Analysis and Visualization of Fractional Reflections

Summary

One can reflect once, twice or m times, where m is an integer. Can m be a real number? In this paper we show how fractional (i.e., not necessarily integer) reflections are performed. The result relies on the fact that a reflection matrix has eigenvalues ±1, and since ${(-1)}^m=\hspace{0.17em}\cos \hspace{0.17em}(\pi m)+i\hspace{0.17em}\sin \hspace{0.17em}(\pi m)$, a fractional reflection may be interpreted visually if the ${\mathbb{R}}^3$ space is augmented by including an imaginary axis.

Additional information

Notes on contributors

Milton F. Maritz

Milton F. Maritz ([email protected]) holds a Ph.D. in Applied Mathematics from the University of the Free State (UFS). He has taught applied mathematics at UFS for 11 years, then physics at UFS for 5 years, and then applied mathematics again at Stellenbosch University for 22 years. His research interests include partial differential equations, image processing, and the mechanics of eccentrically loaded rolling hoops. He has also done research for industry, in particular in the modelling of shaped charge jet formation and penetration.

Marèt Cloete

Marèt Cloete ([email protected]) received a Ph.D. in applied mathematics from the University of Stellenbosch, South Africa. Her academic interests include fluid dynamics, mechanics and struggling with mathematical problems. Currently she is lecturing classical mechanics and PDEs at her alma mater and in her spare time she enjoys all kinds of sporting activities, camping and hiking in the mountains.