Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01691 byguest on 28 September 2021 LEONARDO, Vol. 53, No.3,pp.304–308,2020 304 ABSTRACT rotation or reflection) that, along with cyclic order reversal order cyclic with along that, reflection) rotationor (translation, symmetry global a is there band, the of length the beyond directions both in indefinitely extend to terns as possible in the following specificsense: Imaginingthe pat - when the item is turned over, ent. The symmetry tobe investigated isthe requirementthat differ be mustcolors the butimage) mirror in appear they that (except same the is figures decorating the of shape the other the to face one from where textile, or larger a [4], in bands straps or belts typically textiles, below) (see faced warp- two-faced into worked was symmetry mathematical understanding of planar [2,3]. symmetries operational deep a had designers the that CE), 200 BCE–c. decoration of textiles produced by the Paracas culture (c. 200 Weknow, also from analysis of patterns embroidered the in knots. of means by encoded were numbers where [1] tem sys numerical base-10 a had CE) 1400–1550 (c. Incas the that Weknow mathematics. pre-Hispanic about little very these textiles and from their ceramics. In particular we know from hints besides them about little very know werecords, written no left cultures these Since decoration. weaving textile and of masters all-time world-class, were territories neighboring and Peru pre-Columbian of inhabitants The P in We strip patterns. be assimilarpossiblebyexploitingthesymmetriesofcolored thatthetwofaces withtheaestheticconcern reconciled thisdifference colorontheother.by warpsubstitution,inadifferent Andeanweavers ineachwarplocation,wherefigureswovenononefaceappear,yarns colored double-faced warp-facedbands,withtwoormoredifferently Among thetextilesmadebypre-ColumbianweaversinPeruare in Warp-Faced Pre-ColumbianTextiles and Front-Back Symmetry Colored StripPatterns l a c i n h c e T Anthony with thisissue. See www.mitpressjournals.org/toc/leon/53/3 forsupplementalfilesassociated Website: Stony The aim ofthis work is to show how an additionalkind of a

r Brook, ving ving

e-Colu

Phillips www.math.stonybrook.edu/~tony.

NY n A

(mathematician),

a 11794-3651, nd M nd mb t i l l i h P y n o h A i a r at n Pe n t

U.S.A.

he e l c i Mathematics mat r

Email: u one face is as similar to the other

ORCID: i

cs [email protected].

Department,

0000-0001-7000-7847.

Stony S P

Brook

University,

- - follow convention this specified. unlessotherwise by(physically) flipping acrosshorizontal a axis. Imageswill related are textile a of faces two The warps. the to parallel of the warps. A horizontal reflection is a reflection in an axis Note:follows,whatIn“horizontal” orientation the refersto Pe a different color must appear the on other. complementarily (yarns of more than one color in each warp location), woven woven with two or more different-coloredsets of warps [14] warp-faced called [13]. The warp-faced consideredtextiles here is are fabric the spacing, their or bulk their of because wefts the hide warps the If [12]. warps the under loom, and over the across shuttle the passing by strand, laid, by is strand weft The back-strap. the by held another and bar fixed a between strung are warps The [11]. history weaving online the Andeanconstantthroughout are I) supplementalmaterials (see loom back-strap a of elements main The on We [9,10]. far back as early Paracas (sixth–third centuryBCE). Examplessides. datefromsame [8] both at asthe least look that objects as well as upside-down, read be can that signs pairs”opposed More[6,7]. generally, de- Andeans liked the balances that reciprocity of sense Andean deep-seated the intrinsic reversing quality” [5]. We might say then that for Peruvians, the fabric plane has an color scheme or a reflection ofthe pattern inthe same color. can be a horizontally reflected copy ofthe pattern in reverse cloththe whether oris warp-faced weft-faced,theback side ings as three-dimensional objects. He writes: “Depending on that on other. the to face one on decoration colored the takes cases, some in Preliminary versions of this work appeared in Phillips Phillips in appeared work this of versions Preliminary Franquemont goes on to say: “This process resonateswith Ed Franquemont has emphasized studying Andean weav- r a iodi a ving ving B ac c P k-S w att https://doi.org/10.1162/leon_a_01691 i t trap h Mul h e [15]: Where[15]:onecolor appearsone face, on r n s

L in Ande in oo t i p le W le m arp a n n Se D e ts s ign

©2020 ISAST Pre-Columbian straps and bands are typically decorated terns to construct warp-faced textiles where the two faces with repetitions of a single motif, which may have reflective are as similar as possible, as formulated in the introduction. or rotational symmetry of its own [16]. One copy of the mo- With four-colored patterns, the faces are as similar as pos- tif may be rotated or reflected with respect to the next. The sible up to a possible reversal of the cyclic order of the motifs. combination of these two effects is called a strip pattern. Of See supplemental materials III for a mathematical analysis of the 17 strip patterns listed by Branko Grünbaum and G.C. the phenomenon. Shephard [17], the ones that occur in the textiles considered here are PS1, 2, 4, 6, 7, 8, 11, 12 and 15 in their notation. Two Complementary Colors For our purposes, they can be grouped into four classes The two-colored strip pattern PSk[2], implemented by warp- (see Table 1): faced weaving with two-color complementary warps, gives a. Consecutive images are related by a horizontal two faces as similar as possible. The global symmetry from ­reflection (symmetry group [18] pxax or pxmx). one face to the other depends on the class of the strip pattern as follows [21]: b. Consecutive images are not related by a horizontal reflection but are related by a 180° rotation 1. Class a: translation. (symmetry group px12). 2. Class b: translation and vertical reflection. c. Consecutive images are not related by a horizontal 3. Class c: translation and 180° rotation. reflection nor by a 180° rotation but by a vertical 4. Class d: translation and horizontal reflection. reflection (symmetry group pm11). Figure 1 shows an example from class d. d. Consecutive images are only related by translation See supplemental materials I for patterns for k = 2, 4, 7, (symmetry group p111). 8, 11. Published examples include k = 2 [22]; k = 6 [23]; k = Consecutive images may also be colored differently. A se- 8 [24] and in complementary weft weaving [25]; k = 12 [26] quence of n colors repeating periodically along the strip gives and k = 15 [27]. an n-colored strip pattern. We will use a slight modification [19] of Grünbaum and Shephard’s notation PSk[n] [20] for PSk (as above) with n colors. In those terms, this work gives pictures or citations of examples of textiles with colored strip patterns PSk[2] for k = 1, 2, 4, 6, 7, 8, 11 and 15; as well as PSk[3] for k = 1, 4, 7; PSk[4] for k = 1, 2, 7; and one example of PS4[6], to support the following thesis: Pre-Columbian Andean weavers exploited two- and three-colored strip pat-

TABLE 1. The nine strip patterns manifest in this corpus, listed along with their corresponding frieze symmetry group and their a, b, c, d class. Fig. 1. PS1[2]. A and B: both faces of a belt (cotton, width 13 cm, black and red complementary warps [36]); image C, the horizontal reflection of A, Strip Pattern Class Diagram differs by a translation from B. (Penn Museum item 31487, from Sun Temple, PS1 (translation, p111) d p p p p . . . Pachacamac. Courtesy of the Penn Museum.)

PS2 (glide reflection, p1a1) a p b p b . . .

PS4 (translation, horizontal reflection, a E E E E . . . p1m1) Three Working Colors

PS6 (translation, vertical reflection, c M M M M . . . Anne Pollard Rowe remarks, “The warp-patterned weaves pm11) in which the imagination and skill of Andean weavers are PS7 (translation with 180° rotation, b p d p d . . . most clearly manifest are those involving more than two sets p112) of warps” [28]. An example is their solution to the problem PS8 (translation, 180° rotation, p112) b S S S S . . . of the fabric having only two faces. The weaver makes two passes of the shuttle at each weft level (Fig. 2). On the first, PS11 (translation, vertical reflection, a M W M W . . . translation with 180° rotation, glide she picks at each crossing the warp in that location that will reflection, pma2) show on the bottom of the fabric; returning, she picks the one

PS12 (translation with vertical a p q p q p q that will show on the top. The remaining warps remain hid- reflection, horizontal reflection, den in the interior of the fabric. The resulting structure has 180° rotation, pmm2) b d b d b d been called tubular weave [29], because the spiraling wefts PS15 (translation, vertical reflection, a X X X X . . . enclose a kind of flattened tube. Now there is a choice (warp horizontal reflection, 180° rotation, substitution [30]), at each pick, of which colors are shown pmm2) and which are hidden.

Phillips, Colored Strip Patterns and Front-Back Symmetry in Warp-Faced Pre-Columbian Textiles 305

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01691 by guest on 28 September 2021 For all textiles woven with three or more sets of warps in our corpus there is a terminal area where the decorative pattern is extended by a series of stripes, due to the diffi- culty of performing complicated picking operations when the work is near the far end of the loom [31]; so the direction of weaving is known. (All textiles from here on in this work are shown woven from left to right, unless otherwise stated). It is reasonable then to choose face F so that the permutation Fig. 2. Tubular weave: exploded diagram of six round trips of the weft at one is 1à3à2à1: The color to be woven on the bottom is the color warp location. In this example there are five sets of warps: Pink b and green c of the last motif woven on the top, giving a visual reminder substitute for each other as complementary to red d. Blue yarns (a, a’) serve as outlining, separating blocks of the three working colors [37]. Unused warps of what should be happening on the hidden side of the work are hidden in the interior of the textile. (© Anthony Phillips) [32]. With this arrangement, we can speak of a “last-goes- under” substitution scheme. The three-colored strip pattern PSk[3], implemented by If both faces (call them F, G) of a warp-faced two-faced warp-faced weaving with complementary warps of three col- textile, with yarns representing n different working colors ors, gives two faces as similar as possible. The dependence (1, 2, . . . , n), carry an n-colored strip pattern, then there of overall symmetry on strip pattern class is translation for must be a fixed one-to-one correspondence from the colors class a and translation with horizontal reflection for classes on face F to those on face G; i.e. a permutation of the set {1, b, c and d [33]. Figures 3–5 show examples of classes a, b 2, . . . , n}. Complementarity implies that the same working and d. color cannot appear on both faces: The permutation must be transitive (moving every element). For n = 2 there is only one such permutation, 1ßà2. For n = 3 there are two possibilities, 1à2à3à1 and 1à3à2à1. These permutations are inverses: We observe one or the other depending on which face we take to be face F.

Fig. 3. PS4[3] [38]. The two faces of a strap (camelid fiber, 56 × 4 cm, probably dated 1000–1450 CE [39]), five yarns in each warp location: yellow, red, white with two brown for background). The reverse of the textile has, up to a translation, the same design as the front. (© Anthony Phillips) Fig. 5. PS1[3]. A and B: both faces of a belt (camelid fiber, 58 × 13 cm [40]); C: the reflection of A, differs from B by a translation (the weaver was not systematic in the color of the birds’ eyes). Motif is a diagonal strip of birds, anchored to the background of the next strip. Tubular weave, five yarns in each warp location (yellow, black, pink with two red for outlining). (© Anthony Phillips)

Fig. 4. PS7[3]. A and B: both faces of a belt (camelid fiber, 64 × 8 cm). C: the reflection of A, differs from B by a translation. This belt is faded in places but enough color is conserved to determine a pattern. Each diagonal motif has feline heads and birds on either side of a solid line. Consecutive bands are rotated 180° from each other, so that the feline heads (birds) of one band interlock with the feline heads (birds) of the next. Construction and colors are the same as in Fig. 5. (Note that since the color scheme has order 3 and the Fig. 6. PS1[4]. A and B: both faces of a belt (camelid fiber, 71 × 10 cm [41]). rotation has order 2, the total pattern has period 6.) (© Anthony Phillips) C, the reflection of A, differs from B by a translation. Motifs are diagonal bands representing fish, in the cyclic sequence white-with-black-eyes, black-with-purple- eyes, purple-with-black-eyes, black-with-white-eyes. With red for outlining, five yarns in each warp location. (© Anthony Phillips)

306 Phillips, Colored Strip Patterns and Front-Back Symmetry in Warp-Faced Pre-Columbian Textiles

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01691 by guest on 28 September 2021 Three Working Colors, Four Color Schemes In many textiles a four-colored strip pattern is generated with only three working colors p, q, r by grouping one color (the “distinguished” color, say p) with the other two so as to give Fig. 7. PS2[4]. Both faces of a strap (camelid fiber, 51 × 2.5 cm). The design four different color schemes appearing in the cyclic sequence repeats glide-reflected fish in the coloring sequence yellow-with-black-eyes, (p/q, q/p, p/r, r/p). For example, the fish in Fig. 6 and the birds black-with-yellow-eyes, pink-with-black-eyes, black-with-pink-eyes, etc. With red for outlining, five yarns in each warp location. The two faces are identical up in Fig. 8 are colored, in cyclic order (1) black-with-white- to translation, except that the cyclic order of colorings is reversed. This shows in eyes, (2) white-with-black-eyes, (3) black-with-purple-eyes, the color of the eyes. (The first fish do not follow the pattern, in orientation nor (4) purple-with-black-eyes (so black is the distinguished in color.) (© Anthony Phillips) color). Characterizing a warp substitution pattern as a per- mutation of {1, 2, 3, 4}, then 1 and 3 cannot be interchanged, nor can 2 and 4, by complementarity. Checking cases shows that the only permutations possible are the cyclic permuta- tion 1à2à3à4à1 with its inverse 1à4à3à2à1 and the order-2 permutations 1ßà4, 2ßà3 and 1ßà2, 3ßà4. In the first case, as before, we choose a top face so that the cyclic permutation is 1à4à3à2à1 or last-goes-under. Symbolically, the three sub- stitution possibilities are: I p/q à r/p à p/r à q/p à p/q II p/q ßà r/p p/r ßà q/p III p/q ßà q/p p/r ßà r/p [34]. The four-colored strip pattern PSk[4], implemented as de- scribed above by warp-faced weaving with complementary warps of three colors in four color schemes, gives two faces as similar as possible, just as for n = 2, except that with substitu- Fig. 8. PS7[4]. A and B: both faces of a belt (camelid fiber, 58 × 13 cm [42]). C: the vertical reflection of A, differs from B by a translation. Motifs are similar tion I for strip pattern classes b and c, and with substitutions to those in Fig. 4 except for relative size. (The first four elements have zigzags II and III for classes a and d, the cyclic order of motifs is instead of heads.) Adjacent motifs, rotated 180o from each other, interlock. reversed [35]. Color schemes and their sequence are the same as in Fig. 6. Figure 6 shows color substitution I, class d. Figure 7 shows color substitution II, class a. Figure 8 shows color substitu- tion III, class b.

Acknowledgments /common/image_dup.cfm?catno=41%2E2%2F%205104 [accessed 24 November 2017], probably the back-straps for back-strap looms Thanks to Anne Pollard Rowe for valuable advice and to Bill Casselman according to Adele Cahlander with Suzanne Baizerman, Double- for help with photographs. Woven Treasures from Old Peru [St. Paul, MN: Dos Tejedoras, 1985] p. 24). “Straps” are about 40 cm long and 4 cm wide, and often­occur References and Notes as carrying straps for handbags. See Cahlander and Baizerman p. 27 and plate 10; Ann Pollard Rowe, Warp-Patterned Weaves of the Andes 1 Gary Urton, Signs of the Inka Khipu: Binary Coding in the Andean (Washington, DC: Textile Museum, 1977) p. 19; Denise Y. Arnold and Knotted-String Records (Austin, TX: University of Texas Press, 2005). Elvira Espejo, The Andean Science of Weaving: Structures and Tech- 2 Branko Grünbaum, “Periodic Ornamentation of the Fabric Plane: niques of Warp-Faced Weaves (London: Thames & Hudson, 2015), Lessons from Peruvian Fabrics,” in Dorothy K. Washburn and Don- Fig. 325a; Raoul d’Harcourt, Textiles of Ancient Peru and Their Tech- ald W. Crowe, eds., Symmetry Comes of Age: The Role of Pattern in niques (University of Washington, 1962) Plate 20; Branko Grünbaum Culture (Seattle and London: University of Washington Press, 2004); and G.C. Shephard, Tilings and Patterns (New York: W.H. Freeman for an earlier version, see Symmetry 1 (1990) pp. 45–68. and Company, 1987) p. 68. 3 Anne Paul, “Symmetry Schemes on Paracas Necropolis Textiles,” 5 Ed Franquemont, “Jazz: An Andean Sense of Symmetry,” in Wash- in Dorothy K. Washburn, ed., Embedded Symmetries: Natural and burn [3] p. 83. Cultural (University of New Mexico Press, 2004) pp. 59–80. 6 Franquemont [5] p. 91. 4 “Belts” in this work refers to textiles 50–70 cm long and around 12 7 Rebecca Stone-Miller, To Weave for the Sun: Ancient Andean Textiles cm wide. These are often extended by cords and sometimes linked in the Museum of Fine Arts, Boston (New York: Thames and Hudson, two by two end to end (see the pairs referenced at American Mu- 1992) p. 124. seum of Natural History, anthro.amnh.org/anthropology/databases /common/image_dup.cfm?catno=41%2E2%2F%205103 [accessed 8 The decoration on several pots from that period has almost perfect 15 December 2017], and anthro.amnh.org/anthropology/databases 180˚ rotational symmetry. For example, see the specimen shown

Phillips, Colored Strip Patterns and Front-Back Symmetry in Warp-Faced Pre-Columbian Textiles 307

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01691 by guest on 28 September 2021 here: www.metmuseum.org/art/collection/search/308421?sortBy= 28 Rowe [4] p. 81. Relevance&ft=63232.4 (accessed 15 December 2017). See also Alan R. Sawyer, Ancient Peruvian Ceramics: The Nathan Cummings 29 d’Harcourt [4] p. 46; Emery [13] p. 155. Collection (New York: The Metropolitan Museum of Art, 1966) p. 76. 30 Rowe [4] p. 97. Rowe calls this structure “double-woven, comple- 9 Anthony Phillips, “Simmetrie Fronte-Retro in Manufatti Tessili del mentary-warp weave, with warp substitution.” Periodo Pre-Inca,” in Michele Emmer, ed., Matematica e Cultura 2007 (Springer-Verlag Italia, 2007). 31 Cahlander and Baizerman [4] p. 48; Rowe [4] p. 15. 10 Tony Phillips, “Inside-Out Frieze Symmetries in Ancient Peru- 32 Cahlander and Baizerman [4] p. 132. vian Writings,” American Mathematical Society (October 2008): www.ams.org/samplings/feature-column/fcarc-weaving (accessed 33 See supplemental materials III, proposition 2. 15 December 2017). 34 Substitution pattern III is analyzed and illustrated in Nikkibarla 11 A Moche fineline-decorated bowl, second to eighth century CE, Calonder and Judith Rickenbach, Textilien aus dem alten Peru: Die shows weavers with backstrap looms: www.britishmuseum.org/re Sammlungen der Abegg-Stiftung und des Museums Rietberg(Zürich: search/collection_online/search.aspx?searchText=Am1913%2C1025.1 Museum Rietberg, N.D.), where the working colors are called Mun- (accessed 15 December 2017). For a contemporary weaver, see sup- sterfarben, Fig. 19, p. 463. plemental materials I. 35 See supplemental materials III, propositions 3a and 3b. 12 For a detailed description see Anni Albers, On Weaving (Wesleyan Univ. Press, 1974) plate 5; Cahlander and Baizerman [4] p. 61. 36 Published as Cahlander and Baizerman [4], Chapter 3, Image 7.

13 Irene Emery, The Primary Structures of Fabrics (Washington, DC: 37 Calonder and Rickenbach [34]. Textile Museum, 1966) p. 76. 38 This three-colored strip pattern is labeled PS3[3] in Grünbaum and 14 In complementary warp weaving there can be as many as six; see Shephard [4] p. 414. supplemental materials I. 15 Emery [13] p. 150. 39 Lena Bjerregaard, Pre-Columbian Woven Treasures in the National Museum of Denmark, National Museum of Denmark, 2002 shows 16 Often, as in our Figs 1, 4, 5, 7 and 8, the motif can be extended beyond a strap, item O4223, with the same motif but with pattern PS4[2], the sides of the strip and then have a translational symmetry of its described as “from Pachacamac, 1000–1450 AD.” Also Arnold and own: The strip can be considered a slice of a two-dimensional tiling. Espejo [4] Fig. 325a,b shows a bag with a similar strap using three However, since there are no color changes in that new direction, working colors (yellow, red, purple) and brown outlining, but with nothing is lost by studying the pattern as purely one-dimensional. four color-schemes, pattern PS4[4], dated as Late Horizon Period (Museo Nacional de Etnologia y Folklore, La Paz, item 25769). Only 17 Grünbaum and Shephard [4] p. 224. one side is shown. 18 Dorothy K. Washburn and Donald W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis (Seattle and London: 40 American Museum of Natural History: anthro.amnh.org/anthropol University of Washington Press, 1988) p. 83; Grünbaum and Shep­ ogy/databases/common/image_dup.cfm?catno=41%2E2%2F%20 hard [4] p. 39. 5103 (accessed 15 December 2017), two linked belts, one like this one, described as from “Pachacamac?”; M.J. Jiménez Díaz, Tradición 19 See supplemental materials II for a concordance with the notations de tradiciones: Tejidos prehispánicos y virreinales de los Andes: La of Grünbaum and Shephard [4] and Washburn and Crowe [18] for colección del Museo de América (Museo de América, 2009) p. 311 (Inv. two-colored strip patterns. 02-5-189) identified as “Central Coast, Late Horizon/Inca (1450–1550 AD).” 20 Grünbaum and Shephard [4] pp. 408, 414. 41 Jiménez Díaz [40] pp. 315, 321 (Inv. 02-5-195 and 02-5-197), same 21 See supplemental materials III, proposition 1. identification as Fig. 5. 22 Bi To Kōgei (Beauty in Crafts) 38, Kyoto Shoin (16 January 1958), cover image; also examples of PS1[2] and PS7[2]. 42 American Museum of Natural History: anthro.amnh.org/anthropol ogy/databases/common/image_dup.cfm?catno=41%2E2%2F%20 23 Grünbaum and Shephard [4] p. 106. 5104 (accessed 24 November 2017), published in Cahlander and Baizerman [4] p. 30, and in Rowe [4] p. 97, where they are described 24 Sophie Desrosiers and Ilaria Pulini, Musei Civici di Modena: Tessuti as “Peru, Central Coast, Late Intermediate Period.” Two linked belts, Precolombiani (Modena, Italy: Franco Cosimo Panini, 1992) item 159, one like this one; Jiménez Díaz [40] p. 320 (Inv. 02-5-198), same p. 173 and item 161, p. 174. identification as Fig. 5. 25 Penelope Dransart and Helen Wolfe, Textiles from the Andes (Northampton, MA: Interlink Books, 2012) p. 75. The item is AM1862, 0611.2 in the British Museum. Manuscript received 17 July 2017. 26 Desrosiers and Pulini [24] item 162, p. 174; Arnold and Espejo [4] p. 256. Anthony Phillips is Professor of Mathematics, Emeritus 27 Cahlander and Baizerman [4] plate 10, after p. 38; Rowe [4] p. 51; at Stony Brook University. He received his PhD from Princeton Arnold and Espejo [4] p. 257. University in 1966.

308 Phillips, Colored Strip Patterns and Front-Back Symmetry in Warp-Faced Pre-Columbian Textiles

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01691 by guest on 28 September 2021