In mathematics, structure is the act of equipping sets with additional features. It encompasses concrete objects like geometric shapes, abstract frameworks that model principles like optimization and probability, and even patterns whose only purpose is to embody a sense of balance and beauty.
Beauty in mathematics is not ornament but structure itself. The golden ratio appears in the proportions of a regular pentagon and in the self-similar spirals of a logarithmic curve, while the Platonic solids stand as the purest examples of perfect symmetry in three dimensions. Symmetry, the structure of repetition, acts as both a visual anchor and a mathematical invariant, reminding us that elegance can coincide with rigor.
Optimization provides a different lens on structure. A centroidal Voronoi diagram, for instance, partitions space so that each site is the centroid of its own region, minimizing distortion much like an ideal clustering algorithm minimizes within-group variance. Similar variational ideas drive the design of minimal surfaces, and many other constructions where the best solution emerges by balancing competing constraints.
Rosettes embody all of these structural ideals. Defined as centroidal Voronoi diagrams bounded by a regular convex polygon, Rosettes unite rigid Euclidean geometry with the principles of optimization and symmetry. As we'll see, Rosettes emerge organically from random inputs, their intricate order forming out of chaos. Strictly speaking, a Rosette is any centroidal Voronoi diagram with a regular convex boundary. This includes symmetric configurations as well as asymmetric ones. It's their tendancy toward symmetric arrangements that makes them both mathematically interesting and aesthetically pleasing.
While geometry alone defines Rosettes, color plays an important role in how we experience them. A consistent coloring scheme can highlight a Rosette's symmetry, can be useful for spotting patterns across different Rosettes and may even help reveal a deeper structure or theory behind them.
Rosettes can also be colored purely for aesthetic effect. Custom colorings allow for creative expression and can highlight different aspects of a Rosette, often in surprising and beautiful ways.
As we'll see, each Rosette corresponds to the convergence of an energy minimizer function. Identified up to oriented similarity, the space of Rosettes is therefore discrete and countable despite being infinite. This means that each one can, in principle, be enumerated, labeled and compared. How, then, can we detect or identify novel Rosettes? Can we construct them from more fundamental principles? Can we uncover them through randomized search alone? Perhaps the answer lies in combining the power of automation with human intuition.
The tools to unravel the mystery of Rosettes are now at your fingertips. Dig deeper into their theory, browse the gallery of known configurations, or open the Rosette Studio and find your own.