spiromeetriad

Spiromeetriad is a planar geometric figure defined as the union of three logarithmic spirals that share a common origin and are rotated by 120 degrees relative to one another. If S_k denotes a logarithmic spiral given in polar coordinates by r = a · exp(b(θ − 2πk/3)) for k = 0, 1, 2, then the spiromeetriad is S = S_0 ∪ S_1 ∪ S_2. This construction produces a triadic spiral motif with identical growth along each arm.

The name combines spiro- (spiral) with triad, reflecting the threefold, spiral symmetry of the figure. The term is used to describe a decorative or mathematical object rather than a standard geometric primitive.

Properties of the spiromeetriad include rotational symmetry of order three: rotating the figure by 120 degrees maps it onto itself. Each arm is a logarithmic spiral, so the overall shape exhibits self-similarity in the sense that zooming around the origin preserves the spiral form along all three arms. The figure extends indefinitely as θ increases or decreases, though practical realizations may clip it to a finite region. Parameters a and b control scale and tightness of the spirals, while the choice of θ range can affect the visible extent of each arm.

Variants of the spiromeetriad can be created by modifying the base curve family, such as using different growth rates for the spirals or limiting the angular domain to form finite motifs. It is related to concepts of logarithmic spirals, triadic symmetry, and composite spiral patterns found in decorative design and mathematical demonstrations of symmetry and self-similarity.