Spiral Constructions

While building the contact sheet in Making and Breaking the Grid, I started to research construction methods for different spirals, given that I wanted to build a parametric component which I could configure for different line weights and spacing. There were a few I already knew of from school, but my descriptive geometry days were long behind me. I'll make any processing scripts or grasshopper components available on github.

Overview

Below is a refresher for this family of curves. For all constructions, the following variables can be read as:

• r distance (from origin to any plotted point)
• θ angle in radians
• a scaling factor (spacing between subsequent turns)

Archimedean Spiral

Properties

• Equal spaceing between turns (constant loops)
• Models linear growth

Contruction

r = a + bθ where as θ increases, r increases slowly

Placeholder image

Logarithmic (Equiangular) Spiral

Properties

• Distance between each loop/turn grows exponentially
• Seen in the natural world (nautilus shells, galaxies, hurricanes)

Construction

r = ae^(bθ)

• b tightness of the spiral

Placeholder image

Fermat's Spiral (Parabolic)

Properties

• Used for phyllotaxis (like sunflower seed packing, petal distribution)

Construction

r = a√θ

• As θ increases, r increases slowlv

Placeholder image

Hyperbolic Spiral

Properties

• Turns infinitely towards the center; approaches the origin asymptotically, never actually reaching
• Seen in lens distortion patterns, or for modelling physical systems like gravitational orbits

Construction r = a⁄θ

• As θ approaches ∞, r approaches 0 (winds tighter toward the center)
• As θ approaches 0,r approaches ∞ (grows outward fast)

Placeholder image

Lituus Spiral

Properties

• Variant of the hyperbolic spiral with slower inward decay due to the square root
• As θ increases, r decreases
• Rarely found in nature

Construction r = a⁄√θ where:

• a is a near constant scale

Placeholder image

Golden Spiral

Properties

• Grows exponentially from the center
• Can be approximated using subsequent dividing golden rectangles
• Found in natural patterns like shells and plants

Construction r = ae^(bθ) where:

• b is ln(φ)/θ₉₀, the growth rate (set to golden ratio)
• φ golden ratio, approximately 1.618

Placeholder image

Constructing Archimedean Spirals for Plotting