Rotation and scaling, composed

Parametric equations

Each step multiplies the shape by the rotation-scaling matrix $M=s\cdot R(\theta )$:

θ is the rotation step between neighboring layers. s (< 1) sets how fast the layers shrink toward the center. Lines joining the vertices of the outer frame and the inner layers weave into a grid shaped like a logarithmic spiral.

Interaction

• Rotation step θ: the rotation of each stacked square, which changes how strongly it spirals
• Scale factor s: the scaling at each step, which makes the pattern shrink toward the center or spread out
• Evolution speed ω: drive the periodic change of the parameters and watch the spiral pattern evolve

What to notice

• Rotation and uniform scaling combine into a single matrix. In this setup the two commute, and only repeated iteration builds the spiral.
• When the scale factor is close to 1, the pattern gets denser. When it is too small, the pattern collapses quickly to the center.
• Repeated iteration produces a visual structure like a logarithmic spiral.

Related

• Linear transformation grid
• Affine transformation pattern
• Equiangular spiral

Further reading

• Rotation matrix (Wikipedia)