••••••••••

10Crocheting Algorithms

**Gisela Baurmann**

*is an award-winning architect who has practiced and taught extensively in Europe and the United States, including at Cornell University’s Department of Architecture in Spring 2011. Her work examines cultural techniques as conceptual models of fabrication for application in computational design. Gisela is founding partner of the architecture office Büro NY, based in New York and Berlin. For more information www.BuroNY.com and http://bit.ly/knBIft.*

**Daina Taimina**

*was teaching mathematics at the University of Latvia for 20 years before she came to the United States. Currently she is adjunct associate professor of mathematics at Cornell University. In January 2012, she received the prestigious Euler Book Prize from the Mathematical Association of America for an outstanding book about mathematics for a general audience,*Crocheting Adventures with the Hyperbolic Planes.

••••••••••

*The loop stitch is a noeud coulant: a knot that, if untied, causes the whole system to unravel. It is an element in making stockings, in knitting and crocheting, and the particular way it is formed is dictated by the tools employed and the use intended.[…] I can only say that it is an extremely refined [art] and yields products whose properties can be achieved in no other way. They carry the elements of their richest ornaments in themselves and in their construction. Elasticity and ductility are the specific advantage of these products; this makes them especially suited to close-fitting dressings that embrace the figure and define it without fold.*

Gottfried Semper,

*Style*, 1860.

Crocheting activates a single line—the thread—to generate an elastic surface by moving around and through an empty core. The topology of a crocheted fabric is relatively complex: the thread describes an undulating path along each row, the loops of one row being pulled through the loops of the row subjacent to it.

In crochet, it is possible to create a whole surface of multiple elements from one single thread. Inscribed in the technique is the potential to generate a multitude of volumes, strands, cross-references, and volumetric sequences without ever having to interrupt the fiber’s continuous line. Crocheted or knitted fabric specifies local rules of increase and decrease, of temporary pause and subsequent pick up, thus enabling the fabric to splice open and to reunite again later on. It can generate three-dimensional volumes through separating and subsequently braiding, crossing, and back-referencing its own materiality.

In geometry, the most basic and commonly studied category of surfaces are those with constant curvature. In the early 19th century, C.F. Gauss suggested further categorizing such surfaces by the sign of their curvature—positive, negative, and zero. In the case of a constant positive curvature, the surface geometry is spherical, and in case of a constant negative curvature, it is hyperbolic.

••••••••••

**Crocheted Hyperbolic Plane**, Daina Taimina.

Wool, about 50 cm × 50 cm × 50 cm, 2011.

© Daina Taimina. From the collection of Cooper-Hewitt National Design Museum.

The Euclidean plane (with zero curvature) can be tiled by regular hexagons. At each vertex, there are three hexagons coming together and forming a 3 × 120º = 360º angle. This tiling can be continued infinitely: around each hexagon are six other hexagons. If we start with a pentagon and surround it by five hexagons with the same side length, then at each vertex of this tiling there will be 2 × 120º + 108º = 348º < 360º, and the tiling will form a closed surface approximating the sphere. This tiling describes precisely the way a soccer ball is stitched together. If instead of the pentagon, a seven-sided polygon (heptagon) is chosen, then at each vertex the sum of the angles will be >360º and tiling can be continued indefinitely: this will approximate the hyperbolic plane.[1]

Another way to construct the model of the hyperbolic plane is to attach identical annular strips (portions of strips between concentric circles, with the same width Δ and the same radius as the outer circle), attaching the inner circle of one of the strips to the outer circle of the other. The actual surface with the negative curvature could be obtained by letting Δ → 0.

In crochet, these techniques can be mimicked by uniformly adding an extra stitch after a chosen and fixed number of stitches in a row. This will result in a crocheted hyperbolic plane.[2]

**Diagram to Crochet the Hyperbolic Plane**, Erica Savig, 2007. Courtesy Gisela Baurmann.

••••••••••

What we now call Euclidean geometry was developed in Euclid’s *Elements*around 300 B.C. Among many other hypotheses,

*Elements*contained the so-called Parallel Postulate (also called the Fifth Postulate):

*if a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines (if extended indefinitely) will meet on that side on which are the angles less than two right angles.*

The breakthrough in the study of various forms of the parallel postulate came in the early 19th century when, apparently independently, Janos Bolyai and Nikolaj Lobachevsky developed a new geometry, which later Felix Klein named “hyperbolic geometry.” In 1854, Georg Riemann showed that hyperbolic geometry holds on surfaces with constant negative curvature.[3]

Riemann’s idea was to start with the concept of curvature and to argue that geometry was fundamentally about two types of problems: the intrinsic properties of a surface, and the ways in which a surface is mapped into another surface. Riemann did even more—he showed that the idea of Gaussian curvature can be generalized into higher dimensions. He also mentioned that there were three two-dimensional geometries possible if classified by constant curvature of the surface. By the end of the 19th century, non-Euclidean geometry had become a widely discussed subject in mathematical circles and had great influence on artists, writers, and philosophers.[4]

Since Euclidean geometry had been taught for many hundreds of years, a large number of people truly believed that Euclidean geometry was an exclusive and absolute truth. To accept the existence of non-Euclidean geometries was revolutionary.

In architecture we may apply the syntax of hyperbolic geometry to generate organizations of multiple linkages. When crocheting the hyperbolic plane, the stitches allow for local specificity as well as feedback between local units and the global fabric. This specificity and feedback can be applied to multiple tectonic elements that generate an architectural project: the organization of program units, structural components of varying scale, facade elements, interior dividers, vertical, horizontal and oblique connectors, and many more.

••••••••••

The radical possibilities of “crochet thinking” in architecture and urbanism lie in its ability to seamlessly shift between increase and decrease of units in the overall fabric, while thinking through the generation of each local constituent progressively. This allows the designer to pay attention to the specificity of each particle/component individually, yet produce a complex overall structure. Whether we apply these rules to parameters of architectural, interior, or urban design, or to physical components of them, we inevitably generate a vast array of possible configurations. A collection of physical expressions created in response to a set of parameters affords us to choose “successful” candidates within this array, according to the requirements that we define.In the area of digital output, surface materials can be developed that respond locally to stress analysis, requests for changing transparencies or thicknesses, and so on, by condensing or dispersing the filaments/strands for reinforcement throughout the fabrication piece. Multiple possibilities for the arrangement/gestalt of the linear elements generate program-specific base structures. They can be dipped in or covered with

*pastose*materials for surface generation to produce an abundance of structural, material, and sensory qualities. Through the increase and decrease of stitches in the base configuration, three-dimensional and multidirectional surface structures can be generated.

Current experiments developed in academic settings demonstrate the first in a series of steps that introduce crochet as a conceptual model for architectural fabrication.

In studio projects and material fabrication seminars, students explore crochet as a conceptual method of design and examine its topological traits as well as its ductility. Using diagrams, 3D modeling, and scripting, space formations are created that reference performance articulation and behaviors of crochet stitching. Students employ the properties of the technique, its local rules, continuity of the thread and algorithmic qualities to discover program continuities that suggest spatial assembly. A series of exercises explore rule-based design methodologies through model-making and analytical drawings and provide an introduction to non-Euclidian geometry. These exercises deepen the understanding of a script syntax and its formal output within the confines of a single line trajectory.

••••••••••

**3D Print**, Ke Xu (M. Arch. ’11).

••••••••••

Digital notations of crochet loops generate architectural program diagrams in two dimensions. They are connected following the architectural program’s daily routines. A substitution system and the concept of transposition are deployed to exchange agents in one system with those in the other. Ecology, size, connectivity, and organizational layout of a proposed program activity determine its coupling with a specific stitch notation. Subsequently, the local to global relationship of stitch to fabric are read quite literally: program activities are located, connected, and sized according to the stitch’s spatial and jointing qualities. Qualitative program definitions emerge.The computational stitch construction gets introduced to move from the two-dimensional program diagrams to a three-dimensional structural output.

Variation of input parameters, such as number of stitches and rows, stitch increase, thread diameter, openings, and so on, tests the structural, spatial, and material effects.

Transferring the stitch into computation requires carefully tracing the trajectory of the crochet hook and linking one stitch with its preceding and following ones, as well as with its neighbors in the rows above and below. When crocheting a hyperbolic plane, the main trajectory of the thread is a spiral, while the local units, the stitches, connect one row with the next.

In a series of experiments, specific characteristics of crochet fabric are regenerated as fabrication prototypes through computational tools. The analogue material’s main characteristics—its transparency, structural capacity, ductility, and thickness, as well as its overall shape—are pursued in individual exercises, starting with transparency and structural capacity, as demonstrated here.

While analogue crochet models exhibit emergent qualities, the topology of a computational model is initially generated in a “frozen state,” in order to establish the local connections between units as well as their individual properties. Gravitation, friction, stresses, and pressure will be introduced in a future generation of experiments in order to study the material’s ductility in combination with its tectonic properties and material effects.

••••••••••

**Surface Panelizations**, Huidan Kang (M. Arch. ’11), Ida D. K. Tam (M. Arch. ’12), and Frances Gain (M. Arch. ’11). Computationally constructed stitches populate the surfaces. They are connected and interlaced to generate a continuous fabric.

**Crochet Program Diagrams**, Young Suk Choi and Joshua Freese, 2007. Courtesy Gisela Baurmann. Digital notations of crochet loops generate architectural program diagrams in two dimensions. They are connected following the architectural program’s daily routines.

••••••••••

**Surface Panelization.**Jake Donghyun Kim (M.Arch ’12) and Serena Yuen Ying Lee (M.Arch ’12). Variation of input parameters, such as number of stitches and rows, stitch increase, thread diameter, openings to test structural, spatial and material effect.

••••••••••

**Acknowledgments**

Detlef Mertins encouraged me to conduct the first crochet studio at the School of Design, University of Pennsylvania. Stan Allen and Dagmar Richter granted the opportunity to explore the concept while teaching at Princeton University and Cornell University, respectively. Seok Yoon, James Kerestes, Jungwook Lee, and Jonathan Asher provided conceptual and technical support. Many thanks to Daina Taimina and Jonas Coersmeier, who helped organize my thoughts in many productive discussions. I want to thank all students, who participated in exploring crochet as a generative tool in architecture at Cornell University’s Department of Architecture: Frances Gain, Shujian Jian, Huidan Kang, Donghyun Jake Kim, Han Joon Kim, Yuen Ying Serena Lee, Feng Lin, Lorena Quintana, Dick Kar Ida Tam, Ke Xu, Wendy Yang, Fanbo Zeng, Huang Zheng and at the University of Pennsylvania’s School of Design: Dan Affleck, Emily Bernstein, Leslie Billhymer, Young Suk Choi, Joshua Freese, Kenta Fukunishi, Vincent Leung, Danisha Lewis, Andrew Ma, Lauren MacCuaig, Kara Medow, John Ryan, Erica Swesey Savig, and Alan Tai.

**Endnotes**

1. Daina Taimina,

*Crocheting Adventures with Hyperbolic Planes*(Wellesley, MA: A.K. Peters Ltd. 2009), 135.

2. David Henderson and Daina Taimina, “Crocheting the Hyperbolic Plane,”

*The Mathematical Intelligencer*23, no. 1 (2001): 17–28.

3. More history in Daina Taimina,

*Crocheting Adventures with Hyperbolic Planes*(Wellesley, MA: A.K. Peters Ltd., 2009).

4. Linda Henderson,

*The Fourth Dimension and Non-Euclidean Geometry in Modern Art*(Princeton University Press, 1983).

Go back to 9: Mathematics