate with the visual signatureof constellations? 2) does the sky region associate with the visual signature of constellations? We find that(1) individual cultures are only rarely and weakly thus associated, but the type of culture (by practical use,level of development, and ancestry) show an association. We find clear clusters of cross-culture and cross-typesimilarity in visual signatures, with SE Asian traditions far apart from Mesopotamian, N and S American, Aus-tronesian and Polynesian traditions, which are similar. We also find (2) more diversity of constellation signatureper sky region than expected, with diverse designs around the majority of popular stars.

1 Introduction

For sky watchers through time, the night sky was a canvas to be filled with symbols. They designed constellationsas groups of stars; the constellation figures were named, and assigned a practical utility or a background story. Theyform a visual communication system in some ways similar to characters in a written script [45]; the constellationfigures are more or less complex in form, may or may not resemble the animal, human figure, or object that they werenamed after, and may or may not have been drawn similarly in unrelated cultures [19]. Constellations are now usuallyrepresented as line figures, in which the stars are connected by imaginary lines. There is great diversity among theshapes, sizes, and internal complexity of the line figures across cultures—Fig. 1 shows traditional Chinese [51, 79] andancient Babylonian constellations [26, 25, 27, 79] for the same Southern sky region: where the Chinese drew abstract,short and twisted chains, the Babylonians filled large surfaces with polygons and realistic human or animal figures.

We study 50 sky cultures across the world (located geographically in Fig. 2), with line figures recorded in the literature.For some cultures (such as the Chinese and their area of influence), the line-figure style of representation dates to thebeginning of their astronomical records [51]. For others (such as the Western sky culture and its ancestors), the linefigures are the result of an evolutionary process which started from allegorical pictographs [57, 58] and led to nowglobally recognised figures [56, 28]. For yet other cultures (such as those native to the Americas), the line figures arerecent interpretations for star groups [42, 43, 44, 36, 37, 10, 61].

We define and measure the visual signature of constellation line figures. This signature is a combination of manydimensions: the size of the line figure, the internal structure of the line figure, as well as the brightness of the stars inthe group, all contribute to the visual signature of the constellation. We delineate distinct classes of visual signature,and also internal gradients within a class. We then assess whether the constellation shapes are unique to a sky culture(are giant figures characteristic only to the Babylonians, and abstract chains only to the Chinese?), or to a type ofculture (for example, all cultures with Mesopotamian ancestry, or all cultures where constellations were used foropen-water navigation). All are potential factors to have driven the shape of constellations. The same visual signaturemay also simply be universal in some parts of the sky, so driven by the pattern of stars, rather than by the backgroundof the sky watchers. We investigate these possibilities in a unified way across the cultures. Previous studies lookedat the cognitive basis of forming star clusters and modelling human performance at clustering tasks ([17] for 30

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The network signature of constellation line figures A PREPRINT

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Figure 1: The diversity of constellations line figures. Traditional Chinese constellations (top) and ancient Babylonian (bottom)constellations for the southern sky, visible declinations between 20◦ and the south pole, and right ascensions [90◦, 270◦]. The size,internal structure, and choice of stars to include differs greatly. The names of the brightest stars are marked; some constellationnames are removed for clarity.

constellations, and [30] for asterisms in 27 cultures), but no studies quantify the visual complexity of constellations,nor looked at the quantitative relationship between culture type and visual complexity, on the basis of which culturalsimilarities in terms of visual complexity can be measured.

We first settle the technical terms in the vocabulary, then draw the hypotheses.

Sky culture denotes the astronomical traditions of a society at a point in time. A sky culture varies in how largethe society behind these traditions was (data from one Native American tribe is treated on par as data fromImperial China), when these traditions were documented (from before 0 AD to the 21st century), who theauthor of the constellations was (the society as a collective, or, rarely, a specific individual), and how rich therecords are in line figures (from one to a few hundred).

Constellation or asterism denotes here the popular meaning: a group of stars joined into a figure, rather than theInternational Astronomical Union (IAU) definition [28], namely a bounded region of the sky. The differencebetween constellations and asterisms is the tendency for asterisms to be small in size.

Line figure denotes the dot-and-line representation of constellations, as opposed to the pictograph popular in Westerncultures before the 19th century. Other ways to refer to this in literature include connect-the-dots or stickfigures. Line figures are spatial graphs. The stars are the nodes, placed at specific spherical coordinates onthe sky, and labelled with their magnitude. The links are arcs (or geodesics) between pairs of stars. Thespatial graphs may or may not be planar, or even connected (although the vast majority are).

Visual signature denotes a set of measurable, quantitative features (or statistics) of a line figure. These featuresinclude network features, which measure the structural complexity of the figure (the number of nodes, thesize of components and cycles, measures of node degrees, the connectivity of the spatial graph, and others),spatial features (the diameter of the figure, the length of its edges, the sharpness of the angles, and whether itis spatially planar), and brightness features (statistics over the star magnitudes). We define 19 such features.

We draw two hypotheses.

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Anutan(1998)

11 constell.

Al-Sufi(964)

51 constell.

Arabic moon st.(9th century)

21 ast.

Aztec(16th century)

4 constell.

Babylonian(~1100-700 BC)

50 constell.

Belarusian(19th-21st century)10 constell.Blackfoot

(20th century)4 constell.

Bugis(present)

12 constell.

Chinese medieval(1092 AD)245 ast.

Chinese(1756-1950)

253 ast.Egyptian

(1470 BC and 50 BC)26 constell.

Hawaiian(present)

13 constell.

IAU(105 AD to 20th century)

86 constell.

Inca(1613)

8 constell.

Indian moon st.(before 500 BC)

21 ast.

Inuit(20th century)9 constell.

Japanese moon st.(8th century)

27 ast.

Javanese(19th century)

3 constell.

Kari'na(1980)

8 constell.

Korean(1395)210 ast.

Koyukon(20th century)

1 constell

Lokono(present)10 constell.

Macedonian(present)

16 constell.

Madura(present)

6 constell.Mandar(present)6 constell.

Maori(19th century)

4 constell.

Maricopa(20th century)

4 constell.Maya(15th century)14 constell.

Mi'kmaq(late 19th century)

4 constell.Mongolian

(present)4 constell.

Navajo(20th century)

5 constell.

Norse(13th century)

6 constell.

Ojibwe(present)9 constell.

Pawnee(20th century)11 constell.

Quechua(20th century)

4 constell.

Rey(1952)

80 constell.

Romanian(1907)38 constell.

Ruelle(1786)

70 constell.

Russian(present)4 constell.

Sami(19th century)

3 constell.

Sardinian(present)

11 constell.

Sioux(present)

13 constell.

Tongan(late 19th century)11 constell.

Tukano(1905/2007)20 constell.

Tupi(1614)

8 constell.

Tutchone(20th century)

1 constell

Vanuatu(present)

6 constell.

Western(present)

88 constell.

Western asterisms(present)53 ast.

Zuni(20th century)

9 constell.

Globalsky

cultures:

Figure 2: The location of sky cultures. The 50 cultures are shown with: name, the date of documentation, and the number ofconstellations (or asterisms) with at least one line. Sky cultures with global reach are highlighted at the bottom.

[Hypothesis I] The type of sky culture associates with the visual signature of constellations

The type of a sky culture may relate with, and possible have determined, how constellations look in that culture, theway the complexity of characters in a written script may have been determined by the type of script [45]. Here, weuse mostly associative language (“associates with”) rather than strong causal language (“determines” or “influences”),because a causal link, even when likely, cannot be proven. For this hypothesis we answer four questions:

I.1 The culture itself: Does each culture associate with the visual signature of its constellations? In other words, areits constellations not only homogeneous visually, but also distinct from those of other cultures, as the Chineseconstellations are when compared to the Babylonian?

I.2 The type of astronomy: Do written (as opposed to oral) astronomical traditions associate with the visual signatureof constellations from these traditions?

I.3 The practical use of constellations: Do constellations used as markers for open-sea navigation have a differentvisual signature than those used for political or religious astrology, or for time-keeping in agrarian and hunter-gatherer societies?

I.4 The phylogeny of the culture: Is a common ancestry of cultures associated with the visual signature of constella-tions from that ancestry?

[Hypothesis II] The region of the sky associates with the visual signature of constellations across cultures

Multi-cultural constellation data which overlap in the sky make possible a distinction to be made between the back-ground data (for example, the star pattern around the celestial north pole) and the foreground data (the line figuresdrawn in that region across cultures). Since the background data is fixed, a complementary hypothesis arises: the skybackground itself may be the significant driver of constellation shapes, overpowering the role of the culture typology.For example, variants of the Big Dipper and Orion asterisms recur across cultures [7], and the set of recurring con-stellations also extends beyond these examples and can be cognitively motivated [30]. Some degree of similarity ismotivated by the fixed set of stars in a sky region, but does some diversity remain in the network or spatial features ofthe constellations? We thus answer the question:

II The sky region: Is there recurrent universality in the visual signature of constellations per region of the sky, or, onthe contrary, diversity?

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2 Data: inclusion and limitations

For an overview of the sky cultures in this study, see Table 1. More information on these cultures is provided laterin Sec. 6.1 and 6.2. The most difficult aspect of this research was collecting and verifying existing constellation datafrom a multitude of ethnographic and other sources, and categorising both cultures and constellations by type. Weshare the data not already publicly available at https://osf.io/9gb5a/.

Table 1: Summary of sky cultures. id. provides an identifier, for an. to show ancestry. Under type, astronomical knowledgetransmission is marked as written (w) or oral (o). The (main) practical use(s) (at the time of origin) is for: navigation (nv), religious(re) or political (po) divination, or folk/agrarian/hunter-gatherer time-keeping on land (fo). The count includes constellations withat least one line. References under ref. recover the line-figure data.

location id. an. sky culture timestamp source type count ref.Global I G IAU 105 AD–20th c. standard w re,nv 86 [28, 79]Global I Rey 1952 book w re,nv 80 [56, 79]Global I Western present dataset w re,nv 88 [79]Global I Western asterisms present dataset w re,nv 53 [79]

N Africa Egyptian 1470, 50 BC carving,paper w re 26 [39, 79]W Asia M Babylonian 1100-700 BC tablet,papers w re 50 [26, 25, 27, 79]W Asia In Arabic moon st. 9th c. book,paper w re 21 [35, 79]W Asia G Al-Sufi 964 AD book,dataset w re,nv 51 [2, 79]S Asia In Indian moon st. < 500 BC book,dataset w re 21 [6, 79]S Asia A Bugis present papers o nv 12 [54, 48]S Asia A Javanese 19th c. book,paper o fo 3 [74, 31]S Asia A Madura present paper o nv 6 [20]S Asia A Mandar present paper o nv 6 [54]E Asia C Chinese medieval 1092 AD chart,book w po 245 [51, 79]E Asia C Chinese 1756-1950 chart,book w po 253 [51, 79]E Asia C Korean 1395 chart,dataset w po 210 [79]E Asia C Japanese moon st. 8th c. chart,paper w po 27 [55, 79]E Asia I Mongolian present dataset o fo 4 [79]

Eurasia I Russian present book o fo 4 [66]W Europe I Ruelle 1786 chart w re,nv 70 [60]E Europe I Belarusian 19th-21st c. paper o fo 10 [5, 79]E Europe I Romanian 1907 book,exhibition o fo 38 [50, 69, 79]S Europe I Macedonian present paper o fo 16 [11, 79]S Europe I Sardinian present dataset o fo 11 [79]N Europe I Norse 13th c. verse,book,dataset o nv 6 [29, 79]N Europe Sami 19th c. book o fo 3 [40, 79]

N America nA Maya 15th c. codex,books w re 14 [23, 76, 79]N America nA Aztec 16th c. codices,book w re 4 [14, 15, 4, 79]N America nA Inuit 20th c. book,dataset o fo 9 [41, 79]N America nA Koyukon 20th c. book o fo 1 [44]N America nA Tutchone 20th c. book o fo 1 [44]N America nA Mi'kmaq late 19th c. book o fo 4 [44]N America nA Ojibwe present book o fo 9 [37, 79]N America nA Blackfoot 20th c. book o fo 4 [44]N America nA Pawnee 20th c. book o fo 9 [44]N America nA Sioux present books o fo 13 [36, 44, 79]N America nA Maricopa 20th c. book o fo 4 [44]N America nA Navajo 20th c. book o fo 5 [44]N America nA Zuni 20th c. book o fo 9 [44]S America sA Inca 1613 book o fo 8 [38]S America sA Kari'na 1980 paper o fo 8 [43]S America sA Lokono present dataset o fo 10 [61, 79]S America sA Quechua 20th c. paper o fo 4 [70]S America sA Tukano 1905/2007 book/thesis o fo 20 [33, 10, 79]S America sA Tupi 1614 book,papers o fo 6 [13, 42, 1, 79]

Pacific P Anutan 1998 book o nv 11 [21, 79]Pacific P Hawaiian present website,dataset o nv 13 [67, 79]Pacific P Maori 19th c. paper o nv 4 [49, 79]Pacific P Tongan late 19th c. paper o nv 11 [12, 79]Pacific P Vanuatu present website,dataset o fo 6 [53, 79]

Column location in Table 1 roughly locates each sky culture by continent, with occasionally a culture (such as Rus-sian [66]) spanning continents, and the many cultures in SE Asia separated into S and E Asia. Column id. providesa short identifier to some important cultures: IAU (I), Babylonian (M denoting Mesopotamia), Indian (In), Chinese

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https://osf.io/9gb5a/


medieval (C). Other identifiers (not defined in column id.) include the Greek (G), which would comprise constellationsincluded in the standard IAU culture and is partly of Mesopotamian (M) ancestry, leading to indirect M ancestry for itsdescendants. Others denote cultural regions: Austronesian (A), Polynesian (P), North American (nA)—which unitesthe regional sky cultures of N America from Arctic to the south west, the Great Plains [44] and Mesoamerica—andSouth American (sA) uniting the regional Guianan cultures (Kari'na and Lokono), the co-located Peruvian culturesseparated only by time (Inca and Quechua), and the cultures now in Brazil (Tukano and Tupi), since some tribeswere known to have migrated and inter-related across the continent [42]. With these identifiers, column an. marks theancestry or phylogeny of cultures. For a more detailed account of the phylogeny of sky cultures, see the later Sec. 6.1.

For each culture, column type marks two properties of the culture: whether astronomical knowledge transmission waswritten (w) or oral (o), and what the main practical use(s) were (at the time when the constellations were designed,rather than in modern times): for navigation (nv), religious (re) or political (po) divination, or folk/agrarian/hunter-gatherer time-keeping or orientation on land (fo). For details of the typology of sky cultures, see also Sec. 6.1.

Column timestamp provides the earliest date since when a description of constellations is known, column sourcelists the format(s) in which the culture is documented, and count provides the number of constellations or asterismswith at least one line. In some cases, the first source for the sky culture is an ancient artefact (for example, the firstdocumentation of the Japanese moon stations is a tomb painting with line figures). Even then, the first source providesthe constellation names and some account of their location in the sky. Later work was sometimes needed to identifythe stars and lines precisely—the references under ref. in Table 1 point to modern data sources. The references areof many types (short-form publications from field work or recent studies of medieval codices and other artefacts,books summarising constellation information from prior publications, and open-source datasets from the Stellariumastronomy software [79] which includes a repository of sky cultures). The later Sec. 6.2 provides a timeline of line-figure representations for constellations, across cultures.

Not all cultures with documented astronomies were included, and also not all constellations from a culture wereincluded. The criteria for inclusion of line-figure data in the study are:

Constellations have lines From the entirety of a sky culture, we study constellations with at least one line. Thisexcludes constellations which are either single stars (frequent in E-Asian cultures), or tight star clusters suchas the Pleiades, where lines would be invisible to an observer on the ground.

Line figures are described The line figures come from existing sources, and are not a contribution here. Sources inwhich no line figures were explicitly drawn or described are excluded; this is the case for many N-Americantribes [44] other than those included in Table 1, the Tuareg [8], the Rapanui [18], and many others. Often, thesources simply equate native constellations with modern ones in words, which is too imprecise.

Line figures are justified Sources in which the line figures were drawn, but are unjustified, are excluded. This isthe case for some Stellarium [79] data. We exclude unreferenced Stellarium data, with the following excep-tions: Western and Western asterisms (simpler variants of the IAU inherently defined by public perceptionand issued by various popular magazines), Korean (well documented by star charts with line figures), and theMongolian, Sardinian, Hawaiian, and Vanuatu folk cultures, which are the result of unpublished (or infor-mally published on the Web) field work with or by locals. Sometimes, a reference exists, but it only containsthe star identification, pictographs, star groups, or a textual description of its shape; the modern line figureis tenuous, but follows the perimeter of the pictograph and the stars identified, so captures many of the con-stellation features (network size, spatial size, and brightness properties) that we are interested in, so we retainthem. This is the case for: Al-Sufi, Indian/Japanese moon stations, Maya, and Inuit.

Of the 50 sky cultures, 20 come from ethnographic studies not already available in (or supplementing) Stellariumdata. Of these, 3 sky cultures (Sioux, Tukano and Tupi) are a mix of new data from ethnographic studies and a priorStellarium dataset. All were gathered or extended for this research. The Stellarium datasets were verified against theirsources in most cases, and line figures were occasionally corrected or removed if the sources did not justify them. Veryfaint stars unlikely to be seen with the naked eye (magnitude higher than 7.0), very rarely used, were removed and theline figures reconnected without them.

Not every type of analysis can be performed on this ethnographic dataset, due to its inherent limitations:

Approximate star identification A star may be imprecisely identified from rough sky charts, in the neighbourhoodof the star intended. This is especially the case for E Asian (Chinese, Korean, Japanese) sky cultures, for thenon-determinative (or minor) stars in an asterism [64]. This means that the exact identity of a star in a linefigure cannot be counted on.

Unknown culture age The exact age of most sky cultures is unknown. Written records are the basis of documentationof all cultures, but the time of birth for a tradition is lost in long periods without surviving records. This is the

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case for ancient cultures with written records, as for recent cultures with oral traditions. Thus, age cannot beused as a culture feature, and some questions cannot be asked: those related to the evolution of sky culturesin time, or to the design of constellations in special sky regions, such as around the poles (since the locationof the celestial poles drifts in time due to the natural precession of the earth’s axis).

Unknown culture size The original number of constellations cannot be established for oral cultures, for many rea-sons. Part of the oral knowledge was not documented before becoming lost. There may have been taboosabout sharing this information with strangers [44]. Some of the ethnographers surveying a culture did nothave astronomical knowledge, so did not inquire about celestial traditions, or were vague about the identifi-cation of constellations in the sky. In any case, the result is poor ethnographic records for N American [44],S American [61], European [29, 66] and Russian-language folk [66]. This means that conclusions aboutindividual cultures should not be drawn based on their sizes.

Preference or bias towards bright stars In oral cultures, the constellations that were remembered were likely not arandom sample from a larger, forgotten tradition. Instead, the recollection may have been biased towardsthe more salient constellations, such as those with bright stars. This is not certain; in some oral cultures,bright stars have clear preference: in Polynesian seafaring, only they are named and considered major [21].To allow for the existence of bias in recollection, we consider conclusions drawn based on data from oralcultures alone less trustworthy than those drawn from all cultures.

3 Method: measuring, clustering, and comparing visual signatures

To provide answers to the questions posed, the steps followed are: defining and measuring constellation features intoa multi-dimensional, but interpretable visual signature (Sec. 3.1), clustering constellations by their visual signaturesirrespective of their type, interpreting the clusters (Sec. 3.2), and finally comparing constellations, by measuring theassociation of culture, type, and sky region to the signatures of their constellations, and the similarities betweencultures and types (Sec. 3.3 and 3.4).

3.1 Measuring the visual signature of a constellation

A constellation is formally a spatial network in the astronomical coordinate system. The nodes are stars annotated withtheir locations on the celestial sphere, which are independent of the location of observers on the earth: the declinationδ (equivalent to the geographic latitude) is the north or south angle between the celestial equator and the star; the rightascension α (equivalent to the geographic longitude) is the angle measured along the celestial equator. An additionalannotation is the magnitude, a measure of the star’s visible brightness. A link (or line) is the undirected geodesicbetween two nodes.

We define a set of 19 constellations features or statistics, s1 to s19. For definitions of the constellation featureswhich depend on the network structure (s1 to s11), refer to [47]. All features jointly form the visual signature of theconstellation.

Structural constellations features measure statistics about the internal linking of the line figure:

s1 the number of links, a measure of the network size;s2, s3 the maximum and average degree;

s4 the clustering coefficient (the average of all local clustering coefficients), a measure of how close the networkis to being a triangular mesh;

s5 the maximum core number (the largest k so that there is a maximal subgraph containing nodes of degree k),a measure of whether the network has one or more dense core regions, distinct from sparsely linked peripheralregions;

s6, s7 the number of cycles in the cycle basis (the minimal set of cycles so that any cycle in the network is a sumof these), and the size of the largest cycle in the cycle basis;

s8 the number of connected components (CCs);s9, s10 the average link diameter and shortest path among the connected components (the latter shows whether the

network is small-world);s11 the link connectivity (the minimum number of links which, if removed, disconnect the network).

Spatial constellations features capture geometric statistics:

s12, s13 the spatial diameter of the constellation, and the average link length (both in degrees on the celestial sphere,from the point of view of an observer on the ground);

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s14, s15 the sharpest and the average angle formed by any two links incident at any star (both in degrees);

s16 whether the spatial network, as drawn on the celestial sphere, is planar or not;

Brightness constellations features measure basic statistics on star magnitudes:

s17, s18, s19 the average, minimum, and maximum star magnitude.

Other constellations statistics could be defined; this set was selected to cover fundamental visual aspects of the linefigures, and to exclude redundant measures. For example, the number of nodes in the network is not explicitly in-cluded, but is proportional to s1/s3. These statistics also do not count on the stars being identified precisely (giventhe limitations of the data collection): if a star was mistaken for another in the close neighbourhood and of similarmagnitude, the statistics change little.

We opt for this highly multidimensional definition for the visual signature, instead of defining a small number ofmore abstract or high-level measures of visual complexity, such as the perimetric complexity applicable to writtencharacters [45] or binary images [77], and the algorithmic, representational, or algebraic complexity applicable to 3Dshapes [59] or written characters [45]. Each of our features remain interpretable, are similar to other knows measuresof shape complexity, such as morphological and combinatorial complexity [59], and capture the salient characteristicsof these spatial networks over scattered bright points on a sphere, without reducing them to a single dimension. Thisallows us to understand the word ‘complexity’ in more than one way: a spatially small (s12) but sharp-angled (s14)constellation with closed polygons (s6, s7) such as Cygnus (IAU) can be called complex for reasons different than along (s9), spatially large (s12), but entirely linear (s2) constellation such as Eridanus (IAU).

We provide an overview of the method in Fig. 3, of which the computation of constellation features is the first step.

Figure 3: Summary of the method.

3.2 Clustering visual signatures: nearest-neighbour dimensionality reduction

As a first step, we use manifold learning [24] out of the geometric properties of the high-dimensional constellationfeatures, to optimise the projection (or embedding) of this data into only two dimensions. The objective of the optimi-sation is to preserve the neighbours of a constellation (as given by the Euclidean distance, which captures the similarityacross all features) between the original many dimensions and the final two dimensions. Constellations which are sim-ilar in the original space remain neighbours in the embedded space, and can now be visualised and interpreted. Thisis the second computational step in Fig. 3.

The t-distributed Stochastic Neighbor Embedding (t-SNE) [73, 72] is manifold learning which also separates anyclusters in the data. Conclusions can thus be drawn from the embedding at multiple scales: some constellationswill be local neighbours, while some clusters of constellations will be regional neighbours on a higher scale. T-SNEvisually disentangles data that has internal structure at different scales. The later Sec. 6.3 gives more details on howwe implement, parametrise, and evaluate t-SNE. The latter is via the trustworthiness metric [75, 71] (on a scale from 0to 1), which expresses to what extent the local neighbourhoods are retained in the embedding. This embedding servesas a clustered summary of constellation visual signatures.

For both of our hypotheses, the predictors are categorical variables: the culture name, the type of knowledge trans-mission or type of practical use, the ancestor culture, or the sky region. The questions ask whether these categoricalvariables associate (and may have driven) the numerical constellation features (defined in Sec. 3.1 above). For Hypoth-esis I, the next (and final) step is to measure the association between the categorical predictors and visual signatures.

7


3.3 Measuring association between culture types: assortativity and similarity metrics

The scoring of questions I.1-4 proceeds as follows. Out of the original dataset with all constellation features, wecompute a directed, nearest-neighbour network of constellations: for each constellation, outlinks are drawn to each ofthe p nearest neighbours by Euclidean distance. We set p equal to the average size of our sky cultures. This networkmaintains local neighbourhoods, and has a constant outdegree. Finally, the nodes in this network are marked withthe value of a predictor; for example, for question I.1, each constellation is assigned the name of its culture as nodeattribute. (Note that the network could also be computed from the embedding, with similar results; we opt for theoriginal data, to avoid introducing small errors due to the embedding process.)

The assortativity coefficient. We require a score with a value of 1 for a full association between a predictor anda visual signature. For example, for question I.1, value 1 would signal that all constellations from a culture have aunique visual signature different from that of all other cultures. In other words, the visual signatures segregate alongcultural lines, so any clusters of similar constellations belong to single cultures. The score must instead be 0 for thenull model: when randomly permuting (or mixing, or de-segregating) predictor values over the same manifold. Ascoring method which fulfils the requirements is the assortativity coefficient r by a discrete node attribute (here, apredictor) [46]. This normalised score measures the degree of mixing, and is similar to the intra-class correlationcoefficients used to compare different groups in a population [22]. An r = 0 means that the cultures mix randomly,or that the constellations are equally similar intra-culture as they are inter-culture. A value r > 0 signals instead thatthe constellations are similar intra-culture and dissimilar inter-culture, so the culture is indeed a predictor of the visualsignature. A negative value for r is possible, and signals the opposite, that constellations are more similar inter- thanintra-culture. Sec. 6.3 provides the formal definitions.

When one type of constellations is much less numerous than other types (which happens here, since the cultures arevery heterogeneous in size), the assortativity coefficient treats any constellation equally, so less numerous types arenot weighted more than a more frequent one [46]. This makes for a stable coefficient r, which changes little if anyconstellation or link is added or removed from the data. The maximum possible value of r for a specific dataset may beless than 1 when a class size is smaller than the outdegree [47]. We thus report only the normalised r, after division byits maximum value. We also obtain the expected statistical error on the value of r, denoted σr, by a standard methodusing link removal [46].

We provide both the global assortativity (for example, for question I.1, a mixing score for all cultures), and also aone-versus-others assortativity, which measure how a particular predictor value mixes with all others taken as onegroup (for example, for question I.4, the mixing of all cultures of Chinese ancestry with all other cultures).

The similarity metric. When the results signal positive assortativity, for example that there is some characteristicvisual signature per culture, we also measure the similarity between any two cultures, based on their visual signatures.This similarity metric is a single statistic, defined in this work. Given two classes c1, c2 for which similarity needs tobe computed (for example, between two cultures), we compute r in two situations:

• r: over the constellation network consisting of three labels: c1, c2, and other, and• rm: over the same network with the two class labels c1, c2 merged into a single class c.

We define the similarity ∆ as their signed difference, ∆ = rm − r. ∆ is positive if merging the two classes raises theassortativity of the network, or makes the mixing of classes less random; this signals similarity in the merged classes.∆ is negative if merging the classes makes the mixing of classes more random, so signals dissimilarity. We build agraph of similarities between the predictor values of each question, drawing links between classes only when ∆ ispositive. These links are weighted with the amplitude of ∆.

3.4 Measuring diversity per sky region

We define a sky region not by its boundary coordinates, but by a root star: a sky region is the set of constellations(from any culture) which all include a given root star. This definition is less biased than alternatives: by not limitinga region to certain celestial coordinates, we remove any assumptions over the type of constellation that may be drawnthere, and place no limit on its network or spatial size. We expect that sky regions defined by two neighbouring rootstars will overlap.

For Hypothesis II, the problem is of a different nature than for Hypothesis I, and thus requires a different scoring. Here,we do not compare sets of constellations, but measure how diverse (or, on the contrary, uniform) the constellationsof a single set are, over the clustered embedding of constellations (described above in Sec. 3.2). Assume that theembedding shows k clusters of visual signatures, and the sky region under study is a set of n constellations. Maximumdiversity is achieved when the n constellations are distributed equally over the k clusters, and minimum diversity when

8


all constellations fall within one cluster. We use the classic Shannon diversity (or entropy) index H [63], a commonscore for diversity. The later Sec. 6.3 provides the formal definition. We report the normalised H divided over itsmaximum value, such that H = 0 signals universality (or no diversity), and H = 1 signals maximum diversity.

4 Results

4.1 Statistics over the constellations features

The constellations feature values vary greatly, both within a single culture, and across cultures. Figure 4 shows theculture size, and average and standard deviation of four constellation features per sky culture. The standard deviationstend to be large, meaning that sky cultures are internally diverse. We thus expect that, if an association exists betweenthe individual sky culture and the visual signature of constellations, it is weak.

1 5 32 250num.

constellations

EgyptianAztec

BlackfootInuit

KoyukonMaricopa

MayaMi'kmaq

NavajoOjibwe

PawneeSioux

TutchoneZuniInca

Kari'naLokono

QuechuaTukano

TupiChinese

Chinese medievalJapanese moon st.

KoreanMongolian

BugisIndian moon st.

JavaneseMaduraMandarAl-Sufi

Arabic moon st.Babylonian

RussianBelarusianRomanian

NorseSami

MacedonianSardinian

RuelleAnutan

HawaiianMaori

TonganVanuatu

IAURey

WesternWestern asterisms

2 3 7 20num. links

per constellation

EgyptianAztec

BlackfootInuit

KoyukonMaricopa

MayaMi'kmaq

NavajoOjibwe

PawneeSioux

TutchoneZuniInca

Kari'naLokono

QuechuaTukano

TupiChinese


KoreanMongolian





NorseSami

MacedonianSardinian

RuelleAnutan

HawaiianMaori

TonganVanuatu

IAURey


0 19 40spatial diameter (deg.)

per constellation

EgyptianAztec

BlackfootInuit

KoyukonMaricopa

MayaMi'kmaq

NavajoOjibwe

PawneeSioux

TutchoneZuniInca

Kari'naLokono

QuechuaTukano

TupiChinese


KoreanMongolian





NorseSami

MacedonianSardinian

RuelleAnutan

HawaiianMaori

TonganVanuatu

IAURey


0.07 0.40planar (0) or not (1)

per constellation

EgyptianAztec

BlackfootInuit

KoyukonMaricopa

MayaMi'kmaq

NavajoOjibwe

PawneeSioux

TutchoneZuniInca

Kari'naLokono

QuechuaTukano

TupiChinese


KoreanMongolian





NorseSami

MacedonianSardinian

RuelleAnutan

HawaiianMaori

TonganVanuatu

IAURey


1.00 3.06 5.00avg. magnitude

across constellations

EgyptianAztec

BlackfootInuit

KoyukonMaricopa

MayaMi'kmaq

NavajoOjibwe

PawneeSioux

TutchoneZuniInca

Kari'naLokono

QuechuaTukano

TupiChinese


KoreanMongolian





NorseSami

MacedonianSardinian

RuelleAnutan

HawaiianMaori

TonganVanuatu

IAURey


N AfricaN America

S America

E Asia

S Asia

W Asia

EurasiaE Europe

N Europe

S Europe

W EuropePacific

Global

Figure 4: Constellations features aggregated per sky culture. The culture size (the number of constellations) is on the left.Four constellation features (s1, s12, s16, and s17) are then shown via their averages and standard deviations per culture. The globalaverage of each statistic is marked with a dotted line. The horizontal scales for the first two statistics are logarithmic, and the restlinear.

On the other hand, there are also clear distinctions between sky cultures. Five out of six S-American cultures havevery large constellations (an average of 20.50 links per constellation for Tupi, and 12.40 for Tukano), comparedto an average of 6.88 links per constellation across all cultures. The (Indian, Arabic, and Japanese) moon stationasterisms have few links (averages between 2.42 and 4.11 links per asterism). Large E-Asian sky cultures (Chinese,Chinese medieval, and Korean) are also below average: averages between 4.28 and 4.82 links per asterism. In termsof spatial diameter, it is not only sky cultures with many links per constellation that also have large constellationdiameters: the largest average spatial diameter is Hawaiian (40.27 deg., compared to a global average of 18.85 deg.),although the same culture only has on average 6.31 links per constellation. The fraction of non-planar constellationsis close to zero in N America, but occasionally high for other cultures. N-American, S-Asian, and Pacific culturesconsistently use brighter (lower-magnitude) stars. This raises the expectation that an association may exist between

9


cultures (when aggregated by type, such as by common ancestry, correlated to geographic location) and the visualsignature of constellations.

4.2 The embedding of constellations by their visual features

The embedding learnt by t-SNE (trustworthiness 0.98) has rich internal structure. Figure 5 shows and interpretsthis embedding. All 1591 constellations across 50 sky cultures are projected into a two-dimensional “map” of con-stellations. The orientation of the embedding, the measurement units, and the concrete values on the axes are notmeaningful [73, 72] so are not shown; instead, it is the gradients of each of the 19 original constellation features whichhelp to interpret these reduced dimensions. Twelve of them are overlaid on the embedding, at the top of Fig. 5; theremaining seven are less important to the interpretation. There are distinct “islands” (or clusters) of constellations, andalso local and global gradients for the constellation features.

Figure 5: The mapping of constellation features to two embedded dimensions. All plots show the same embedding. Onepoint represents one constellation; all 1591 constellations from 50 sky cultures are plotted. (top) The gradient of each constellationfeature, as was projected in the low-dimension space by the manifold-learning process. Twelve constellation features are shown, oftype: relational (s1 to s11), spatial (until s16), and brightness (s17). (bottom) A summary of the clusters.

The embedding of structural features. Constellations with a large number of links (s1) are located right of centre andare distributed among a number of clusters; those with only one link (the simplest possible shape) form an isolatedcluster. The peaks for the average degree (s3), the number of cycles (s6), and the link diameter (s9) are close to thetop-right corner, and that for clustering (s4) is at the extreme right, with smooth gradients for all these features acrossthe embedding. The maximum core number (not shown) is low and only takes two values; almost all constellationswith a 2-core are embedded in the top-right quadrant. There are relatively few disconnected constellations (s8), andthey are embedded close to (but slightly isolated from) connected constellations which have otherwise similar statisticsin dimensions other than s8.

10


The embedding of spatial and brightness features. The average angle formed by two links in a constellation (s15) alsohas a global gradient, from 360◦ (meaning the constellation has no angles) throughout the leftmost cluster, to sharpangles throughout the rightmost. Almost all non-planar constellations (s16) are isolated from the rest, in a centralcluster. The remaining statistics have local gradients; in particular, the average magnitude (s17) has a distinct gradientwithin many of the clusters.

The same Fig. 5 (bottom) summarises intuitively the shape of the line figures, as grouped in major clusters. ClusterC1 (the smallest possible line figures, namely single segments) comprises 12.7% of all constellations. C2 (line-and tree-like constellations) is the largest cluster at 46.5% and has small subclusters: disconnected line- and tree-like line figures in (a), and spatially small but well branched line figures in (c). C3 (spatially large and very brightfigures) comprises only 14 constellations. C4 groups almost all of the non-planar constellations, at 3.7% of the total;besides the common property of non-planarity, these constellations are otherwise very diverse in shape. C5 (largeconstellations with complex internal structures of both cycles and tendrils) is the second-largest cluster with 20%. C6

(triangles and triangular meshes, no tendrils) groups 3.7% of the figures. Finally, C7 (cycles larger than three lines,no tendrils) comprises 12.2%.

In the embedding, the clusters are oriented such that constellations in different clusters, but with some features incommon, remain relatively close. For example, constellations in C4 with fewer links (s1) are oriented towards C1, andthose with most links towards C5. Large clusters have clear internal gradients in feature values. We provide examplesas to where well known IAU constellations, and less known constellations from other cultures, are embedded, in Fig. 6:black data points mark some of the IAU constellations, whose line figures are available elsewhere [28], while red datapoints and bold names mark constellations from other cultures, and their line figures are shown to scale.

Figure 6: Examples of constellations over the embedding. In the background, in light blue, the embedding of all constellations,the same as in Fig. 5. In the foreground, we show example constellations: (1) black data points mark some IAU constellations,whose line figures are available publicly [28]; (2) red data points and bold names mark constellations from other cultures, with linefigures drawn to scale.

4.3 [Hypothesis I] The type of sky culture associates with the visual signature of constellations

The questions we posed can now be answered by introducing, one by one, the hypothetical predictors of visual signa-tures, and measuring the extent to which they agree with the embedding of visual statistics.

11


I.1. The culture itself

Cultures rarely and weakly associate with the visual signature of constellations. If the culture itself were to associatewith the visual signature of its constellations, then constellations from the same culture would be (1) internally homo-geneous, or close neighbours in the embedding, but also (2) externally heterogeneous, or separated from constellationsof other cultures. When the predictor is the culture itself, this is only weakly the case. We obtain a positive, but lowglobal assortativity r = 0.079 (σr = 0.00122). Although in some cultures many constellations are visually similarand thus embedded closely together, in those cases there are similarities also with other cultures. This is shown for sixof the largest sky cultures in Fig. 7 (top), where each plot emphasises the embedding of a single culture. The E-Asiancultures (Chinese and Korean) are internally homogeneous (with many constellations in clusters C1 and C2), butare also similar cross-culture. The same holds to a lesser extent for the Babylonian and Western cultures, for whichthe focal clusters are C4 to C7. The similarity among these cultures may be naturally explained by their commonancestry. The constellations of many other sky cultures (not shown in the figure) are scattered in the embedding, sothose cultures are not even internally homogeneous.

Al-Sufi

Anutan

Arabic moon stations

Aztec

Babylonian

Belarusian

Blackfoot

Bugis

Chinese

Chinese medieval Egyptian Hawaiian

IAU

Inca

Indian moon stations

Inuit

Japanese moon stations

Javanese

Kalina

Korean

Koyukon

Lokono

Macedonian

Madura

Mandar

MaoriMaricopaMaya

Mikmaq

Mongolian

Navajo

Norse

OjibwePawnee

Quechua

Rey

Romanian

Ruelle

Russian SamiSardinian

Sioux

Tongan

Tukano

Tupi

Tutchone

VanuatuWestern

Western asterisms Zuni

the similarity betweenindividual cultures

Figure 7: Visual signatures by culture (I.1). (top) The constellations from one culture are shown in the foreground, over thebackground of all other constellations. Five examples are shown. (bottom) The similarity graph for cultures. The node size isproportional to the number of constellations per culture, and the edge width to the similarity ∆.

We also zoom in locally, on one culture at a time, to assess how that culture mixes with all other cultures taken asone. The two Chinese sky cultures have the highest association with the visual signatures of their constellations; thisreaches r = 0.131 for Chinese medieval. The next-largest sky culture in terms of the number of its constellations isKorean; this is much less assortative, r = 0.067. Other cultures also have weak but significant internal homogeneity:Al-Sufi (r = 0.117), Ruelle (r = 0.079), Rey (r = 0.066), IAU (r = 0.060), Western asterisms (r = 0.058), andBabylonian (r = 0.053). All these examples have σr < 0.00231. The other sky cultures mix almost randomly withall others.

Cultures cluster by similarity, and the clusters only partially follow phylogeny. The graph of similarities (metric ∆)computed between all pairs of cultures shows ‘families’ of cultures. Figure 7 (bottom) draws this graph. Only positivevalues for ∆ are used (namely, similarities rather than dissimilarities), and the graph is laid out with a force-directedlayout guided by the similarity as edge weight. The strongest similarity is between Chinese and Chinese medieval(∆ = 0.099). The lowest similarity value drawn as an edge in the figure is 0.003; this threshold for plotting waschosen because it preserves the top half of the edges, and sparsifies the graph for clarity.

The Chinese and Korean cultures are the most similar and form their own isolated cluster (on the left of Fig. 7 bottom).All other cultures are clustered together: they are weakly, but very frequently similar. This large cluster has internal

12


structure: a distinct group (in the centre) is formed by the ‘classical’ cultures close to the ancestral roots of Westernconstellations (IAU and its derivations, Al-Sufi, Babylonian, and Egyptian). Romanian is the folk culture with thestrongest similarity with these classical cultures. It is very similar to IAU in its global statistics over constellationfeatures (as seen before in Fig. 4), and contains large, intricate constellations (embedded in clusters C5 and C7).Another large and distinct group (on the right of Fig. 7 bottom) is formed by mostly folk astronomies with oraltraditions, covering all continents. The asterisms in the three moon-station cultures are similar, but isolated from allothers except for some similarity with the Western asterisms.

This clustering only partly follows ancestry (column an. in Table 1): the cultures of Chinese origin are strongly related,as expected (with the exception of the Japanese moon stations), and so are some of those with Greek origin. However,many European folk cultures (also of Greek origin) are as similar with N-American, S-American, Austronesian, andPolynesian cultures as they are among themselves. We thus expect that some phylogenies are dissimilar from allothers in visual signature, but some are closely related despite the geographical distance. Later, question I.4 will testphylogeny as a predictor.

I.2. The type of astronomy

A minority of the sky cultures in this study had a written astronomical tradition (marked in column type in Table 1). Inthese cases, the constellations were documented early on a longer-lasting medium, such as a codex, chart, book, stoneor clay tablets. Unsurprisingly, the number of constellations which have survived from written cultures is higher: the16 written cultures have 1299 constellations, and the 34 oral cultures only 292. For this predictor, we obtain a positiveassortativity coefficient of r = 0.231 (σr = 0.00237), which signals a significant, positive association.

Oral astronomies use brighter stars. This positive association is not due to any characteristic constellation featuresfor the written cultures, which include both those of Chinese ancestry, and the classical Western cultures: they arecomplementary in visual signatures (in Fig. 7 top, the first four examples are all written cultures). When groupedtogether, the written cultures span all clusters in the embedding uniformly. The positive association is due instead tothe characteristics of oral astronomies, most of which formed the rightmost similarity cluster in Fig. 7. The oral groupconsists of some Eurasian, almost all native American, Austronesian, and Polynesian cultures. Their constellationsare also present in all clusters of the embedding, but preferentially in regions where bright stars are used (towardsthe centre of the embedding, as shown by the average magnitude, constellation feature s17 in Fig. 5. However, it ispossible that this association is due to the limitations of data collection from oral cultures (specifically, a possible biastowards brighter constellations, introduced in Sec. 2).

I.3. The practical use of constellations

While only four cultures had political divination as practical use (marked in column type in Table 1), these 736constellations include the distinct Chinese-Korean similarity cluster (from question I.1, shown in Fig. 7). We thusexpect that political divination retains a unique visual signature.

There are three other uses (276 folk, 198 navigation, and 308 religious constellations, with the remaining uncate-gorised). Whether or not they also have an association with the visual signature is not clear from the answer toquestion I.1 alone. In particular, the Western cultures are mixed: they contain some constellations with an originallyreligious role (those inherited from Mesopotamia via the Greek [57, 3]), and some originally designed by navigatorsaround the northern pole, on the celestial equator, and in the southern skies [58, 3, 62]. The practical use of a stargroup also changed with the evolution of cultures: while Ursa Major was originally a navigator’s constellation in theGreek tradition [58], its Big Dipper (or Plough) asterism is part of many agrarian and hunter-gatherer folk cultures [7].On the other hand, the seafaring cultures (Austronesian, Polynesian) are geographically apart from most others, sosome uniqueness in visual signature is expected.

Navigation, religious, and folk constellations are similar. For question I.3, the result only partly follows our expec-tations: we obtain a positive assortativity coefficient of r = 0.256 (σr = 0.00212), but this is mainly due to a strongassociation between Chinese-Korean cultures (with political use) and their visual signature: r = 0.444 (σr = 0.00312)when testing the mixing of political use with all other uses. The location of political-use constellations over the em-bedding (Fig. 8, left) shows the expected segregation of this use away from the centre of the embedding. On the otherhand, all other uses (also in Fig. 8) appear similar so mix almost randomly. The similarity graph is in the same figure.Only three similarity links exist, and they are comparable in value: the strongest is between navigation and religiousconstellations (∆ = 0.080), which is close to the similarity between the Chinese and Chinese medieval cultures (I.1).We thus conclude that, as expected, political constellations have a strong associated signature, while, on the contrary,all other use types are similar.

13


folk

navigation

political

religious

the similarity betweenpractical uses

Figure 8: Visual signatures by practical use (I.3). (top) Constellations per use are shown in the foreground, over the backgroundof all other constellations. (bottom) The similarity graph between practical uses. The node size is proportional to the number ofconstellations per use, and the edge width to the similarity ∆.

I.4. The phylogeny of the culture

For phylogeny as a predictor, we obtain a positive assortativity coefficient r = 0.296 (σr = 0.00234), after havinggrouped the sky cultures into nine ancestry groups (by column an. in Table 1). The Mesopotamian ancestry groupsBabylonian, and all descendants of Greek and IAU cultures. The Sami and Egyptian cultures have no known ancestry,and each forms their own group. Of the resulting nine groups, the largest six are shown in Fig. 9. While r is fairlyhigh, signalling an association between phylogeny and visual signature, as for question I.3, this is due to a strongassociation for only some of the phylogenies: Chinese and Mesopotamian.

Austronesian

Chinese

Egyptian

Indian

Mesopotamian

N AmericanPolynesian

S American

Sami

the similarity betweenphylogenies

Figure 9: Visual signatures by phylogeny (I.4). (left) Constellations with common phylogeny are shown in the foreground,over the background of all other constellations. The smallest three phylogenies are not shown: Sami (3 constellations), Egyptian(26), and Austronesian (27). (right) The similarity graph between phylogenies. The node size is proportional to the number ofconstellations per phylogeny, and the edge width to the similarity ∆.

Chinese and Mesopotamian ancestries associate with a visual signature of constellations. From Fig. 9, it becomesclear that constellations of Chinese ancestry self-group to a large extent, and dominate the regions of constellationclusters (from Fig. 5) with the simplest visual shapes over faint stars: a large part of C1 (isolated line segments), alarge part of C2 (line- and tree-like constellations), and part of C7 (single-cycle constellations)—all with relativelyfaint stars. Chinese ancestry mixes relatively little with all others, r = 0.446 (σr = 0.00312), as expected from theresults for question I.3. Constellations of Mesopotamian ancestry self-aggregate to a lesser extent, and dominate theclusters defined, on the contrary, by the most complex visual shapes over bright stars: C4 (non-planar constellations),C5 (constellations with both cycles and tendrils), C6 (triangular meshes), and part of C7 (constellations with cycleslarger than a triangle)—all with relatively bright stars. Mesopotamian ancestry mixes more with all others, r = 0.299(σr = 0.00300), in particular with non-Chinese and non-Indian ancestries. The other phylogenetic groups have adegree of internal homogeneity; for example, the N-American ancestry (overlapping with the Mesopotamian) has adiversity of shapes over bright stars, but, due to the overlap, no distinct signature. This then raises the expectation thatthere are clusters of visually similar phylogenies.

14


American, Austronesian, Egyptian, Mesopotamian, and Polynesian ancestries are similar. The similarity graph be-tween phylogenies (Fig. 9, right) shows all edges with positive similarity values. The strongest pairwise similarity isbetween Mesopotamian and N-American ancestries (∆ = 0.040). The outlier Sami culture (with only three constel-lations) bares some resemblance with Polynesian and American cultures. However, it is not internally homogeneous:the three constellations have little similarity among themselves, which leads to weak assortativity for this culture inany test. In the similarity graph, clusters of similarity are apparent: a Chinese-Indian cluster, but also a tight clusterof six phylogenies. If the former cluster may be explained by geographical vicinity, the latter cannot: it spans allcontinents. This signals that complex visual shapes over bright stars, using cycles, tendrils, and combinations, may bea natural universal preference in cultures outside SE Asia.

4.4 [Hypothesis II] The region of the sky associates with the visual signature of constellations across cultures

Some stars are much more frequently linked into constellations than others. From among the stars linked in these 1591constellations, 67 stars are present in 20 or more constellations each. Occasionally, a culture has constellation variants,or simply different constellations which overlap; in these cases, the same star is present in two or more constellationsof the same culture. This is relatively rare; the majority of constellations per root star come from different cultures.We take these 67 stars as root stars, each defining a sky region, or constellation sets. When two stars are close inthe sky, their constellation sets overlap, but are rarely identical. The most frequent star is ζ Ori, in Orion (IAU), with56 constellation variants; outside the belt of Orion, the next most frequent star is η UMa, with 41. We measure thediversity for each of the 67 sets.

Over the embedding, we have delimited seven clusters of visual signatures (Fig. 5). Although they are of unequal sizes,with C3 in particular very small (14 constellations or 0.87% of the total), they have comparable presence from the rootstars: 20 out of 67 frequent stars have at least one constellation in C3, 23 in C1, and only 14 in C6. On the other hand,all root stars have constellations in C2, and 62 (almost all) in C5. We compute the normalised diversity index H , persky region defined by root star, over these seven clusters, and provide the results in Fig. 10. Per root star, this showsthe diversity index H among all constellations of that star (on the left axis), and the number of constellations withthat star (on the right axis). The dotted line marks the middle of the diversity range, 0.5, and the diversity markers arecoloured differently above and below this; although this middle value has no particular meaning, it helps to separatethe results into high and low.

And

Aql

Aur

CM

a C

Mi

Cas

Cas

Cas

Cas

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Figure 10: Diversity by root star (II). Per root star, the normalised diversity index H among all constellations with that star(on the left y axis with triangles), and the number of constellations with that star (on the right y axis with dashes). Value 0.5 fordiversity is marked with a dotted line, and the diversity markers are coloured differently above and below this. The mean diversityis H = 0.536. The alternating background emphasises root stars from the same IAU constellation.

There is a broad range of diversity among the root stars. Root star θ CrB has the lowest diversity, H = 0.217. Itsconstellations can only be found in two clusters, C2 for smaller and larger variants of the curved chain of stars familiarfrom Corona Borealis, CrB (IAU) plus one larger, branched form, and C7 for variants with the chain of stars closedinto a loop. In general, all CrB root stars maintain a diversity H < 0.5, as do all stars from Scorpius, Sco, and mostfrom Cassiopeia, Cas and Ursa Major, UMa (all IAU constellations). This amounts to 27 out of 67 root stars, or 40%.Among the root stars, there is no strong relationship between the number of constellations per star and the value of thediversity index.

The majority have high diversity. The mean diversity is H = 0.536, with 60% of root stars having high diversity.The peak is at α Ori (one of IAU Orion’s shoulders), H = 0.834. The 30 constellations over this star span all sevenclusters, although not equally. Figure 11 provides some as examples, to scale, with their embedding location. Not all

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could be included. A small number are identical to those already shown. Another constellation appeared previouslyas an example over the embedding in Fig. 6: Wintermaker (Ojibwe) in C5. The high diversity is not only in shape,but also in constellation semantics (although not the subject of this study); many of the shapes do not depict a humanfigure, as the Mesopotamians and the Greeks did.

Figure 11: The diversity of constellations over the root star α Ori (II). In the background, in light blue, the embedding of allconstellations, the same as in Fig. 5. The IAU constellation Orion is shown at the top left, with αOri emphasised. In the foreground,there are examples: black stars mark all constellations over α Ori, of which some are also shown, to scale, and with the root staremphasised.

5 Discussion and conclusion

We summarise our findings, discuss implications and possible use, and the limitations of this study.

Sparse data, many cultures, and results stable We assembled a diverse dataset of line figures from old and newastronomies around the world, but most cultures have an incomplete record, with their own astronomical traditionssupplanted by the global modern astronomy (Sec. 6.1 emphasises the cultures with such incomplete records). Cultureswith few line figures are unlikely to provide interesting statistics, such as an assortativity coefficient far from the base-line of the null model, because those few line figures are rarely even similar among themselves. The ideal ethnographicdata for a study of this type has a complete record for all cultures. Nevertheless, aggregating many small cultures bytheir type (with the most convincing case that of culture phylogeny) lead to large, relatively balanced groups of dataand many significant similarity links and clear similarity clusters (in Fig. 9). The results shown are relative to thecurrent dataset of constellations, since the notion of close neighbours in the constellation feature space is crucial to ourmethod of computing assortativity and similarity, and additional data modifies the neighbourhood graph. However,a small amount of additional data would not change the results significantly: the variance σr, based on link removalfrom the neighbourhood graph, is low, and the results have remained consistent when we found additional data andreran the computation.

Constellation complexity is multidimensional We worked with a highly multidimensional definition for the visualsignature, composed of 19 morphological features (networked, spatial, and brightness), instead of a combined metricfor visual complexity. The morphological features are individually interpretable, and the embedded map of constella-tions in two dimensions still retains this, if the feature gradient is overlaid (as in Fig. 5). The concept of complexity forstar constellations is debatable, and instead of fixing it into one dimension, we provided clusters of visual signatures,and a map (manifold) of constellations, as they are related by this multidimensional signature. This map can serve as

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a global taxonomy for constellations. Our choice of morphological features for constellations has its limitations: itcaptures the spatial-network properties, but may consider two constellations dissimilar when human perception wouldconsider them similar. For example, adding a link to close a cycle in a constellation makes a large difference in networkstructure, but, if the link is very short, a only a small difference in perception.

In-depth study possible with different parametrisation This study provided global statistics across cultures, withno in-depth focus on a particular cultural region. A focused study would be possible, by retaining only the interestingpart of the data, and re-parametrising the manifold learning. A smaller perplexity value p (Sec. 6.3) would separatethe large clusters of constellations into more definite subclusters, which could lead to more detailed, local insights.

Relatively unique visual signatures, clusters of phylogenies We obtained an assortativity r = 0.446 when mea-suring how constellations of Chinese ancestry “mix” with all others (question I.4). This, as expected, tells that the twoancestries are far from similar in signature. No other culture or type of culture is as dissimilar from another as Chineseancestry is from any other ancestry. Mesopotamian ancestry also has its own signature, with r = 0.299, but this iscloser to that of American, Polynesian, and Austronesian cultures, which altogether form an unexpected similaritycluster, drawing large, intricate visual shapes over bright stars.

Diversity exists The majority of popular stars form visually diverse constellations (Sec. 3.4). Others shows a con-sistency of shape, with constellations embedded in only a few clusters in the manifold. Because of this diversity,our map of constellations, with its clusters of shapes, can help to guide any future work which aims to find accurategenerative models to explain constellation shapes from cognitive principles, such as follow-ups to [17, 30]; classes ofsuch models would be more accurate than single models.

6 Supplement: In-depth information on data and method

This section supplements the information in Sec. 2, which described the data only briefly, and Sec. 3, to which itadds a formal description of the metrics in the method. The first subsection below describes the cultures, as a generalintroduction. The second specifically motivates the line figures in this study, and underlines the limitations in collectingthem. The third expands the method.

6.1 Sky cultures, their type, and phylogeny

Each paragraph introduces a related group of astronomical cultures from Table 1, their early sources, timeline, whetherthe culture had written or oral traditions, their practical use (column type in the table), any limitations in their docu-mentation, and motivates the ancestry noted (column an. in the table. Each culture or group of cultures is emphasisedin bold in the text when first mentioned. This is a too broad a subject to cover in depth here, so this general summaryremains brief.

Primitive astronomies: moon stations The moon stations, also known as lunar lodges or lunar mansions, weregroups of stars on the ecliptic, close to the moon during its revolution cycle of 27 solar days and 7 3

4 hours. Between27 and 28 moon stations (naks. atras) or lunar mansions were documented during the Indian Vedic period (before500 BC), some of which are single stars; if there was external influence in this early period, the literature is toolate to provide information [6]. Arabic asterisms for time-keeping and orientation (the anwa’) also predate the Arabs’knowledge of other astronomies. At an unknown time, the Arabs received the Indian system of 28 lunar mansions [32],and each mansion was then identified with one of the Arabic anwa’. After the spread of the Greek astronomy, theseasterisms were documented, in the 9th-century Book of Anwa’ by Ibn Qutaybah [35]. Both systems were used fordivination [35].

The names of the 28 Chinese xiu (lunar lodges) were listed systematically around 433 BC on a chest discovered ina tomb; they are likely much older (as old as the 3rd millennium BC), but evidence for this dating is lacking [65].The question whether the lunar mansions in Chinese astronomy were influenced by the Indian naks. atras or the otherway around remains open [78]; there may have been no influence. Japanese astronomy closely followed the Chinesetradition. The Japanese sei shuku (lunar lodges) are based on their earliest sky chart, showing the 28 lunar lodges onthe ceiling of a tomb from ~700 AD, identified with the help of a later 17th-century chart [55]. The Indian, Arabic,and Japanese moon stations are recorded in this study in individual datasets; the Chinese moon stations are insteadpart of two sky-wide datasets: one called Chinese medieval, and the later Chinese.

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Early astronomies: Mesopotamia and Egypt The Babylonian sky is based on MUL.APIN [57], a clay-tabletcompilation of Babylonian star catalogues produced up until that time. These were developed in stages from ~3200BC in two overlapping traditions, to represent gods and their symbols (the twelve signs of the zodiac and associatedanimals), and rustic activities used in the local farming calendar (workers, tools, and animals). “Many constellationsbelonged to both traditions, but only the divine were transmitted to the West” [57]. We mark this culture as havingprimarily religious usage. Many pictographic representations of the zodiacal signs inform the line figures: on cylinderseals (from ~3200 BC onwards), boundary stones (~1350–1000 BC), in the Seleucid zodiac (clay tablets, with somecopies surviving from the last few centuries BC) and the circular zodiac at the temple of Hathor in Dendera (anEgyptian ceiling bas-relief, ~50 BC, merging the Mesopotamian zodiac with Egyptian constellations) [57].

For the ancient Egyptian sky, which was connected with religion [3], two pictographic references are used: theastronomical ceiling of the tomb of Senenmut at Deir el Bahari in Luxor (~1470 BC), and the Egyptian figures on theDendera zodiac [39]. In classical times, these native Egyptian constellations were combined with the Mesopotamian,producing the standard sky of the Greco-Roman period. We only use the older, native Egyptian constellations.

Early astronomies: The Mediterranean The classical Greek sky (not a separate culture in this study, but includedin the Western cultures) had 48 constellations and was derived from two pre-Greek cultures. The twelve signs of thezodiac and associated animal constellations were Mesopotamian, then adopted by the Greeks in 500 BC1. Other largeconstellations around the pole and equator of that time date from ~2800 BC, probably originate from the Mediterraneanregion, and were designed as markers for sea navigation [58]2. The Greeks assembled these traditions during ~540-370 BC, with the definitive documentation by Ptolemy in the Almagest, ~150 AD. The Al-Sufi data from the Bookof Fixed Stars [2] is a revision of the star catalogue in the Almagest with corrections and the addition of indigenous,Arabic astronomical traditions.

Western and Asian sky cultures of Greek ancestry have this mix of constellations of (originally) religious and nav-igational use. The practical use is thus per constellation, not per culture. The constellations with religious use areMesopotamian zodiac and their associates, Orion, and constellations associated with Greek mythology and their as-sociates 3. Those with navigational use are the Mediterranean constellations, and, for more recent cultures, the con-stellations surrounding the south pole, attributed to various navigators [62]. Not all Western constellations had a clearpurpose when formed; those which were the result of desk research are left uncategorised.

Early astronomies: East Asia Chinese astronomy is well documented; refer to [64, 65] for an extended description,of which we provide a summary. Chinese astronomy developed independently of European, Arab or Indian influencesuntil as late as the 17th century. Some star groups were mentioned in East Asia before 1000 BC: the Big Dipperasterism on rock carvings, the name Dou (the Ladle, presumably for the Big Dipper) on bone inscriptions, and fourasterisms including the Ladle, explicitly described, in folk songs. Qualitative descriptions of about 100 xingguans(asterisms) were given in a catalogue from the 1st century BC, and later catalogues list 280 asterisms. The onlysurviving celestial map until the 10th century is the Dunhuang star chart (a manuscript on paper dated around 700AD), which draws the entire night sky visible from China (1345 stars in 257 asterisms). The astronomer Su Song’sprinted star chart from the Song period (1092 AD, with 1,464 stars in 283 constellations, which have become standardfigures) is the oldest printed chart to survive. In comparison with Western constellations, the Chinese asterisms aresmaller, and name terrestrial items (the imperial family, officials, domestic animals, crops, and buildings). They wereused for astrological predictions at the imperial court [64].

The Chinese medieval dataset draws the Song-period chart, Xinyi xiang fayao (New design for an armillary andglobe), using books of accurate stellar measurements by officials from 1052 AD; this information was compiled ina book [51]. The later, more complete Chinese dataset draws the skies based on a star catalog revised by Qing-period officials with the help of Western astronomers, finished in 1756 [51], and later additions to this. The Westernastronomers did not supplant the Chinese asterisms with Western ones, but measured star positions more precisely,occasionally added stars to asterisms, and added southern stars near the south pole which were invisible from China.

1The Greek classical zodiac constellations of Babylonian origin are, from oldest to newest: Taurus, Leo, Scorpius, and Aquarius(~3000 BC, when they marked the cardinal points), Gemini, Virgo, Sagittarius, Pisces, and Capricornus (3rd or 2nd millenniumBC). The remaining three, Aries, Cancer and Libra, the least bright, were accepted late, in classical times [58].

2The Greek classical constellations of Mediterranean origin are bears (Ursa Minor, Ursa Major), serpents (Draco, Hydra, Ser-pens and Cetus), giants (Hercules, Ophiuchus, Boötes, Auriga), and some large southern marine constellations (Eridanus, repre-senting a river meandering southwards). Likely, this set includes also Ara, Centaurus, Argo Navis, and Lupus [58].

3Greek mythology characters and their vassals were: Cepheus, Cassiopeia, Andromeda, Perseus, Pegasus, Coma Berenices,Corona Australis, Corona Borealis, Canes Venatici, Delphinus, Lyra, Canis Major, Canis Minor, Lepus, Crater, Sagitta, Triangu-lum [58, 3].

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The Korean sky culture follows the earliest surviving Korean chart, a marble stele dated 1395 AD, Ch’onsang yolch’apunyajido (Chart of the regular division of the celestial bodies). This chart is a reproduction of an older Chinesechart. Some line figures differ from the medieval Chinese counterparts, and modern measurements of the star po-sitions suggest a date around 30 BC—so this may preserve traditions older than the surviving Chinese charts [65].The identification of the stars on both of these maps is tenuous, due to the irregular projections and imprecise starplacement.

Western sky cultures The International Astronomical Union (IAU) standardised a list of 88 constellations withnames and three-letter abbreviations (at its first General Assembly in 1922) based on the classical Greek sky and laterdiscoveries in the southern hemisphere. At that time, the constellation figures only had vague and variable perimeters.Standard constellation boundaries along vertical lines of right ascension and horizontal parallels of declination werepublished in 1930 [16]. Line figures had been sketched by a French astronomer, Alexandre Ruelle, on the first modern-looking star chart with line figures and without pictographs [60] (dated 1786), including constellations which are nowobsolete (such as Cerberus and Musca Borealis). We transcribe this early chart in a dataset. No line figures werepublished widely until much later, in the 1930s. The first popular line figures were H. A. Rey’s [56] (1952) intuitivefigures of the objects they are supposed to represent. They largely adhere to previous traditions, but also sometimesdeviate from the figures described since the Almagest. For example, Rey’s bear in Ursa Major is oriented the oppositeway compared to Ptolemy’s description. The IAU line figures, “Alan MacRobert’s constellation patterns [from the Sky& Telescope magazine], drawn in green in the charts, were influenced by those of H. A. Rey but in many cases wereadjusted to preserve earlier traditions” [28]. The Western constellation set is a simpler variant used by the popularastronomical software Stellarium [79]. Separately, we use the set of popular Western asterisms which do not respectIAU constellation lines, such as the Spring and Summer Triangles (with various sources provided for the figures [79]);we mark this is a folk culture.

The Belarusian folk sky is compiled by a local ethnoastronomer [5] from sources in the 19th and 20th centuries,with added material collected 2004–2007. Most lines are a subset of the IAU constellations, with different names andmeanings. The Romanian sky is the result of an early study (1896) by a local mathematician and teacher [50], whosent copies of a sky map to teachers throughout Romania, requesting them to ask the oldest peasants about their beliefsabout constellations. In 39 of the papers returned, there were new accounts of constellations of mixed origins (agrar-ian, religious, or linked to historical events); this makes the collection unique in depth among the few documentedastromythologies of Europe. Macedonian ethnoastronomical research began in 1982, with 140 villages visited forsurveys. Given the symbolism of the constellations, the roots of this sky culture are likely in early agricultural com-munities of the Neolithic in the Balkans [11]. The Sardinian data is the result of unpublished research by locals [79].Only four constellation shapes are identified well across Russian-speaking republics from the early 20th century on-wards; the collectors of Russian folklore were not familiar with the sky, so their polls were focused on the most famousof the constellations, leading to poor ethnographic records [66].

While not geographically Western, the present Mongolian constellations are inspired by the Western ones (althoughthe Mongolian mythology has its own identity [79], and earlier cultures may have been influenced by the Chinese).Knowledge transmission in this culture remains essentially oral; this data was collected in 2014 from field work.

All these folk cultures had oral traditions, an agrarian iconography, and partial influence from the IAU constellations.

Norse (particularly Icelandic) astronomy may have been well developed for navigation and time-keeping, but little hasbeen preserved. Greek and Latin names for constellations supplanted local ones in the medieval period, and continentalinfluence is likely. The only old sources are literary: the Eddas (from 13th century Iceland), and a compilation oftimetelling verse [29] give Norse interpretations to parts of modern constellations. The Sami sky culture may be veryold, but written records for this oral tradition of reindeer herders exist only since the 19th century [40]. There are fewconstellations, all connected to Sarva the Elk. The Sami sky culture doesn’t resemble the Western, and there is nodocumented influence.

North American sky cultures In Mesoamerica, the ancient Maya sky is due to research [23, 76] based on impreciseinformation, namely the partial pictographs of animal constellations in the native Paris Codex, dated to around 1450.Unlike the Maya sky, the Aztec sky is based on line figures and textual descriptions from 16th-century codices (linefigures in Primeros Memoriales [14], and descriptions in the Florentine Codex [15]; the challenge was in locating thestars and constellations [4], which is necessarily approximate. Both Maya and Aztec cultures kept written astronomicalrecords, and served a divinatory function.

The Native American sky cultures outside Mesoamerica are all oral traditions documented recently, between the late19th century and the present, with a practical use as time-telling tools for farming, hunting, and gathering [34]. Sincethe tribes live independently, their traditions retained some originality, with star mythology related to hunting and

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planting [44]. There are occasionally striking commonalities: natives in the Northeast, Southeast, western Subarctic,and the Plateau all have myths about the Big Dipper asterism in Ursa Major as a bear and hunters. Many nativeconstellations either remain uncertain (and this study includes the more likely ones), or are mentioned by the nativesbut not located in the sky (and then are not included here): “In many cases the information is slim at best; many of theethnologists who studied these tribes were not familiar with the stars and did not inquire about them [or were] vagueabout identification” [44], and “many of the [Inuit] elders interviewed insisted that the information they possessedabout astronomy was meagre compared to that of their parents or grandparents; the present generation of elders is thelast repository of a more or less detailed knowledge on this subject” [41].

South American sky cultures The traditions are oral and represent the natural world also in South America. Likethe Maya sky, the Tupi (on the coast of Brazil) have constellations documented in an 1614 book [13], which werelater identified by 20th-century studies via comparisons with the constellations of other tribes [42, 1]. These state that,due to migrations, there is influence between the coastal Tupi, Amazonian, Guianan and Andean tribes [42]. In theAndes, the Inca constellations are from a 1928 interpretation [38] of a 1613 astronomical drawing in Cuzco, Peru; theinterpretation uses textual descriptions from co-located tribes, Aymara and Quechua, to make an identification. Fieldwork from 1978 also documents the little that remains of these constellations among the Quechua tribe [70], with thisdata heavily Spanish-influenced. On the coast, the Lokono territory borders that of the Kari'na; though unrelated, theyshare a Guianan culture, and the ethnoastronomical traditions have similarities. The Lokono constellations came outof recent ethnographic work [61]. Those of the neighbouring Kari'na were documented from before 1900 to 1980 (thelatter, field research in three Carib villages from Suriname [43]). The researchers state that the Kari'na sky traditionsare fading fast: many of the constellations which were mentioned by Indians in a 1907 study were unknown by 1980.For the Amazon, Tukano constellation data was provided in research reported in 1905 [33] and 2007 [10], with allfigures in a similar style. We add them up as the Tukano sky culture (10 constellations from either study).

Polynesian sky cultures Stars and constellations are the most important tool for long-range oceanic navigationfor the Polynesians (Hawaii, Maori, Tonga) and outlying islands which are culturally Polynesian (Anuta, Vanuatu), allwith ancestry in a common seafaring Polynesian culture. Their knowledge was transmitted orally. Some constellationswere still known by Anutan sailors at the end of the 20th century, and were identified and then drawn during fieldresearch in 1972-73 and 1983 [21]. The Hawaiian “star lines” were lost by the 1970s, then reconstructed with helpfrom a Polynesian navigator from another island [67]: a star line is a group of main stars that explicitly connect intolines and simple shapes, some of which point directly to main cardinal points as wayfinding instruments. Tongan datais entirely due to a synthesis of the sailing directions written by a high Tongan chief in the late 19th century [12]; thisreport also explains that “Few if any living Tongans are able to point and name more than a very small proportionof the stars. This rapid decay of the old astronomy is doubtless due to the universal use of the mariner’s compass”.Maori data is due to a synthesis of 19th-century information, plus fieldwork carried out at the end of that century [49];“what is remarkable”, the researcher wrote, “is that so little was recorded, for 2,500 naked-eye stars were visible tothe Maori”.

The Vanuatu data comes from field work with locals on the Tanna island in Vanuatu by an amateur astronomer in2019, published informally; the researcher also states that “in many places of Vanuatu this ancestral knowledge is nowalmost forgotten” [53]. This data is an outlier for this geographical region: all constellations represent agriculturalconcepts.

Austronesian sky cultures All four are oral cultures, recently documented, with three from seafaring traditions. TheMandar [54] and the Bugis [54, 48] are two neighbouring ethnic groups on the island of Sulawesi in Indonesia. Bothhave a strong seafaring tradition due to the Austronesians which migrated into Sulawesi. The Mandar constellations arefading from memory: the data was collected “through interviews with retired fishermen [..] known to have knowledgeabout Mandar star patterns. We noted that the younger fishermen did not use star patterns for their navigation” [54].The Bugis data is mostly recorded in [48], which provides a unified drawing of constellations from prior sources.On the island of Madura, Indonesia, off the coast of Java, the locals also have a maritime tradition, likely also ofAustronesian origin. The constellation data was collected recently via a survey of about 100 families [20]. Althoughit is nearby on the island of Java, the Javanese astronomy serves an agrarian society, cultivating rice. Informationfor one constellation (The Plough in IAU Orion) was first provided in person to the author of a 1885 study [74] bya religious figure from an English mission on the island. This was confirmed and supplemented with two others ina recent paper [31] with data from 2019 interviews; the authors also confirm “widespread similar asterisms in thearchipelago”.

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6.2 Line figures: both ancient and modern representation for constellations

The line-figure representation is new in many cultures, but native in others. We summarise the timeline (also inFigure 12) and the sources for line figures, emphasising any limitations in their collection.

Figure 12: Regional timeline for line figures. The centuries AD are marked at the bottom, on a non-linear scale. Dark blue timeintervals are recent periods, during which line figures are documented. They are preceded by periods of development or transition(in light blue), during which important sources for line figures are marked. In E Asia, the circle-and-line representation is native. InEurope and W Asia, line figures developed out of pictographs, alongside the development of modern astronomy. In the Americasand the Pacific, they were likely introduced under Western influence.

From ancient pictographs or text to recent line-figure interpretations The ancient Babylonian and Egyptianskies are based on pictographs without stars clearly identified and overlayed, so the line-figure identification is tenuousin both cases. The results of the identification differ among researchers (for both the Babylonian [57, 27] and theEgyptian sky [39]). We use the most recent and extensive identifications [26, 25, 27, 39]. Of pictographic Greekorigin, the IAU constellations never had standard line figures, despite many early variants and current wide use.

The line-figure identification is tenuous also for the Inca, Maya, and Tupi skies, for which the early 15th- and 17th-century sources do not identify stars and rarely draw line figures: the Inca source draws the Stove constellation(equivalent to IAU Crux) with lines, but the rest are from recent identification research. A slightly easier case, theAl-Sufi source [2] draws pictograms and lists the stars with their astronomical data (latitude, longitude, magnitude)and text ordering and placing each star in the pictogram (α UMi is “the star at the end of the tail” [68]). The Al-Sufiline figures link the stars based on this text [79]. The Norse constellation Aurvandil’s Toe is mentioned in the Eddas(in verse instead of a pictograph), and its identification with IAU Corona Borealis is likely, but not certain [79]. Theremaining five Norse constellations [29, 79] use bright stars and simple line figures. The Indian [6] and Arabic [35]moon stations consisted of a few well-identified stars, without lining; the dataset draws very basic lines to link thesestars (in most cases a chain).

From ancient oral traditions to recent line-figure interpretations The traditions of native tribes in N Americaoutside Mesoamerica were almost entirely oral, with few old artifacts recording star groups: pictographs on the ceil-ings of Navajo caves show constellations from as early as 1700; early 1900s Navajo and Tipai sandpaintings and anundated Pawnee star chart on buckskin show recognisable star groupings without lines [9, 44]. The line figures areinterpretations of the researcher. The same is the case for Lokono in S America: the researchers write that “manyLokono, when drawing constellations, did not connect the stars with lines and that in some cases, there was littleagreement among speakers as to which star within a constellation corresponds to which parts of the plant or animal itrepresents; the Lokono tradition allows for more flexibility in interpreting particular star groups” [61]. The Tukanoalso drew their constellations in the form of star clusters without lines; it is the researchers who added the lines: Koch-Grünberg as a printed star chart in his 1905 book [33] and a dataset was contributed to Stellarium based on the clustersin a 2007 dissertation [10]. This process was similar for the Quechua [70]. The Kari'na line figures “have been drawnin the sky by villagers and referred to Western sky-maps by us” [43].

All the Indonesian, plus the Anutan and Vanuatu constellations were identified with locals then drawn by the re-searcher [20, 74, 31, 48, 20, 21, 53]. The oldest explicitly drawn line figure is from the 1885 Javanese source [74]. The

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Maori and Tongan were described only in text, but simple lines correspond well to this text [49, 12] (for example,“Toloa means a wild duck, and Tongan imagination pictures the cross as a duck whose head is γ and tail α, the wingsbeing β and δ” [12]).

Some oral sky cultures of the Europeans and Asians (Russian, Sami, Belarusian, Macedonian, Romanian, Sar-dinian, and Mongolian) were documented relatively recently (in the 19th century or later, with the researchers usingsky charts), and described all line figures at collection time. Most researchers provide line drawings directly. Othersdescribe them in words, such as the Belarusian report [5] (constellation The Cross is “α, γ, η, β Cygnus – a verticalbeam, ε, γ, δ Cygnus – a cross beam”).

Native line figures of all ages In E Asia, line figures were and remain the only standard representations for starasterisms; on a Chinese tomb ceiling (uncertain date BC), the moon stations are depicted with stars as small circles,joined into groups by straight lines, as in most Chinese and Western star charts. The Dunhuang star chart, as well asall its successors in China, Korea, and Japan, draw line figures, with the stars drawn of roughly equal size regardless ofmagnitude, and annotate the star names. No pictographs or other symbolic representations are used until the Westerninfluence. In the Americas, the Primeros Memoriales codex [14] drew line figures for the Aztec sky in the 16th-century, and one Inca constellation is lined in the 17th-century source [38]. In the Western world, the first completestar chart with line figures (Alexandre Ruelle [60], 1786), and subsequent variants, did not catch on with the public.Only H. A. Rey’s line figures [56] became widely adopted from 1952, followed by variants, such as the IAU and Sky& Telescope magazine’s [28] and the simpler Western sets [79]. All are included in this study.

6.3 The method: implementation details

Tuning the manifold-learning process We run the t-SNE [73, 72] implementation from scikit-learn [52] andparametrise it as in Table 2. We aim for two dimensions, which can be visualised clearly. The distance used issquared Euclidean distance, which is calculated after scaling the data feature-wise with a standard scaler, which re-moves the mean of the distribution and scales the feature to unit variance. First, t-SNE converts the closeness of datapoints into joint probabilities, and minimises their Kullback-Leibler (KL) divergence in the original vs. the embeddedspace. The gradient-descent algorithm does not guarantee that it will reach the optimum KL divergence, so we trydifferent random seeds (that is, do multiple restarts) and select the best divergence found. We use approximate (ratherthan exact) gradient calculation, since the algorithm is computationally heavy—the runtime of manifold learning with20,000 iterations is on the order of 10 minutes, even though the number of data points (constellations) is relativelysmall, in the thousands. An exact gradient calculation would guarantee that two identical constellations are embeddedat exactly the same coordinates. With the approximate calculation, we occasionally see small differences betweenthese coordinates, but identical constellations remain very close in the embedding.

Table 2: Parameters for the t-distributed Stochastic Neighbor Embedding (t-SNE) in its scikit-learn implementation [52].PCA stands for Principal Component Analysis.

t-SNE parameter valueembedding dimensions 2initialisation PCAgradient calculation algorithm approximatelearning rate 50number of iterations 20,000perplexity p 32

The perplexity parameter p makes a difference: it is essentially the number of nearest neighbours to consider whengenerating the conditional probabilities. We set this to the average culture size (the number of constellations). Thismakes regional structure visible. A smaller value for p would be useful if splitting large clusters is desirable.

Evaluating the embedding The trustworthiness metric [75, 71] expresses to what extent the local structure (theneighbourhood of all constellations) is retained by the embedding. Given a number of nearest neighbours as parameter(here, equal to the perplexity value from Table 2) and a type of distance (here, Euclidean), the trustworthiness is avalue in [0, 1] which reaches 1 if all constellations have retained all their nearest neighbours after the embedding. Wereport the trustworthiness value to evaluate the embedding itself.

Interpreting the embedding For t-SNE, the interpretation of the embedding has some caveats. The scale of the twoembedded dimensions has no meaning (similar to the dimension of a distribution after standard scaling), so is nevershown. The size of clusters in the embedding (for example, the area of the convex hull around a cluster) also has

22


no meaning, because the t-SNE algorithm adapts to regional density variations in the data, and equalises these. Thecomposition of the clusters is meaningful, although dependent on the perplexity parameter.

The assortativity coefficient r This is a classical (2003 [46]) measure for assortative mixing in a graph. We repeatthe definition here for completeness; examples and a discussion are available in [46, 47]. Given a directed networkwhose nodes have a categorical type marked (denoted i or j), a square mixing matrix can be formed out of this graph.This matrix has as many rows as there are distinct node types. A cell, denoted eij , contains the fraction of directededges in this network which connect a vertex of type i to one of type j. The sum of all cells in the mixing matrix is 1.The fraction of edges which have as source any node of type i is denoted ai, and likewise bi for destinations of type i.The expressions for ai, bi, the assortativity coefficient r, and the expected standard deviation for r are:

ai =∑j

eij , bj =∑i

eij , r =

∑i eii −

∑i aibi

1 −∑

i aibi, σ2

r =∑k

(rk − r)2,

where rk is the value of r for the network in which the kth edge is removed.

Intuitively, r compares the diagonal of the mixing matrix (the total fraction of edges between nodes of the same type,regardless of the type) with the expected value for this total fraction, as given by random chance: aibi is the expectedfraction of same-type edges, given the relative fraction of those types in the data. This definition is normalised, so themaximum value is 1. However, this maximum value of 1 is not reachable for any network, for example when the nodeoutdegrees are larger than the number of nodes from one type [47]. When this occurs, we report r normalised, dividedby its empirical maximum value.

The Shannon diversity index H Also known as entropy [63], the diversity index H measures how much “choice”there is among the values of a discrete distribution. In this case, it measures how constellations from a given setdistribute among discrete clusters of visual signatures. Assuming that the embedding shows k clusters of visualsignatures (indexed by i), and the sky region under study is a set of n constellations, the diversity index is:

H = −∑i

pi log pi,

where pi is the fraction from the n constellations which fall into cluster i. H = 0 if all constellations fall into onecluster. The maximum value, log k, is reached if the constellations distribute equally among all clusters. We report Hnormalised, divided by this maximum value.

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