By Dewan Mukto Browse All
Licensed under CC BY 4.0 （Unless specified otherwise)
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