Integer-Digit Functions: An Example of Math-Art Integration

M athematics and the visual arts mutually reinforce one another [10]. On the one hand, many mathematical objects appear in artistic or decorative works [5, 6]. In particular, mathematical curves and art have a long-standing connection through the application of geometric principles [13, 20]. Simple curves such as the catenary are ubiquitous, for example in the work of the Catalan architect Antoni Gaudı́, as well as in ancient [19] and in modern architecture [12]. On the other hand, many mathematical objects display artistic appeal per se. The vast list includes, among others: knots [1, 2], mosaics and tiles [3, 21], Fourier series [9], topological tori [15], and fractal curves [4], all of which produce visual patterns of undeniable beauty. Such connections between mathematics and the arts are explored and celebrated annually by the Bridges community formed by mathematicians and artists [8, 18]. This article introduces a new family of curves in the plane defined by functions transforming an integer according to sums of digit-functions. These transforms of an integer N are obtained by multiplying the function value f N ð Þ by the sum of f ai ð Þ, where ai are the digits of N in a given base b and f is a standard function such as a trigonometric, logarithmic, or exponential one. These curves display a few attributes, such as beauty, symmetry, and resemblance to natural environments, which make them attractive for artistic purposes. Digit Sum with ‘‘Memory’’ A nonnegative integer N can be represented in a given base b according to the following expression