Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (2012)

Pages 183–190

Natural (or intrinsic) coordinate systems parameterize curves based on their inherent properties such as arc length and tangential angle, independent of external reference frames. They provide a convenient means of representing many organic, flowing curves such as the meandering of streams and ocean currents. However, even simple functions written in natural coordinates can produce surprisingly complex spatial patterns that are difficult to predict from the original generating functions. This paper explores multi-frequency, sine-generated patterns in which the tangential angle of the curve is related to the curve's arc length through a series of sine functions. The resulting designs exhibit repeating forms that can vary in subtle or dramatic ways along the curve depending on the choice of parameter values. The richness of the "pattern space" of this equation suggests that it and other simple natural equations might provide fertile ground for generating geometric, organic and even whimsical patterns.

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