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Cornu Spirals and the Triangular Lacunary Trigonometric System

Department of Chemistry, Concordia College, Moorhead, MN 56562, USA
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Fractal Fract 2019, 3(3), 40; https://doi.org/10.3390/fractalfract3030040
Received: 13 June 2019 / Revised: 8 July 2019 / Accepted: 9 July 2019 / Published: 10 July 2019
This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed. View Full-Text
Keywords: Fresnel integral; Cornu spiral; lacunary trigonometric systems; triangular numbers; Gaussian summations Fresnel integral; Cornu spiral; lacunary trigonometric systems; triangular numbers; Gaussian summations
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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.

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