The term ‘natural calligraphy’ is introduced to describe a class of dynamic lines found in nature. These lines are well-defined although sometimes short-lived. They occur in different natural processes and on different scales. A selection is presented in this paper so as to better understand the class. They possess, for instance, cusps in density, involved topology and sweeping curves. Their occurrence is sometimes surprising because diffusive processes lead to smoothening of densities, the weakening of strong gradients, and the erasure of edges and sharp boundaries. As they change over time, forms of the line - ‘characters’ - emerge that seem particularly striking but these can disappear as quickly as they appear. As these characters evolve from and to less remarkable forms, we have a better chance of understanding the characteristics that make them appear compelling or beautiful. The lines exist in higher-dimensional spaces and we present two-dimensional projections of these spaces (photographs and images from simulations) as illustrations in this paper. Features that we describe as striking are often a result of this projection so that the natural form and the viewer both play a part in creating a ‘work of art’. We distinguish two types of experiment that we can conduct simultaneously in natural calligraphy. First, there is the scientific study of the natural processes creating the line. These processes possess both predictable and random (chaotic) elements. They are therefore not strictly repeatable although some features of the process are robust. Second, there is our placement in, and subjective selection from, the process as artists. We discuss these two types of experiment in more detail in relation to our examples. The exciting possibility suggested by these natural processes is a dynamic evolving calligraphy with a continuum of forms and symbols. Modern developments in the concept of trajectory in science have enriched the simple idea of a line and the extremal principles at the heart of these developments underlie the aesthetic of natural calligraphy. This paper aims to give an introduction to these ideas that might be of interest to both to scientists and practitioners in the arts.