Rope rosettes

Rope rosettes comes in many forms. They have their foundations in the seafaring tradition, and have many similarities with other traditional interlace patterns, like the Sona patterns from Angola and neighboring regions, Kolam patterns from South India and Celtic decorations, to mention a few (Rossing & Krifel, 2003, ch.2).

Properties

Rope rosettes have some important properties that can be exploited in the construction of an analysis tool. Although it is possible to synthesize many different patterns, some of these are hard or impossible to realize in rope form. As the goal is to analyze physical rope rosettes, some simplifying assumptions can be made:

Mathematical foundation

As discovered by Rossing, any rope rosette can be described by two periodic functions with several components, one each for the x- and y-direction (Rossing, 2012). One can visualize the periodic functions as a series of rotating vectors, connected end to end to each other. Each vector has a length, a direction of rotation, a speed of rotation and a starting point. The visual properties of the rosette can all be determined by analysis of the mathematical description, and vice verse. These properties of the periodical functions and the rope rosettes and how they are related to each other are summarized below. The mathematical property is listed first, and the rosette property in bold italic font. A more in-depth description can be found in (Rossing & Krifel, 2003, ch.7, 9).

The mathematical description of a rope rosette is as follows; two periodical equations, one for x- and one for y-direction. Following the convention of Rossing and Krifel, they have the general form of the equation below, where ai is the amplitude and fi the frequencyof component i. Note that for some rosettes, the sines and cosines might be switched in some components.

Number of components
Defines the order of the rosette.

Rosettes are categorized after their order, which is the minimum number of Fourier components needed to describe the rosette. For instance, the Turk’s head rosette is of order 2, and the Eye rosette of order 5.


Amplitude ai
Defines the type of the rosette.

A rosette is one of three main types; closed and overlapping, open and ring-shaped or non-overlapping. A rosette is closed if the bights encloses the center, which makes the center hole very small or disappear. An open, ring-shaped rosette lies as a ring around the center. A non-overlapping rosette has bights which do not enclose the center. The figure above shows the different types.

Frequency fi
Defines the general shape of the rosette. Below follows some descriptions of the specific frequency properties.


Pairwise frequency differences
The bights of the rosette.

A bight is a loop on the outer edge. In this setting, a bight equals a sub-pattern, such that the frequency differences are the number of sub-pattern repetitions. See the figure above.

Fundamental frequency
The largest frequency, defining the rosette slotting.

The slotting is the number of strands crossing an imaginary straight line from the rosette center, see the figure above. Slotting is not well defined for a non-overlapping rosette.

Base frequency
The smallest, positive frequency, determining the skipping of the rosette, that is how the sub-patterns are interconnected.

With a skipping factor of 1, each sub-pattern is connected to its direct neighbors. With a skipping factor of 2, each sub-pattern is connected to its second nearest neighbors, and so forth. The skipping factor is the same as the number of circuits traced around the center before reaching the starting point.

References

Rossing, N. K. (2012). How to make rope mats and rosettes.

Rossing, N. K. & Krifel, C. (2003). Matematisk beskrivelse av taumatter (2nd ed.). NTNU-trykk.

Wikipedia Commons. (2007). Image of celtic knot. Retrieved June 29, 2015, from https://commons.wikimedia.org/wiki/Category:Celtic_ knots?uselang= nb#/media/File:Celtic-knot-insquare-transp.png

Wikipedia Commons. (2012). Image of kolam pattern. Retrieved June 29, 2015, from https://commons.wikimedia.org/wiki/File:Kolam-Attur_town-2012- Salem-Tamil_Nadu-11.JPG