A self-similar helix as a metaphor for continuous time with nested periodicity

Humans perceive time as both linear and circular, which is reflected in the dichotomous way we typically organize and visualize temporal information in science: as either time series or periodic plots. These seemingly incompatible perspectives can be reconciled with a helical representation of time. Helices have been underused as a visualization of temporal information, in part because two-dimensional visualization of three-dimensional structures leaves portions of the data hidden and in part due to inherent disadvantages of radial visualizations over linear ones. Nonetheless, the helical metaphor of time brings along an important advantage: it offers an intuitive way to depict continuous time with periodicity at multiple scales. Here, I introduce a representation of time via a model of self-similar, or hierarchical, helices. Unlike conventional spiral and helix visualizations that represent periodicity at a single temporal scale, the proposed framework extends the helix concept to nested periodic structures, allowing multiple temporal scales to be represented simultaneously within a continuous temporal trajectory. This model can serve as a cognitive map or mental image of time, as well as an immersive data visualization approach for exploring patterns in timed data with nested periodicity. I demonstrate the model with a few examples (astronomical light phases, air temperature profiles, tidal cycles, and wildlife activity patterns) and provide R functions to generate interactive displays of temporal data projected onto hierarchical helix structures.