## Clifford Water Drop Wave by Hamed Baghal Ghaffari

A plot resembling a ripples in water created with Clifford Legendre Polynomials

Clifford Water Drop Wave

Hamed Baghal Ghaffari and University of Newcastle at Australia (UON)

Abstract

In this work we first write a peice of Maple code which returns the Clifford Legandre Polynomials. Then we use these polynomials to plot a water drop wave.

Keywords: Clifford Legendre Polynomials.

 Preliminary Material The even and odd Clifford legendre polynomials (CLPs) introduced in [1] can be calculated respectively as in below (see [2]). C  C where m is the dimension, Y
 Maple Code Even Clifford Legandre Polynomials.   evencliffordlegendrewithoutyk := proc(r, N, k)

Plots

Here we show some initial plots. For the main art work, see the end of the file.

 >

It is easy to play a bit with the parameters to ger a wave-like picture.

 >

Now we will plot a nice Clifford water drop wave.

 > Hamed := piecewise( x^2 + y^2 < 10, 1/10*evencliffordlegendrewithoutyk(sqrt((x/3)^2 + (y/3)^2), 10, 1), -10 ):
 > p := plot3d([Hamed], x = -5 .. 5, y = -5 .. 5, color = blue, style = surface, grid = [200, 200]): plots:-display(p, view = [-2 .. 2, -2 .. 2, -1 .. 1],axes = none);

Odd Clifford Legandre Polynomials.

oddcliffordlegendrewithoutyk := proc(r, N, k)

And just some more plots with the odd Clifford Legandre polynomials for fun.

 >
 >
 Acknowledgements The author is supported by the Australian Research Council through Discovery Grant DP160101537 and Lift-off fellowship from Australian Mathematical Society.

References

 1 Delanghe, R., Sommen, F., & Soucek, V. (2012). Clifford algebra and spinor-valued functions: a function theory for the Dirac operator (Vol. 53). Springer Science & Business Media, https://doi.org/10.1007/978-94-011-2922-0
 2 Ghaffari, H. B., Hogan, J. A., & Lakey, J. D. (2022). Properties of Clifford-Legendre Polynomials. Advances in Applied Clifford Algebras, 32(1), 1-25, https://doi.org/10.1007/s00006-021-01179-8