I am always fascinated by things that are symmetric. In symmetrical things we see beauty and thereby mathematics. It is therefore a quest and adventure for an inquisitive mind to go deeper and deeper to study the building blocks of things that are symmetric. The images that I produce show the ubiquity of mathematics and how mathematics is a key to understand the nature that we live in, in particular, and how it unlocks the secrets of our universe in general.

The usual elementary functions and their compositions can generate sophisticated graphs which are shown. The structures (or patterns) are superimpositions of polar surfaces resulted from several compositions of tilts and turns on the coordinate axes in three dimensions. When they are viewed from a different turn and tilt they generate a totally different, fascinating structure.

The structures are a few from my collection, which numbers more than 70. For those who are interested to see them, check my website and click on the gallery button.

— *Dejenie A. Lakew *

The superimposition of two polar surfaces: $\rho = 2\sin 4\theta$, $\rho = 5/3\cos 4\theta$ (wire-framed) with some compositions of tilts and turns. The two polar surfaces are generated in such a way that one is a derivative surface of the other but with different polar radius.

Superimpositions of polar surfaces. Equations: $\rho = 10\sin 8\theta$, $\rho = 10\cos 8\theta$ with a number of compositions of tilts and turns.

Superimpositions of polar surfaces. Equations:

$\begin{array}{ll}\rho = & 2\cos(3\exp(\sin 3\theta))*\exp (\sin 3\theta)*\\ & \exp(\sin 3(\exp (\sin 3\theta)),\end{array}$

$\begin{array}{ll}\rho = & \exp(\sin 3(\exp (\sin 3\theta)),\end{array}$

$\begin{array}{ll}\rho = &-3\cos(3\exp (\sin 3\theta))*\exp(\sin 3\theta)*\\ & \exp(\sin (3(\exp\sin 3\theta)))\end{array}$

with compositions of tilts and turns. Here the first polar surface is the derivative of the second surface and the third surface is a spatial reflection of the first through the origin with wire frames, but with a larger spatial radius.

Superimposition of polar surfaces. Equations:

$\begin{array}{ll}\rho = & 2\cos(3\exp(\sin 3\theta))*\exp (\sin 3\theta)*\\ & \exp(\sin 3(\exp (\sin 3\theta)),\end{array}$

$\begin{array}{ll}\rho = & 3 + 2\cos(3\exp (\sin 3\theta))*\exp(\sin 3\theta)*\\ & \exp(\sin 3(\exp\sin 3\theta)) \end{array}$

with several compositions of tilts and turns.

Superimposition of three polar surfaces. Equations:

$\begin{array}{ll} \rho = & 2\cos(3\exp(\sin 3\theta)*\exp (\sin 3\theta)*\\ & \exp(\sin 3(\exp (\sin 3\theta)),\end{array}$

$\begin{array}{ll} \rho = & 3 + 2\cos(3\exp (\sin 3\theta))*\exp(\sin 3\theta)*\\ & \exp(\sin 3(\exp\sin 3\theta)),\end{array}$

$\begin{array}{ll} \rho = & -3\cos(3\exp (\sin 3\theta))*\exp (\sin 3\theta)*\\ & \exp(\sin 3(\exp (\sin 3\theta)) \end{array}$

followed by many compositions of tilts and turns.

Superimposition of three polar surfaces. Equations:

$\begin{array}{ll} \rho = &2\cos(3\exp(\sin 3\theta)*\exp (\sin 3\theta)*\\ & \exp(\sin 3(\exp (\sin 3\theta)),\end{array}$

$\begin{array}{ll} \rho = &3 + 2\cos(3\exp (\sin 3\theta))*\exp(\sin 3\theta)*\\ & \exp(\sin 3(\exp\sin 3\theta)), \end{array}$

$\begin{array}{ll} \rho = & 6\;\end{array}$ (the outer sphere wire-framed)

with many compositions of tilts and turns as rigid transformations.