Unlocking the underlying information of golden ratio in the Hogarth curves

Purpose Curves with various profiles have been demonstrated to be more attractive and decorative than the straight lines by William Hogarth. Among all kinds of curves, Hogarth proposed seven serpentine lines as the most beautiful curves, i.e., Hogarth curves. Those seven Hogarth curves are subsequently applied in a wide range of fields, e.g., sculpture, painting, architecture and fashion design, indicating their significance to the development of the formal beauty. Recently, the beauty of Hogarth curves has been suspected to be induced by their special-designed curvature, which could have the potential relationship with the Golden Ratio. The purpose of this paper is to explore the relationship between the Hogarth curves and golden ratio by comparing the curvature of curves with the Fibonacci sequence. Design/methodology/approach Each of the Hogarth curves was fully restored and divided into two parts according to the turning point of the curvature; the ratios of span, curvature and angles between these two parts were compared with the Fibonacci sequence. Findings The experimental results disclosed that the ratio of the fourth Hogarth curve, which was considered as the most beautiful line by Hogarth, was infinitely approaching the golden ratio. Based on the relationship between the fourth Hogarth curve and the golden ratio, the ratios of each curve were employed to define and normalize these curves, providing a quantitative way to redraw the Hogarth curves. Originality/value This research work unlocked the information of the relationship between the Hogarth curves and golden ratio, and proposed an effective and convenient mathematic way to quantify the Hogarth curves. The experimental findings disclosed the underlying mechanisms of the beauty of the forth Hogarth curves. Such a fundamental study will promote the establishment of the normalized methods for evaluating the beauty of arts and provide novel ideas for researchers and industrial technologists to use the Hogarth curves.