Ramanujan's Butterfly Digital Print 2019

A family of thirteen triangles with vertices based on the roots of the Ramanujan simple cubic. All of the vertices of these triangles lie along the graph of the function $$n(x) = \frac{1}{1-x}$$.

### Background and Inspiration

Let $$r_1$$, $$r_2$$, and $$r_3$$ be the roots of the Ramanujan simple cubic $f_B(x) = x^3 - \left(\frac{3+B}{2}\right) x^2 - \left(\frac{3-B}{2}\right) x + 1,$ then the function $$n(x) = \frac{1}{1-x}$$ permutes the roots: $n(r_1) = r_2, \quad n(r_2) = r_3, \quad\text{and}\quad n(r_3) = r_1.$ I realized that because the function $$n(x)$$ permuted the roots, the three possible pairings $$(r_i,n(r_i))$$ would create triangles. Furthermore, the triangle vertices lie along the graph of $$n(x)$$. For a particular $$B$$ value, consider the triangle with vertices that are the ordered pairs of roots $$\{(r_1,r_2), (r_2, r_3), \text{and } (r_3, r_1)\}$$. The triangles associated with $$\pm B$$ are symmetrically located with respect to the $$B=0$$ triangle. The artwork is a visualization of these triangles for $$B$$ values in the set $$\{0, \pm 4, \pm 8, \pm 12, \pm 16, \pm 20, \text{and} \pm24\}$$.

The artwork was inspired by the article Cubic Polynomials, Linear Shifts, and Ramanujan Simple Cubics by Dresden et al in the December 2019 issue of Mathematics Magazine.

A version of this work was originally done for use as cover art for the December 2019 issue of Mathematics Magazine. This is one of 25 original artworks I created for the journal Mathematics Magazine during 2015–2019.

I was also inspired by Albion College alumnus and University of Illinois professor Bruce C. Berndt, who has written extensively on the life and works of Srinivasa Ramanujan. I organize a colloquium series at Albion College where Berndt has given several fascinating talks on Ramanujan.

### Publication History

• Libellule, fiori, origami: Momenti artistici al Joint Mathematics Meetings 2020 by Maria Mannone, Nuova lettera matematica 1 , p. 91-95, June 2020.
https://www.unipapress.it/it/book/nuova-lettera-matematica-1_242/.
• 2020 Joint Mathematical Meetings Exhibition of Mathematical Art, p. 77, Edited by Robert Fathauer and Nathan Selikoff, ISBN: 978-1-938664-33-5, Tessellations Publishing, 2020.
• Mathematics Magazine, Cover Art, Vol. 92, No. 5, December 2019.

### References

• Cubic Polynomials, Linear Shifts, and Ramanujan Simple Cubics by Greg Dresden, Prakriti Panthi, Anukriti Shrestha, and Jiahao Zhang. Mathematics Magazine, Vol. 92, No. 5, 374-381, December 2019.
doi:10.1080/0025570X.2019.1655310.