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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

Eqn : \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

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

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

Example

9) Saint wings & halo

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

Example

10) Warrior's Cardioid

Eqn : 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

Eqn : 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

Eqn : 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

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

Example

14) Thorny Aster

Eqn : 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

Eqn : 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

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

Example

17) 4D Wormhole

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

Example

18) Dragonscale bracelet

Eqn : -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

Eqn 1 : 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
Eqn 2 : 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

20) Among Us compressed

Eqn 1 : 1.4x^{2}+4.2\left(y-1\right)^{2}\le0.18
Eqn 2 : r\le\left(\sin\theta\right)^{5}+0.5
θ (Theta) range : -2π < θ < 2π

Example

21) Arcaneus Tourbillion

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

Eqn : 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

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

Example

29) Fantasy Cross

Eqn : 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

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

Example