Curlicues

Figure 6. General structure for phosphazenes with mesogenic side groups. Example is a mixed substituent polymer (VII) where R represents the trifluoroethoxy group and the mesogen with flexible spacer is represented by the curlicue and rectangular box.

Fig. 2. (a) Front view of the conformation of the peptide chain of the inhibited nuclease at high resolution (Type II crystals), (b) Right side view of the chain conformation at high resolution. The curlicue at the upper left center represents the inhibitor pdTp. 

Use a standard type font where possible. Most standard type styles have about the same level of legibility. As a font becomes more exotic—a lot of curlicues, flourishes, graphic elements, etc.—legibility decreases and even simple information may be difficult to read or take longer to read. 

I soon saw, by the way in which the white man s track doubled and redoubled again, that the fellow could not be cutting such curlicues for nothing. 

Curlicues are fantastic spirals or twists that seem to be everywhere we look. When we think of examples of curlicues, both the mundane and exotic easily come to mind, such as the gentle curl of a fern tendril, the shape of an octopus s retracted arm, the death-form assumed by a centipede, the spiral intestine of a giraffe, the shape of a butterfly s tongue, the spiral cross-section of a scroll, the shape of the yellow brick road in Munchkinland in the film classic The Wizard of Oz, and even the characters of several written languages such as Farsi and Tamil. 

In this chapter I am interested in the mathematical curlicues, and in particular will explore infinite, recursively spiralling patterns. This class of curlicues is quite fascinating to geometers not only because of their beauty and complexity but also because of their renormalization symmetries and various fascinating mathematical properties. ( For Further Reading, at the end of this book, offers some interesting technical papers for the mathematically inclined reader.)... 

I tested a range of famous transcendental and irrational constants that control the angle that each step takes in the curlicue. After 10,000 iterations (steps), the golden curlicue, defined by the golden ratio... 

From a theoretical standpoint, the curves in this chapter are infinite. In other words, as they are generated by taking more and more steps on a plane, their lengths grows to infinity. I have computed the temperatures for the curves up to 10,000 steps. Would the temperature trends change if I were to use 1 million steps Does the behavior at 10,000 steps reflect the limiting behavior near infinity How does the finite precision of computers affect our ability to accurately draw curlicues ... 

Mathematically inclined readers may be interested in other recursively spiralling patterns T created in the complex plane. One such curlicue can be generated using... 

Berry, M., and Goldberg, J. (1988). Renormalization of Curlicues. Nonlinearity, 1 1-26. Markowsky, G. (1992). Misconceptions about the Golden Ratio. College Mathematics Journal, 23(1) (January) 2-19. 

Moore, R., and van der Ponten, A. (1989). On the Thermodynamics of Curves and Other Curlicues. McQuarie University Mathematics Reports 89-0031, April. 

In Table 2.1 the bicomponent fibers (which often have half the cylinder as polymer I and the other half as polymer II) are of interest because of the differences in thermal coefficients of expansion. On cooling, the differential rates of contraction cause a certain coiling or curlicue formation in the fibers. 

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