Explore 30+ Beautiful Mathematical Shapes | Parametric & Polar Curves | Hearts, Fractals & More
Parametric Equations define curves using a parameter (usually t): x = f(t), y = g(t). This allows creating complex shapes that aren't functions in the traditional y = f(x) sense.
Polar Equations use distance r and angle θ: r = f(θ). Perfect for circular patterns, spirals, and symmetric designs like roses and cardioids.
| Curve Name | Type | Equations | Range | Description |
|---|---|---|---|---|
| ❤️ Hearts & Symbols | ||||
| Heart | Parametric |
x = 16sin³(t) y = 13cos(t) - 5cos(2t) - 2cos(3t) - cos(4t) |
0 ≤ t ≤ 2π | Classic Valentine heart shape using trigonometric powers |
| Heartbeat | Parametric |
x = t y = sin(t) + 0.5sin(5t) + 0.3sin(10t) |
0 ≤ t ≤ 4π | ECG-like waveform simulating heartbeat rhythm |
| Smiley | Multi-Trace |
Head: x = 5cos(t), y = 5sin(t) Eyes: circles at (-2,2) and (2,2) Smile: y = -0.5x² - 2 |
0 ≤ t ≤ 2π | Happy face with parabolic smile |
| Pac-Man | Parametric |
x = 5cos(t) y = 5sin(t) |
0.5 ≤ t ≤ 5.78 | Iconic game character - partial circle with mouth |
| Rainbow | Parametric |
x = 10cos(t) y = 10sin(t) |
0 ≤ t ≤ π | Semi-circular arc (upper half circle) |
| 🌹 Roses & Flowers | ||||
| Rose (5 petals) | Polar |
r = 5sin(5θ) Alternative: r = 5cos(5θ) |
0 ≤ θ ≤ 2π | Pentagonal symmetry - odd n gives n petals |
| Rose (8 petals) | Polar |
r = 5sin(4θ) Alternative: r = 5cos(4θ) |
0 ≤ θ ≤ 2π | Even n gives 2n petals |
| Clover (4-leaf) | Polar | r = 3|sin(2θ)| | 0 ≤ θ ≤ 2π | Four-lobed clover using absolute value |
| Sunflower | Polar |
r = 0.5√θ Based on golden angle (137.5°) |
0 ≤ θ ≤ 80 | Phyllotaxis pattern - Fibonacci spiral arrangement |
| 🌀 Spirals | ||||
| Archimedean Spiral | Polar | r = 0.5θ | 0 ≤ θ ≤ 20 | Linear growth - constant spacing between turns |
| Logarithmic Spiral | Polar |
r = e^(0.1θ) Also called: Equiangular spiral |
0 ≤ θ ≤ 15 | Exponential growth - found in nautilus shells |
| Galaxy Spiral | Polar | r = e^(0.15θ) | 0 ≤ θ ≤ 12 | Models spiral galaxy arms |
| Cardioid | Polar |
r = 4(1 + cos(θ)) Alternative: r = 4(1 - cos(θ)) |
0 ≤ θ ≤ 2π | Heart-shaped curve - epicycloid with 1 cusp |
| 🎵 Lissajous & Harmonic Curves | ||||
| Lissajous (3:2) | Parametric |
x = 5sin(3t) y = 5sin(2t) |
0 ≤ t ≤ 2π | 3:2 frequency ratio creates figure-8 pattern |
| Harmonograph | Parametric |
x = 5sin(2t) + 3sin(3t) y = 5cos(3t) + 2cos(5t) |
0 ≤ t ≤ 2π | Complex harmonic motion - pendulum-like drawing |
| 🦋 Butterflies & Nature | ||||
| Butterfly | Parametric |
x = sin(t) · (e^cos(t) - 2cos(4t) - sin(t/12)⁵) y = cos(t) · (e^cos(t) - 2cos(4t) - sin(t/12)⁵) |
0 ≤ t ≤ 12π | Fay's butterfly curve - transcendental function |
| DNA Helix | Parametric 3D |
x = 3cos(t) y = 3sin(t) (z = 0.5t for 3D view) |
0 ≤ t ≤ 8π | Double helix structure (2D projection) |
| ⬡ Geometric Shapes | ||||
| Ellipse | Parametric |
x = 8cos(t) y = 5sin(t) Standard form: x²/a² + y²/b² = 1 |
0 ≤ t ≤ 2π | Elongated circle - conic section |
| Star (5-pointed) | Polar | r = 3 + 2cos(5θ) | 0 ≤ θ ≤ 2π | Pentagram with rounded edges |
| Torus Knot (3,2) | Parametric 3D |
x = (2 + cos(2t))cos(3t) y = (2 + cos(2t))sin(3t) (z = sin(2t) for 3D) |
0 ≤ t ≤ 2π | Trefoil knot - simplest non-trivial knot |
| 🔲 Fractals & Chaotic Attractors | ||||
| Sierpinski Triangle | Chaos Game |
Vertices: (0,0), (1,0), (0.5,√3/2) xn+1 = (xn + vx)/2 yn+1 = (yn + vy)/2 |
8000 iterations | Self-similar fractal - dimension ≈ 1.585 |
| Dragon Curve | Iterative L-System |
Rules: R → R+L+, L → -R-L Angle: 90°, 12 iterations |
12 iterations | Paper-folding fractal - tiles the plane |
| Mandelbrot Set | Complex Iteration |
zn+1 = zn² + c z0 = 0, c = x + yi Plot if |z| < 2 after 100 iterations |
-2≤x≤1, -1.5≤y≤1.5 | Most famous fractal - infinite complexity at boundary |
| Julia Set | Complex Iteration |
zn+1 = zn² + c z0 = x + yi, c = -0.7 + 0.27i Plot if |z| < 2 after 100 iterations |
-2≤x≤2, -2≤y≤2 | Mandelbrot's twin - each c value gives different set |
| Lorenz Attractor | Differential Equations |
dx/dt = σ(y - x) dy/dt = x(ρ - z) - y dz/dt = xy - βz σ=10, ρ=28, β=8/3 |
5000 steps | Chaotic weather model - "butterfly effect" |
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