Wild knots

doi:10.1016/j.cag.2006.08.021

Computers & Graphics

Volume 30, Issue 6, December 2006, Pages 1027-1032

https://doi.org/10.1016/j.cag.2006.08.021 Get rights and content

Abstract

Wild knots, a class of knots which cannot be represented by polygonal paths in 3-dimensional space, are investigated as an embellishment to the Celtic style of ornamental knotwork. Wild knotwork designs are compared to fractal knotwork designs and the traditional technique of N-interlacement. It is shown that all three styles may co-exist in one design.

Introduction

Celtic knotwork is an ornamental art style in which interlaced cords trace complex and well-structured designs to fill a given area. These designs consist of closed cords of finite length with an alternating weave, and it is a mark of the artist's skill to create complex designs from as few cords as possible, ideally a single cord. See [1] for a brief summary of the key points of this style of artwork, and [2], [3] for a more in-depth treatment.

Fig. 1 (left) shows a simple plait in the Celtic style. Note that the design consists of a single cord and that the grid upon which it is based is rectangular with sides in the ratio 3:4; the number of cords in a simple plait will equal the greatest common divisor of the grid's number of horizontal and vertical units.

In the more pragmatic terms of knot theory, a knot is a closed curve with no self-intersections that cannot be unknotted to produce a simple loop [4], [5]. A link is a set of one or more knots with mutual entanglements; hence, a knot is a link with one component. Traditional knotwork designs are generally knots, although this is not always the case; in this paper, the term design shall be understood to mean a link.

Section snippets

Wild knots

Mathematicians Ralph Fox and Emil Artin introduced the concept of wild and tame embeddings in 1948 [6]. Curves that are equivalent to polygons are described as tame, and curves that are not are described as wild [5], [7]. Similarly, if we define polygonal knots as those knots equivalent to piecewise-continuous polygons with a finite number of vertices and no self-intersections, then tame knots are those equivalent to polygonal knots and wild knots are those that are not (please see [5] for

Tame fractal links

Fig. 4 shows two recursive designs constructed using similar principles, although in these cases the generator sets are themselves tame knots without attachment points, and the interaction between levels comes through interleavement of the tame cords at successive levels. These designs are fractal in the sense that the design is similar at all scales; it will look the same no matter how far the view is zoomed in.

Note that while Fig. 1 (right) consists of a single wild knot, the two designs

N-interlacement

N-interlacement is a traditional technique for adding complexity to knotwork designs by splitting cords into two or more parallel subcords, which are then subsequently rewoven to ensure an alternating weave [3]. Fig. 5 shows the basic 3:4 plait as a 2-interlaced design (left) and a 3-interlaced design (right).

N-Interlacement may be achieved simply by offsetting the main cord path using the attenuation rules above to give N parallel subcord paths, then offsetting these, again applying the

Breaks and crossovers

A break consists of splitting two cords at the point where they cross and rejoining the cross-cords to inflect relative to either of the two possible axes of reflection; these will be the horizontal or vertical axes in the case of the rectangular grid.

Fig. 6 (left) shows a design on the 3:3 grid consisting of three cords—note that the greatest common divisor of (3, 3) is 3. The middle figure shows this design reduced to two cords by adding a vertical break (bottom center) and the rightmost

Putting it all together

Fig. 7 shows a hybrid design that incorporates the techniques described in the previous sections. The intergenerational reduction factor for this design is approximately 59%.

The main cord paths consist of a recursively repeated tame knot (sharp corners) interwoven with a wild knot (rounded clover shape). The main cord paths have been 2-interlaced, and the resulting paired subcords shaded light and dark for both the wild and tame components.

The total number of cords C in this design, for

Conclusion

This paper demonstrates how the principles of wild knots and fractal links may be introduced as an embellishment to knotwork designs in the Celtic style. The construction of wild knot designs may be achieved by the recursive application of a simple generator set of pre-defined cord segments, provided that their end points satisfy certain geometric constraints.

These principles, together with the traditional knotwork technique of N-interlacement, may be used to embellish a design and add

Acknowledgement

Thanks to Paul van Wamelen for suggesting corrections and improvements to the original draft of this paper.

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Chaos and GraphicsWild knots

Abstract

Introduction

Section snippets

Wild knots

Tame fractal links

N-interlacement

Breaks and crossovers

Putting it all together

Conclusion

Acknowledgement

Computers and Graphics

Computer-Aided Design

Celtic art: the methods of construction

Celtic knotwork

Chaos and Graphics
Wild knots