# Cornu Spirals and the Triangular Lacunary Trigonometric System

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## Abstract

**:**

## 1. Introduction

## 2. Properties of Triangular Numbers

## 3. Lacunary Sequences

## 4. Fresnel Integrals and the Cornu Spiral

## 5. Results and Discussion

#### 5.1. Isometry and Quasi-Symmetry

#### 5.2. Series Relations

#### 5.3. Large n and Self-Similarity

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Left panel: Set of partial summation values for ${\mathcal{F}}_{50,1}$. The values of the partial summation for ${\mathcal{F}}_{50,1}$ closely follow a Cornu spiral. The values appear equally spaced along the curve. In this case, the spacing is $\frac{1}{2\sqrt{50}}$. Furthermore, the length of the curve is $\frac{2n+1}{2\sqrt{n}}$. Right panel: ${\chi}^{2}$ values for $n=1$ to $n=150$. The Cornu spiral models the sequence of partial summations better with increasing n.

**Figure 2.**The sequence of partial summations for ${\mathcal{F}}_{250,3}$ (top, left panel), ${\mathcal{F}}_{251,3}$ (top, right panel), and ${\mathcal{F}}_{252,3}$ (lower, left panel). These data sets represent the three basic shapes that occur when $q=3$. As n progresses, the shapes cycle through these q (in this case 3) canonical shapes (the space between adjacent datum decreases as n increase). Note there are three spirals present in the top two graphs. This is a manifestation of a general characteristic that there are q Cornu spirals. The exception seen in the lower left graph is also a general characteristic. In the case shown, $q=3$ evenly divides $n=252$. The result brings the data set into the ${\mathcal{F}}_{n,1}$ class. If q is prime, this will happen only once. If q is composite, then there will be several values which divide into n. Finally, in all cases, inversion symmetry is preserved as must be the case. The lower right panel shows the members of this canonical family superimposed on one another. There is a three-fold local quasi-symmetry around the primary spiral centers (see text for details). The green trifold is set to guide the eye. Formula for the isometry of the trifoil from the origin is given in the text.

**Figure 3.**Analog to Figure 2 for the cases of ${\mathcal{F}}_{n,5}$ where n runs from 651 to 655. The canonical shapes each have $q=5$ Cornu spirals except for 655 into which five divides. The bottom right panel shows the members of this canonical family superimposed on one another. Now the local quasi-symmetry is five-fold. The black pentifoil guides the eye.

**Figure 4.**A striking manifestation of the self-similarity that can exist in these lacunary systems. The graphs shows the case of ${\mathcal{F}}_{2831,67}$. This is one of the two most ordered of the 67 canonical members of this family (the other is rotated by $\pi /2$). The full shape is composed of $q=67$ individual Cornu spirals. Interestingly, the inflection points of each Cornu spiral falls along the blue curve which is precisely a rotated and inverted Cornu spiral of the ${\mathcal{F}}_{2831,1}$ case. Furthermore, the individual Cornu spirals alternate as tangent or normal to the blue curve.

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**MDPI and ACS Style**

Vogt, T.; Ulness, D.J.
Cornu Spirals and the Triangular Lacunary Trigonometric System. *Fractal Fract.* **2019**, *3*, 40.
https://doi.org/10.3390/fractalfract3030040

**AMA Style**

Vogt T, Ulness DJ.
Cornu Spirals and the Triangular Lacunary Trigonometric System. *Fractal and Fractional*. 2019; 3(3):40.
https://doi.org/10.3390/fractalfract3030040

**Chicago/Turabian Style**

Vogt, Trenton, and Darin J. Ulness.
2019. "Cornu Spirals and the Triangular Lacunary Trigonometric System" *Fractal and Fractional* 3, no. 3: 40.
https://doi.org/10.3390/fractalfract3030040