The Khipu As a Space/Time Coordinate System

The Khipu As a Space/Time Coordinate System by Gary Urton Introduction Andean Studies has always had a problem with time. This is primarily due to the absence of a system of writing in the ancient Andes, from which we might be able to read their accounts of what happened in their past, who succeeded whom and for how many years each Inka ruled…in short, their history, Thankfully, the same cannot be said for space, about which we know a good deal, as the dimensionalities of space are evident in abundance in the shapes of objects preserved from many ancient Andean societies. This paper is an attempt to develop an understanding of space and time as constructed in a particular Andean object that incorporates a unified referencing of space/time. Since societies have existed in the Andes beginning some 13,000 years ago down to the present day, I should clarify that I will focus on the time period in Andean pre-history associated with the florescence of the Inka Empire (ca. 1450-1532 C.E.) – the last, great empire of the pre-Columbian Andes. The object that I will be concerned with is a topic in material cultural studies that I have spent the past 30 or so years researching and working on – the khipu (Quechua: “knot”), the device used for administrative and other forms of record keeping in the Inka Empire (Figure 1; for studies of khipus in museum collections, see Ascher & Ascher, 1978, 1997; Pereyra, 2006; and Urton, 2003, 2017; see also the website of the khipu database project: https://web.archive.org/web/20210126031628/https://khipukamayuq.fas.harvard.edu/). 2 Figure 1 – An Inka Khipu said to have been obtained by Hiram Bingham in Cuzco, Peru (Peabody Museum, Yale University; composed of two fragments: ANT.019326 [large fragment] and ANT.029912 [small fragment]) The khipu is a topic of particular interest and relevance with respect to issues of time and space because, as an instrument or device for recording information, this status places the khipu in the category of other known systems of record-keeping, such as chronicles, annals, time sheets, and a whole world of other devices for recording and chronicling events. Most such recording systems contain such information as when and where events occurred, which event preceded or followed which other event, who were the actors, and other such bits of quantifying and qualifying information. Now, it must be said here that, if we are going to stack such requirements and expectations on the shoulders (as it were) of the Inka khipus, we will need to be confident these objects can bear the burden of such expectations. In fact, this is a complicated proposition insofar as khipus are concerned. These were, after all, pieces of knotted, sometimes colorful, string, the contents of which remain largely undeciphered today. Therefore, one issue that confronts us from the beginning of this project is how can we investigate such a complicated matter as the space/time contents of these devices without 3 knowing with certainty their contents, or even the general nature of the information they record? As we proceed, we will see that from their structural properties, as well as from deductions we can make on the basis of their organization of cords, colors, knots, etc., that we can, with some degree of confidence, attribute specific meanings to these knotted string records that is relevant for considerations of time and space. In any case, this is the challenge I take up in this paper. The Dimensionalities of Khipu Structure I will first describe the structures – the spatial arrays of constituent elements – of the khipu itself. An understanding of the structure of these devices will be critical to our explication of how both space and time were configured within these objects. Figure 2 displays the basic structures of khipus. Figure 2 – The Structures of Khipus 4 What we could term the “backbone,” or “spine” of the khipu is the element labeled “primary cord” in Figure 1. This is usually the thickest and structurally most complex cord composing a khipu, as it is made up of multiple spun, plied and re-plied cords and is often wrapped with additional cords in complex and colorful arrays. To the primary cord are attached a variable number of other cords, so that when the primary cord is extended, horizontally (as in Figures 1 and 2), some cords are pendant from the primary (= pendant cords), while others leave the primary cord upward (= top cords), in the opposite direction from the pendant cords. Both pendant cords and top cords may have one or more levels of secondary, or “subsidiary,” cords attached to them. Finally, it is important to note that cords are generally knotted in a complex base-10, place value knotting system. Most khipus contain numerical information recorded by Inka state administrators, such as census figures, tribute and storehouse accounting data, etc. (Ascher & Ascher, 1997; Urton, 2003, 2017). As for dimensionality, khipus thus have the horizontal dimension of the primary cord and the up/down, vertical dimension of the pendant and top cords. On the subject of dimensionality, I also note that the object depicted in Figure 2 is a two-dimensional drawing of an object that, in reality, occupies three-dimensional space. That is, as noted, khipu cords are made of spun and plied threads of either cotton or camelid fibers occupying three-dimensional space. With this recognition of the various components of khipu spatial structures, we have established the core elements of a three-dimensional spatial coordinate system: left/right, up/down and depth. Beyond its three-dimensionality, however, we see the potential indicators of a fourth dimension, time, in the labelling of elements in Figure 2. That is, at the left side of the primary 5 cord, we see the label “End Knot.” On the right side, we see the label “Dangle End.” Now, it is the case, from long and close study of the organization of information recorded on the onethousand or so khipus that exist in museum collections around the world today (see Urton, 2017:261-4 for an inventory), that what is lableled in Fig. 2 “end knot” can more accurately be identified as the “beginning knot.” This is the case because it has been shown with considerable certainty that the “reading” of information on any given khipu began at what is the left side in Figure 2. This is the portion of the primary cord in which pendant cords are attached closest to the end knot (Note: If this khipu were to be rotated 180o in the horizontal plane, the beginning end would be on the right-hand side). On the opposite end of the primary cord, in the section referred to in Figure 2 as the "dangle end," there is on most samples a good deal of primary cord which is “empty;” that is, a section of the primary cord which does not carry pendant cords and, therefore, this end “dangles” some distance beyond the last pendant cords. It is known from various colonial sources written by Spaniards who viewed former Inka recordkeepers, called khipukamayuqs (“knot makers/organizers/animators”), manipulating these devices (e.g., see Garcilaso de la Vega, 1966 [1609-1617]:329-333; Cobo, 1979:253-255), as well as what can be viewed in several colonial era illustrations showing khipus being held by the khipu-keepers, that the dangle end was at the “end,” or termination, of what was recorded on the khipu. In addition, this dangling portion was employed as a final spiral wrapping for the khipu when it was turned on itself, in a spiral, for transport or storage (see the khipu in the right hand of the individual shown in Figure 3). A never-unwrapped, spiral khipu is shown in Figure 4. 6 Figure 3 – Provincial Administrator with an Open Khipu (in his left hand) and a Spiral-Wrapped khipu (in his right hand; from Guaman Poma de Ayala, 1980 [1585-1615]: 348) Figure 4 - Spiral-wrapped khipu (Dumbarton Oaks, Conklin Donation; PC.WBC.2016.070) With this recognition of the “beginning” and “end” of a khipu record, we have an indication of time, or a relation of temporality (beginning/end), as well as a relationship of precedence, or succession, which is embedded in the structure of the khipu itself. Adding the dimension of time to the spatial dimensions of horizontal (left/right) and vertical (up/down) 7 space, along with depth, we have established that the khipu was a four-dimensional structure – a mobile, manipulatable, micro-cosmic system of space/time coordinates. The examples we have viewed of khipus to this point have been relatively simple, in terms of their size and complexity. However, we can see how complex these objects can become by viewing Figure 4. In this sample, the primary cord runs horizontally through the center of the image. Many pendant cords hang down from the primary cord in differently colored cord groupings; one top cord arises from each group of pendant cords. All cords shown in oblique angles are secondary, or subsidiary, cords attached either to the pendant cords or to the top cords. Figure 4 – A complex Khipu (Conklin Donation; Dumbarton Oaks, Washington, D.C.; PC.WBC.2016.072. UR297 in Harvard Khipu Database) 8 Although I have stated that the reading and interpretation of khipus, at least as practiced by modern researchers, goes from the beginning knot end of the primary cord to the dangle end, we cannot say for certain that khipus were either recorded or read in such a strict, linear fashion. There is enough evidence – e.g., in the sequencing of cords by grouping and color spacing and, to some extent, in the organization of numerical values knotted into samples (see below) – to give us some confidence in this reading order; however, we have no deeply informed early Colonial sources who, from their familiarity with Inka khipu-keepers, inform us specifically with respect to the reading and/or recording directions. It may also have been the case that the recording and reading of information along a primary cord proceeded in different directions – e.g., from both ends to the center. In addition, readings might have proceeded leftto-right, or right-to-left, or even by skipping from cords in one section to those in another, in a seemingly random manner. We cannot be certain of how cords on any given khipu were composed; that is, whether the cords were produced in some pre-determined plan of recording and were then tied to the primary cord all at once, or if, perhaps, the different cord groupings of any given khipu accumulated over time. All we have are the completed cords that were present on each sample at the time it was discovered and transferred into a museum collection (e.g., see Loza, 1999, on the large khipu collection in the Ethnologische Museum, in Berlin). Given that we cannot unambiguously “read” anything other than the numerical data recorded on 99.9% of khipus, we are often left having to rely on a form of structural logic to determine recording/reading order, or, in some cases, by following the logic of numerical information (e.g., addition going in one direction, subtraction in another, etc.; see below). 9 To appreciate the problems of interpreting khipu configurations outlined above, the reader may view an even more complex sample than that shown in Figure 4 (see Figure 6). The sample in Figure 6 is a very large cotton khipu, composed of a total of 332 pendant cords, in the collections of the Museum of World Cultures, in Gothebörg, Sweden. This khipu, which I studied in 2005, is said (in museum records) to have been recovered in Nazca, on the south coast of Peru. It is identified as UR113 in the Khipu Database. Figure 6 – Cotton Khipu from Nazca, south coast of Peru (Museum of World Cultures; Gothebörg, Sweden; #1924.18.0001) Figure 7 is a schematic drawing showing the different cord groupings of the khipu in Figure 6. In the schematic, the total length of the khipu will be realized if we attach the left end 10 of the bottom row of images and notations onto the right side end of the top row. Figure 7 – Schematic of cords on Khipu UR113 from Gothebörg, Sweden (shown in Fig. 5) Some explanation of notations in the schematic of Figure 7 may be helpful to the reader. The term “banding,” which appears along the top of several cord sections, refers to a kind of cord color organization, in which one sees groups of cords in bands of different colors (e.g., 6 white cords, followed by 6 light brown cords, then 6 dark brown cords, etc.). Color banding has been shown to contain cord groupings with low numerical values. It is thought that such values pertained to individual-level data (e.g., household census information). Such color patterning in pendant cords is opposed to that displaying large, aggregate- or group-level numerical values, which are indicated by cord "seriation" -- series of repeating cord colors, such as: dark brown, medium brown, light brown, while; repeat (see Clindaniel, 2018:94-114). 11 Immediately below the schematic drawing of the various cord groupings in Figure 7 is an indication of the general magnitude of numerical values knotted into the cords in the respective sections. Below these is a series of numbers from 1 to 31 indicating the serial order of different cord groupings composing this khipu; below that is an indication of the number of cords within each cord grouping; and, at the bottom, the serial numbering of all cords from #1 (top left) to #332 (bottom right). As we know from previous khipu studies (e.g., Ascher & Ascher, 1997; Clindaniel, 2018; Urton, 2003, 2017), much of the information encoded in khipus took the form of numerical data knotted into the cords that pertained to various administrative functions and protocols of the Inka state bureaucracy. Although we do not have something like “time stamps,” indicating when any given cord or group of cords was/were attached to a primary cord, there is no reason to insist dogmatically on one position or another – that is, that the cords were attached all at once, or that they accumulated over time, and that they may have even changed in some more dynamic fashion, with some being added, others removed, etc., over time. The point is that, in principle, there is no reason a priori to insist on, nor to eliminate, any of a number of possible modes of cord formation in khipu construction. What we may conclude from the above observations is that there is no reason to assume that these records were stable, fixed and static -- i.e., that they would not have been manipulated and changed over time (this position has been argued cogently by Frank Salomon; personal communication, 2010). If that were to have been the case, this would make the task of their interpretation considerably more straightforward, in that whatever configuration of cords we encountered in viewing a khipu in a museum collection could be presumed to have been the 12 complete record entered in one recording session by an Inka administrator. Instead, we have to allow for the equal possibility that each khipu was created in a process that was dynamic and ever-changing, producing different configurations over time but which got frozen in time, when the sample was placed in a burial or was otherwise taken out of active use. Given this uncertainty, we are cautioned against becoming dogmatic on any given position concerning the temporality of khipu formation and production. The point here with respect to our theme of time/space is that khipus, while static and fixed today, may very well have been much more changeable and dynamic in their construction and use in Inka times. If we do allow for the various positions outlined above, the question becomes whether or not there are any examples in which we see clear, concrete, interpretable expressions of time and space in one or more khipus preserved today? I argue below that there are, and that from study of a few such examples, we can gain unique insights into indigenous Andean understandings and representations of space and time. Calendrical Time One of the most straightforward expressions of time in material culture takes the form of calendars – an organized construction for counting or marking off the passage of days over monthly and annual time periods. We have one extraordinary such calendar in a khipu found in the north of Peru, in a cultural region known as Chachapoyas, at the site of Lake of the Condors, in Amazonas, northern Peru (Guillen, 1999; Urton 2008; von Hagen, 2002). In a rock overhang high above the lake, local workmen found a group of seven chullpas (“burial chambers”) containing some 250 mummy bundles in addition to 32 khipus and a wealth 13 of other burial goods (textiles, wooden sculpures, etc.; see Urton, 2001; von Hagen, 2002). I will focus here on one main khipu sample, which I designate as UR6 (see Figure 8), as well as two additional, smaller khipus (UR9 and UR21). These three khipus will give us unique insights into the construction of calendrical time (esp. by months and years) in the Inka world. Figure 8 – Khipu UR6 (the “Calendar khipu”) from Chachapoyas, northern Peru The principal form of cord organization in UR6 is in paired sets of pendant and what I have termed "loop pendant" cord groups. The latter are groups of pendant cords attached not directly to the primary cord, but rather to a looping thread the two ends of which are attached to the primary cord (Figure 9). 14 Figure 9 - The Organization of Pendant and Loop Pendant Pairs on Khipu UR6 Pendant groups are usually composed of 20-to-22 cords, while loop pendant groups are composed of 8-to-10 cords suspended from loops hanging down from the main cord. The most common arrangement is 21 (pendants) + 9 (loop pendants); this produces, on average, 24 groups of about 30. The numbers of cords in the 24 pendant and loop pendant sets along the primary cord of UR6 is shown in Figure 10. (The several groups of cords circled at the bottom of the khipu in Fig. 9 show a different organization from those included in the two-year calendar count.) 15 Numbers of cords in pendant & loop pendant groupings Total cords: 754 Figure 10 – The layout of pendant cords on UR6 16 As I have noted, at the lowest level of cord organization, the 730 cords are the product of 24 paired sets of “pendant” and “loop pendant” cord groups, which average 21 cords for the former, 9 cords for the latter. Together, these two cord groupings average 30 cords/set. As 30 is close to the number of days in the synodic lunar month (= 29.5 days), 24 such units total 720 (days). However, as we see in Figures 9 and 10, several of the groups contain more than 30 cords/days, such that the total number of days/cords is 730, which is very close to two annual solar years (365 x 2 = 730). The two-year calendrical structure of UR6 is shown in Figure 11. Figure 11 – The Organization of cord groups in the Two-Year Calendar of Khipu UR6 17 The two-year period of 730 cords/days composing this khipu is divided into two unequal “annual” periods of 362 and 368 cords/days. These two cord sections are each further subdivided into two half-year periods of (respectively) 179 and 183 cords/days for what I have termed “Year One,” and two half-year periods of 185 and 183 cords/days composing “Year Two.” I would also note that both the pendant and loop pendant cords are knotted with numerical information in the base 10 (Quechua) system of registering decimal-based information used by Inka administrators. I will discuss these numerical data later. The construction of the two-year Chachapoya calendar shown in Figure 11 reflects a structure and organization of calendrical periods common to Western calendars (e.g., 12 approximate synodic month periods = one year). The principal reflection of an Andean sensibility in this particular calendar construction is the fact that the calendar is not of a single year – as is true, for instance, of the calendar we might have hanging on our wall. Rather, it is a two-year, or dual annual period construction. As anyone familiar with Andean forms of organization will attest, dualism was a ubiquitous feature of Andean forms of organization. This value, therefore, is represented in the calendrical construction in Figure 11. That said, there is in fact another way to understand the construction of the calendar of khipu UR6 that is even more profoundly dual – and Andean. I pursue that description in what follows. The Construction of the Chachapoya Two-Year Calendar in an Andean Mode A remarkable development in our studies of the khipu archive at the Lake of the Condors site, was that two other khipus, designated UR9 and UR21, are close copies of each other (i.e., in terms of numbers of cords and values knotted into the cords) and both display virtually 18 identical cord and knot information recorded in a four-month section in the second year of the calendar khipu, UR6 (Figures 12 and 13). Cord groups tied together in khipus UR9 and 21 record essentially the same data recorded in the sections tied as "loop pendants" in khipu UR6. Fig. 12- UR9 Fig. 13- UR21 From the recognition of the status of khipus UR9 and UR21 as recording data almost identical to those in one four month period of UR6, I have hypothesized that a total of six pairs of “source” khipus would have provided the data from which UR6 was constructed. That is, since UR9 and UR21 provided the data for one four-month period of the two-year calendar, logically, there should be a total of six pairs of khipus in all, each pair of which would have been related to (i.e., the source for) a different four-month period, totaling 24 months (or two years). My hypothetical reconstruction of these circumstances and values is shown in Figure 14. Figure 14 – The Source Pairs for the Six Four-Month Periods of the Calendar Khipu I should clarify that the khipu pairs shown with white labels (i.e., without khipu identifications) in Figure 14 were not recovered from the Lake of the Condors site. Thus, these are hypothetical (yet to be recovered?) pairs of khipus. 20 When we re-present UR6 as shown in Figure 14 recognizing its construction as a product of six pairs of khipus, each accounting for one four-month period, we derive an alternative calendrical construction and organization to that shown earlier, in Figure 11 (see Figure 15). Year 1 Year 2 Figure 15 – The two-year calendar as a composition of six four-month periods In Figure 15, we see a hypothetical reconstruction of the six pairs of khipus, each of which provided information for one four-month period of the complete calendar of UR6. As we know from many other sources and circumstances (e.g., Urton, 2017:39), pairs of khipu-keepers were a common feature of accounting in Andean societies. In such a pairing, one would usually be the dominant member of the pair (= hanan, “upper”), the other would be subordinate (=hurin, “lower”). Such a dual pairing probably applied to the two years of the Chachapoya twoyear calendar, as well – one would have been considered upper, the other lower (it is not clear to me now which would have been which in the UR6 calendar). We see in Figure 15 that each year in this calendrical construction was composed of three four-month periods, rather than the one or two annual periods of the Western-like calendar organization shown in Figure 11. Although the two different ways of organizing time in the Chachapoya calendar of Lake of the Condors shown in Figures 11 and 15 both yield the same overall two-year periods (i.e., 362 + 368 = 730), I strongly suspect the construction in Fig. 15 would have made much more sense to either a Chachapoya or an Inka calendar specialist. The calendar would have been the social construction of six pairs of khipu-keepers, each of which was responsible for providing the information for one four-month period in the two-year calendar cycle. The calendar construction shown in Figure 15 is essentially a dual and quadripartite social construction, very much like the social and political organization of the space of Cuzco, capital of the Inka empire (Urton, 2022; Zuidema, 1964), as well as that in settlements across the empire as a whole. If this was indeed the case, we can conclude from this section that, at least in this one rather dramatic case, the structures of time as formulated in a khipu were (or could be) convergent, or homologous, with the structures of Inka space and society. 22 Pendant cord count on UR6 I have thus far discussed only the feature of the number and arrangement of pendant and loop pendant cords on khipu UR6, which are convincingly organized on the model of a twoyear calendar. However, in addition to their numbers and arrangement, the cords are also knotted in the decimal place value system of numerical registry discussed earlier. What numerical values are knotted into the cords? These are given in Figure 16. Before discussing the total numerical values on cords of the two years, I would note one curiosity, which is that the largest numerical values are registered on the cords of loop pendants, not on the pendant cords. I do not at present have an explanation for this circumstance. As we see in Figure 16, when summing the numerical values only on those cords organized by the (ideal) 21 + 9 patterns of pendants + loop pendants, the sum for Year 1 is 2,028; that for Year 2 is 934, which totals 2,962. When we add in the values knotted into cords not organized by the 21 + 9 pattern (see values with asterisk * in Fig. 16), this brings the total to 3,042. What I suggest is important here is the approximation of the values 1,000 and 3,000. In the Inka system of decimal administration, an important unit of organization -- especially in the organization of mit'a ["state laborers") -- was the waranka, a unit of 1,000 laborers. In an earlier study of khipu UR6 (Urton, 2001), I argued that the total numerical values registered closely approximated three warankas = 3,000. We have ethnohistorical testimony that a lord of the local Chilchos population, the cacique Guaman, was said to have been the lord of 3,000 Chilchos tributaries. Thus, I have argued, and re-state here, that I think khipu UR6 was a record of the two-year organization of tribute labor by the 3,000 Chilchos subjects of the Inkas in the Chachapoyas region. Thus, we have a measure in this khipu of labor time. 23 Hypothetical numbers of mit'a laborers in each four-month section of khipu UR6 Figure 16 - numerical values knotted on cords of UR6 24 I would clarify that what the above interpretation implies is that labor time was being accounted by pairs of khipukamayuqs; each pair of accountants was responsible for organizing laborers over a four-month period in the two years accounted for in the calendar in UR6. I suggest that the pairs of accountants would have been related to each other by the common Andean/Inka principle of asymmetrical dualism: one accountant would have pertained to the upper ranked (Hanan) moiety, the other to the lower ranked (Hurin) moiety. The final accounting -- in UR6 -- would have been the values adjudicated, and duly registered, by the head khipukamayuq, who would have maintained khipu UR6. Time and Space in the Khipus on a Smaller Scale We have to this point looked at full khipu constructions. I want to turn now to constructions and operations that occur inside of khipus to argue that the space/time of khipus occurred not only in such long-term constructions as annual calendars, but in what we could term micro-processes – that is, in operations that take place inside the strings, knots and colors of khipus. These include, for instance, arithmetic and mathematical formulations. I begin with the simple proposition that time is implicated in the simplest arithmetic formula. For instance, the statement 150 – 10 = 140 is a formulation of an arithmetic operation that implies, or incorporates, space, directionality, and time. Now, it is true of course that in such a simple arithmetic statement as that just given, the result of this statement of subtraction takes virtually no time to calculate; one can perform the operation virtually unconsciously. Such an observation, however, belies the actual operation that must occur (although it may be in a split second) as the eye and brain process the quantities placed into such a formula. One cannot 25 know what is happening to this “150” until one is given the information that “10” is to be subtracted from it. In the manner in which Western arithmetic writes such a subtractive statement, the directionality of the operation is from left to right – the original value is on the left, the action to be performed on that value (“subtract 10”) is in the center, and the product, or the present state of what has become of that “150” now becoming “140,” is to the right. Time, space and directionality are all part of such a subtractive statement. We have a good deal of evidence of such statements, or of cord constructions representing such operations, in the Inka khipus. These occur in the layout of knots reflecting such arithmetic operations as that discussed above, as well as in the color coding of the directionality of such operations. I have stated above that the cords of khipus are knotted in complex numerical arrangements, the number base of which is a base-10 place-value system of knotting. This is illustrated in Figure 17. Figure 17 – The Base-10, Place-Value System of Knotting Khipu Cords 26 To be brief, summing numerical totals along any single cord is the result of the operation of addition, but now in the vertical dimension. Beginning at either the top or bottom of a cord, one sums the knots along the cord in their place value and derives the sum of the addition (Fig. 18). Figure 18 – Summing of knot values in the khipu base-10, place-value system In Table 1, I display four columns of series of numbers recorded on successive cords on two different “matching” khipus from the site of Inkawasi, in the Cañete Valley, on the south coast of Peru (Urton and Chu, 2015, 2019). I refer to these as “matching” khipus because they contain, for the most part, virtually identical numerical values reflecting certain arithmetic operations; those operations, however, are different, yet reciprocal, between the pairs of khipus. That is, as seen in Table 1, what is displayed as a subtractive operation among three values on one khipu, are also subtractive, but with a rearrangement of those same three values, on the “matching” khipu. For example, what we see in the shaded section of the second column, beginning at the top, is a sequence of three values on successive cords (first column) of khipu UR267A as: 141 [-] 15 [=] 126; a subtractive statement using these same values appears on three successive cords (fourth column) of khipu UR255, as: 141 [-] 126 [=] 15 (third column). 27 Table 1 – Matching Inkawasi Khipus with Complementary Arithmetic Statements Chu and I have argued that such pairs of khipus were formulated in a system of checksand-balances at the site of Inkawasi, which was a major storage center located in an Inka military installation in the Cañete Valley (Urton and Chu, 2015). The point here is that the numerical values constituting these arithmetic statements are composed of knots tied along cords that are spaced in the base-10 place-value knotting system of the khipus. As I have argued above, any such arithmetic formulation as those knotted into the khipus in Figure 18 both occupy space and require time in the statement of their arithmetic operation. Here, again, the khipu is understood to be a space/time construction. In his study of the Inkawasi khipu, Jon Clindaniel (2018) has shown that such operations as those outlined and discussed above were also composed in arrangements of cord colors. An example appears in Table 2. 28 Cord Number Cord Color 1 2 3 w AB AB:MB 8 w w 9 AB:MB 11 12 13 14 15 16 w w 7 AB:MB w w w Color Meaning Addition, + Subtraction, Result, = Addition , + Subtraction , Result, = Addition , + Subtraction , Result,= Addition , + Subtraction , Result, = Number on Cord 106 15 91 161 15 140 *Broken 206 15 191 238 15 223 Table 2 – Cord Color and Numerical Operations in Inkawasi Khipu UR267A (Clindaniel, 2018:85) Table 2 shows several cord series on the same khipu (UR267A) shown in the first two columns of Table 1, but from the beginning cords of that khipu (cords 1 to 16). Column 2 of Table 2 gives the cord colors of these 16 cords; column 3 states the arithmetic operations – i.e., “color meaning” – performed on these cords; and column 4 shows the numerical values knotted into the respective cords. As Clindaniel states the nature of the operations displayed in Table 2: …in cords 1-3, you can see that the color white (W) was used to designate addition (credit), the color amber brown (AB) was used to designate subtraction (debit), and the mottled color combination of amber brown and medium brown (AB:MB) was used to designate the result of the arithmetic operations…Therefore, the operations as designated by the colors would be: “+106-15=91” (2018:86). In his study, Clindaniel discovered that colors were commonly used in a system of marked and unmarked categories, with the former being more specialized, or unique, while the 29 later were more general, or common. In general, he found that lighter colors were unmarked, while dark colors tended to be marked. He found also that mottled colored cords (i.e., ones displaying a mixing of two or more colors/hues), were the highest ranked of the various cord color combination options available to a khipu-keeper (2018:86-88). Clindaniel went on to identify many more combinations and permutations than those shown above, in Table 2. The point here is that the use of spatial positioning in making arithmetic statements (e.g., one cord with a high value, the next cord with a low value, and the third cord with the result of subtracting the smaller value from the higher value) was complemented by a complex system of color marking -- together indicating operations on materials existing in space/time. Conclusions I began this study by stating that Andean studies has always had a problem with time, and explaining that circumstance by noting the absence of a system of writing in any PreColumbian civilization. The absence of writing notwithstanding, we subsequently went on to examine closely the structures and operations of the main recording device used in the Inka Empire, the khipu. Although I would not claim for the khipu the status of being a writing system, it is nonetheless the case that, on the basis of close studies of khipus in museum collections over the years, researchers have succeeded in showing that these objects held very detailed and complex forms of information (especially numerical) in their strings, knots and colors. Recent studies have carried our understanding of khipu recording to an even higher level, demonstrating that khipu recording involved not only complex arrays of numerical data, 30 but that by means of a complex system of “markedness” relations in color coding (see esp. Clindaniel, 2018; also see Urton, 2003 on markedness in khipu coding), the khipu contained even more complex data encoding than has been thought up until very recent times. Our focus in this paper, however, has been on the question of the ways in which we can identify space and time in khipu structures and coded data. In the course of this paper, we have seen unified space/time constructions at different levels. On a micro-level, we have seen how khipus are structured as four-dimensional objects which have the capacity to encode very complex forms of data. At a more general level, we have also seen how space/time coordinates were structured in a bi-annual calendar system found in an archaeological site in Chachapoyas, in the northern highlands of Peru. In short, I believe we can say without exaggeration that the khipu was a very efficient time/space coordinate system for constructing representations of space and time in the Andean world. I suspect that there are even more complex formulations of space/time embedded in these knotted-string devices that await discovery in the future. 31 References Ascher, Marcia, and Robert Ascher 1978 Code of the Quipu: Databook I and II. https://courses.cit.cornell.edu/quipu------. 1997 Mathematics of the Inca: Code of the Quipu. 1981; New York: Dover. 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