I had been playing around and experimenting with polar equations (and lots of other types of equations) for an unhealthy duration of time. As a result of my deep delve into the world of mathematics, I discovered several wonderful equations whose graphs portray a high level of creativity by nature itself.

I began summarizing my findings in a book entitled "AA4MM - Amusing Art for Mathematical Minds"

Equations and their links to graphs.

The equations have been provided in Desmos notation, so you can directly use the keyboard shortcuts [Ctrl+C] and [Ctrl+V] to copy/paste them into Desmos. Or if you\'re feeling lazy, simply hit the links to explore them individually.

Have fun!

1) Butterfly

$\theta r= -\phi\theta\sin\left(\theta\phi\right)$
θ (Theta) range : -4 < θ < 4 \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$ \ Example

2) Alien rocks

$r\theta^{-2}=\phi\ln\left(e^{\theta}\log\theta\right)+\left(\phi\theta\sin\theta-\cos\phi\theta\right)^{2}$
θ (Theta) range : -2499 < θ < 2499 \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$ \ Example

3) Cherry

$r\theta= -\phi\ln\left(e^{\theta}\log\theta\right)-\left(\phi\theta\sin\theta\right)$
θ (Theta) range : 0 < θ < 12π \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$ \ Example

4) Paper Windmill

$r\theta\ =\ \phi\ln\left(e^{\theta}\log\theta\right)+\left(\phi\theta\sin^{2}\left(\phi\theta\right)\cos^{2}\left(\phi\theta\right)\right)^{2}$
θ (Theta) range : -29 < θ < 100 \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$ \ Example

5) Pickaxe

$r\theta^{3}=\phi\ln\left(\cos\theta\right)$
θ (Theta) range : -2499 < θ < 2499 \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$ \ Example

6) Spider / Stickbug

$r\theta^{3}=\phi\ln\left(\cos\theta\right)$
θ (Theta) range : -2499 < θ < 2499 \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$ \ Example

7) Maple leaf

$r\le\sin\theta+\left(\sin\left(\frac{9\theta}{2}\right)\right)^{2}$
θ (Theta) range : -200π < θ < 200π \ Example

8) Fan palm leaf

$r=\sin\theta+\left(\sin\left(\frac{9\theta}{2}\right)\right)^{9}$
θ (Theta) range : -200π < θ < 200π \ Example

9) Saint wings & halo

$r=\sin\theta+\left(\sin\left(\frac{91\theta}{2}\right)\right)^{4003}$
θ (Theta) range : -2π < θ < 2π \ Example

10) Warrior's Cardioid

$r=\sin\theta+\left(\sin\left(9\theta\right)\right)^{67}+\sin\left(0.1\theta\right)$
θ (Theta) range : -20π < θ < 200π \ Example

11) Flower of Damnation

$r=\sin\left(4\theta\right)+\left(\sin\left(\frac{13\theta}{2}\right)\sin\left(\frac{\theta}{18}\right)\sin\left(90\theta\right)\right)^{6}$
θ (Theta) range : -20π < θ < 200π \ Example

12) Abstraction 787

$r=\sin\theta+\left(\sin\left(9\theta\right)\right)^{67}+\sin\left(0.1\theta\right)$
θ (Theta) range : -2π < θ < 2π \ Example

13) Assassin's Creed logo

$r=\sin\left(\theta\right)^{2}+\left(\sin\left(3\theta\right)\right)^{8}$
θ (Theta) range : -2π < θ < 2000π \ Example

14) Thorny Aster

$r=\sin\left(\frac{7\theta}{2}\right)+\left(\sin\left(9\theta\right)\right)^{60}$
θ (Theta) range : -2π < θ < 200π \ Example

15) Toxic Orchid

$r=\sin\left(4\theta\right)+\left(\sin\left(9\theta\right)\right)^{67}+\sin\left(0.1\theta\right)$
θ (Theta) range : -20π < θ < 200π \ Example

16) Cursed Diamond

$r=\sin\left(\theta\right)+\left(\sin\left(3\theta\right)\right)^{8}$
θ (Theta) range : -2π < θ < 2000π \ Example

17) 4D Wormhole

$r=\cos^{2}\left(\theta\right)+\left(\cos\left(80\theta\right)\right)^{2400}$
θ (Theta) range : -200π < θ < 200π \ Example

18) Dragonscale bracelet

$-r=\sin\left(\theta\right)+\left(\sin\left(10\theta\right)\right)^{2}-\cos\left(4\right)$
θ (Theta) range : -2π < θ < 2000π \ Example

19) Batman logo compressed

$r\le\cos\left(\cos\left(6\theta\right)\right)+\left(\tan\left(0.9\cos\left(\theta\right)\right)\right)^{4}$
\ θ (Theta) range : 0 < θ < π \ \ $r\le\cos\left(\cos\left(3\theta\right)\right)+\left(\tan\left(0.9\cos\left(\theta\right)\right)\right)^{4}$
θ (Theta) range : -π < θ < 0 \ Example \ Or this one

20) Among Us compressed

$1.4x^{2}+4.2\left(y-1\right)^{2}\le0.18$ \

or

$r\le\left(\sin\theta\right)^{5}+0.5$
θ (Theta) range : -2π < θ < 2π \ Example

21) Arcaneus Tourbillion

$r=\sin\left(2\theta\right)^{2}+\left(\sin\left(10\theta\right)\right)^{3}$
θ (Theta) range : -2π < θ < 2000π \ Example

22) Gem of the Shadows

$r=\sin\left(2\theta\right)^{100}+\left(\sin\left(4\theta\right)\right)^{2}+\cos\left(2\right)^{2}$
θ (Theta) range : -2π < θ < 2000π \ Example

23) Meridian Obelisk

$r=\sin\left(2\theta\right)^{2}+\left(\sin\left(\theta\right)\right)^{12229}$
θ (Theta) range : -2π < θ < 2000π \ Example

24) Unknown but Iconic

$r=\sin\left(2\theta\right)^{60}+\left(\sin\left(\theta\right)\right)^{20\pi}$
θ (Theta) range : -20π < θ < 2000π \ Example

25) Digital Tribal

$r=\sin\left(2\theta\right)+\left(\sin\left(3\theta\right)\right)^{2}$
θ (Theta) range : -200π < θ < 2000π \ Example

26) Fantasy sword / Glaive

$r=\sin\left(2\theta\right)^{2}+\left(\sin\left(2\theta\right)\right)^{9}$
θ (Theta) range : -2π < θ < 2000π \ Example

27) Darth Vader Bunny

$r=\sin\left(2\theta\right)^{10}-\left(\sin\left(\theta\right)\right)^{45}-\cos\left(2\right)$
θ (Theta) range : -2π < θ < 2000π \ Example

28) Low-poly rabbit mesh

$r=\sin\left(2\theta\right)+\left(\sin\left(3\theta\right)\right)^{7}$
θ (Theta) range : -2π < θ < 2000π \ Example

29) Fantasy Cross

$r=\sin\left(2\theta\right)^{10}-\left(\sin\left(\theta\right)\right)^{45}-\cos\left(6\right)$
θ (Theta) range : -2π < θ < 2000π \ Example

30) Golden Ratio

$r\theta=\phi$ \ Φ (Phi) definition : $\phi=\frac{1+\sqrt{5}}{2}$
θ (Theta) range : -180< θ < 99999 \ Example