Cyclic Twill-Woven Objects Ergun Akleman1, Jianer Chen2, YenLin Chen2, Qing Xing3, Jonathan L. Gross4 1Visualization Department, Texas A&M University 2Computer Science Department, Texas A&M University 3Architecture Department, Texas A&M University 4Computer Science Department, Columbia University Abstract Classical (or biaxial) twill is a textile weave in which the weft threads pass over and under two or more warp threads, with an offset between adjacent weft threads to give an appearance of diagonal lines. This paper introduces a theoret- ical framework for constructing twill-woven objects, i.e., cyclic twill-weavings on arbitrary surfaces, and it provides methods to convert polygonal meshes into twill-woven objects. It also develops a general technique to obtain exact triaxial-woven objects from an arbitrary polygonal mesh surface. 1. Motivation Beyond its use in fabric design, weaving provides a wide variety of ways to create surface patterns that can be embodied in sculpture and in innovative architec- tural design. We focus here on twill-weaving, which can provide strength, durability, and water-resistance, along with interesting diagonal patterns. We describe Figure 1: Three biaxial twill-woven objects ob- methods to convert polygonal meshes into twill-woven tained by applying Catmull-Clark subdi- sculptures. vision to the same initial mesh. It has recently been shown (Akleman, et al. [3, 4]) how any given polygonal mesh can be transformed into 2. Introduction objects woven from ribbons of varying width, such that the ribbons cover the underlying surface almost com- It has recently been described [3] how any arbitrary pletely, except for small holes. The ribbons can be man- twist of the edges of an extended graph rotation system ufactured inexpensively by using laser-cutters on thin induces a cyclic plain-weaving on the corresponding metal sheets. The corresponding plain-woven sculp- surface. The method works for all meshes and is very tures are constructed physically by weaving these metal simple: just twisting all edges in the same way. On the ribbons. Mallos [26] has created large-scale plain- other hand, in order to construct other weaving struc- woven objects. tures we had to use two different types of twisted edges, With the design and construction of more and more characterize conditions to design the desired weaving unusually shaped buildings, the computer graphics structures, and develop algorithms to create exact and community has started to explore new methods to re- approximate versions of the weaving structures. Herein duce the cost of the physical construction for large we extend the mathematical model to twill-weaving, shapes. Most of the currently suggested methods fo- which is used in fabrics such as denim or gabardine. cus on reduction of the number of differently shaped Classical twill is a biaxial textile weave in which each components [31, 11, 12]. There exists a contemporary weft (filling) thread passes over and under two consec- interest among architects to explore weaving as an al- utive warp threads, and each row is obtained from the ternative construction method [23, 13, 18] based on tra- row above it by a shift of one unit to the right or to the ditional bamboo-woven housing [19, 20, 29]. This sug- left. The shift operation creates a diagonal pattern that gests how weaving with ribbons from thin metal sheets adds visual appeal to a twill fabric or to a twill-woven can also be useful for economical construction of com- object, as illustrated in Figure 1. plicated shapes. In what follows, we define a twill-weaving as a type Preprint submitted to SMI 2011 March 20, 2011 of cyclic weaving structure on general surfaces. Based symmetry group acting on the strands. Twill weav- on this definition, we identify three mesh conditions ing belongs to a certain family of isonemal fabrics in that are collectively necessary and sufficient to obtain a which each weft row of length n (the period) is obtained twill-weaving from a given mesh. In fact, many meshes from the weft row immediately above it by a cyclical do not satisfy these three conditions, which implies that shift of s units (the offset) to the right, for some fixed it is impossible to obtain an exact twill for them. In- value of the parameter s, such that n and s are relatively tuitively, we may expect that a mostly-(4; 4) mesh, i.e., prime. If s = ±1, the fabric is called twill [17]. More a mesh with large areas of quadrilaterals with 4-valent generally, the resulting fabrics are called (n; s)-fabrics. vertices, would admit a reasonably good twill. Indeed, This family of fabrics also includes plain-weaves (with we have developed an edge-coloring algorithm that will n = 2; s = 1) and satins. create an exact twill whenever the mesh is twillable. Twills are widely used in clothing fabrics, for in- Even if the mesh is not twillable, the output of the al- stance, in denim or gabardine. Their characteristic di- gorithm satisfies most of the twill conditions, and it agonal pattern that makes the weaving visually appeal- exhibits the characteristic diagonal pattern for mostly- ing. Since twill-weaving uses fewer crossings than (4; 4) meshes, as shown in Figure 11(c). plain-weaves, the yarns in twill-woven fabrics can move Our generalized definition of twill leads us to iden- more freely than the yarns in plain-woven fabrics. This tify a previously unknown weaving pattern that we call property makes twill-weaving softer, more pliable, and triaxial twill. Triaxial twill patterns are created from better draped than plain-weaving. Twill fabrics also meshes that are populated with (3; 6) regions (i.e., tri- recover better from wrinkles than plain-woven fab- angles with 6-valent vertices). Such meshes can be ob- rics. Moreover, yarns in twill-weaving can be packed tained by triangularp schemes such as mid-edge subdivi- closer. This property makes the twill-woven fabrics sion [25] or 3 subdivision [24]. We prove that every more durable and water-resistant, which is a reason why mesh obtained by mid-edge subdivision [25] is twill- twill-fabrics are often used for sturdy work clothing or able. Triaxial twill patterns are visually interesting and for durable upholstery. reminiscent of some of the tilings of M. C. Escher, as A special family of twills is characterized by two in- shown in Figure 11(d). We note that a triaxial twill tegers a and b, where a is the number of over-crossings, does not demonstrate the characteristic diagonal pattern and b the number of under-crossings of a weft thread as of classical biaxial twill. it crosses the warp threads. Each twill-weaving pattern Obtaining a biaxial twill that exhibits diagonal pat- in this family can be expressed by a triple a=b=s, where terns depends on having a proliferation of (4; 4) re- s = ±1 and where b=a=-1 is a 90◦-rotated version of gions in the mesh. Quad-remeshing schemes such as a=b=1. For instance, 2=2=1 and 2=2=-1 in Figure 2 de- the Catmull-Clark [6] or Doo-Sabin subdivisions [10] fine 90◦-rotated versions of exactly the same twill struc- can achieve that proliferation, and they can make the tures. Similarly, 3=1=1 and 1=3=-1 define exactly the number of crossings in each cycle divisible by 4, an im- same twill-weaving structures. This is expected since portant property of twill weaving. In x8, we show the one over in weft must correspond to one under in warp existence of meshes that continue to be biaxially twill- and one under in weft must correspond to one over in able after application of quad-remeshing schemes. warp. Thus, the total number of overs and the total num- ber of unders should be equal. 3. Textile Twill Most textile weaves are 2-way weaves, also called bi- axial weaves. They consist of row and column strands, called weft and warp respectively, at right angles to each 2/2/1 2/2/-1 3/1/1 1/3/-1 other. They are also 2-fold, which means that there are never more than two strands crossing each other. The popularity of biaxial weaving comes from the fact that Figure 2: All possible biaxial twills for period n = most textile weaving structures are manufactured using 4. loom devices by interlacing the two sets of strands at right angles to each other. The basic purpose of any In this paper, we focus mainly on 2=2=1 twill, simply loom device is to hold the warp strands under tension, called 2/2 twill, but our results generalize to other a=b=1 so that the weft strands can weave under and over warp twills. The 2/2 twill pattern is quite common, and its strands to create a fabric. Using a loom, it is possible characteristic pattern of twill is not limited to fabrics. to manufacture a wide variety of weaves by raising and Brick walks and hardwood floor tiles sometimes exhibit lowering different warp strands. twill patterns. (See Figure 3(c).) Grunbaum and Shephard [16] formally investigated This work can also be useful to study the covering of the mathematics behind these 2-way, 2-fold woven fab- an arbitrary surface with hexagonal tiles. The visual re- rics. They coined the phrase isonemal fabrics [17] lationship between the 2=2 twill pattern and the regular to describe 2-way, 2-fold fabrics that have a transitive hexagonal tiling is shown in Figure 4. [28] [27] 2 b over b c a a under r c over (a) (b) (c) (a) Cycle condition (b) Offset condition Figure 3: Non-fabric examples of twill pattern.