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

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Colored Strip Patterns and Front-Back Symmetry in Warp-Faced Pre-Columbian Textiles 305 T e c h n i c a l a r T i c l e Colored Strip Patterns and Front-Back Symmetry in Warp-Faced Pre-Columbian Textiles A n T hony Philli ps Among the textiles made by pre-Columbian weavers in Peru are in some cases, takes the colored decoration on one face to double-faced warp-faced bands, with two or more differently colored that on the other. yarns in each warp location, where figures woven on one face appear, Ed Franquemont has emphasized studying Andean weav- by warp substitution, in a different color on the other. Andean weavers ABSTRACT reconciled this difference with the aesthetic concern that the two faces ings as three-dimensional objects. He writes: “Depending on be as similar as possible by exploiting the symmetries of colored whether the cloth is warp-faced or weft-faced, the back side strip patterns. can be a horizontally reflected copy of the pattern in reverse color scheme or a reflection of the pattern in the same color. We might say then that for Peruvians, the fabric plane has an WeAving And Mathematics intrinsic reversing quality” [5]. in PRe-ColumbiAn PeRu Franquemont goes on to say: “This process resonates with The inhabitants of pre-Columbian Peru and neighboring the deep-seated Andean sense of reciprocity that balances territories were world-class, all-time masters of weaving opposed pairs” [6,7]. More generally, the Andeans liked de- and textile decoration. Since these cultures left no written signs that can be read upside-down, as well as objects that records, we know very little about them besides hints from look the same from both sides. Examples [8] date at least as these textiles and from their ceramics. In particular we know far back as early Paracas (sixth–third century BCE). very little about pre-Hispanic mathematics. We know that Preliminary versions of this work appeared in Phillips the Incas (c. 1400–1550 CE) had a base-10 numerical sys- [9,10]. tem [1] where numbers were encoded by means of knots. We also know, from analysis of patterns in the embroidered WeAving WiTh MulTiPle Warp Sets decoration of textiles produced by the Paracas culture (c. 200 on A Back-Strap looM BCE–c. 200 CE), that the designers had a deep operational The main elements of a back-strap loom (see the online understanding of planar symmetries [2,3]. supplemental materials I) are constant throughout Andean The aim of this work is to show how an additional kind of weaving history [11]. The warps are strung between a fixed mathematical symmetry was worked into two-faced warp- bar and another held by the back-strap. The weft is laid, faced (see below) textiles, typically belts or straps [4], or strand by strand, by passing the shuttle across the loom, bands in a larger textile, where from one face to the other over and under the warps [12]. If the warps hide the wefts the shape of the decorating figures is the same (except that because of their bulk or their spacing, the fabric is called they appear in mirror image) but the colors must be differ- warp-faced [13]. The warp-faced textiles considered here are ent. The symmetry to be investigated is the requirement that woven with two or more different-colored sets of warps [14] when the item is turned over, one face is as similar to the other (yarns of more than one color in each warp location), woven as possible in the following specific sense: Imagining the pat- complementarily [15]: Where one color appears on one face, terns to extend indefinitely in both directions beyond the a different color must appear on the other. length of the band, there is a global symmetry (translation, rotation or reflection) that, along with cyclic order reversal PeRiodiC PatteRnS in AndeAn deSign Note: In what follows, “horizontal” refers to the orientation Anthony Phillips (mathematician), Mathematics Department, Stony Brook University, of the warps. A horizontal reflection is a reflection in an axis Stony Brook, NY 11794-3651, U.S.A. Email: [email protected]. Website: www.math.stonybrook.edu/~tony. ORCID: 0000-0001-7000-7847. parallel to the warps. The two faces of a textile are related See www.mitpressjournals.org/toc/leon/53/3 for supplemental files associated by (physically) flipping across a horizontal axis. Images will with this issue. follow this convention unless otherwise specified. 304 LEONARDO, Vol. 53, No. 3, pp. 304–308, 2020 https://doi.org/10.1162/leon_a_01691 ©2020 ISAST Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01691 by guest on 28 September 2021 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.
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