This book in a single contribution for the first time examines the historical development of the concepts of vortex, vorticity, knot, chirality, and helicity between 1517 and 1969. It considers their origins in early pictorial and conceptual studies, their subsequent formalization in mathematical analysis, and their eventual consolidation as interconnected elements within fluid mechanics, mathematical physics, geophysics, and astrophysics.

Attention is given to the progression from qualitative representations to formal definitions, including the emergence of vorticity, curl, and the gradient operator, as well as to the introduction of knot, chirality, and helicity as new means of describing physical and geometrical structure. The discussion emphasizes the extent to which these notions evolved in parallel, with recurring intersections between geometry, physics, and analysis.

The volume is intended for undergraduate and postgraduate students and researchers in applied mathematics, fluid mechanics, dynamo theory, topology, knot theory, geophysics, astrophysics, and the history of mathematical physics.

John Z. Shi obtained a Bachelor of Science (B.Sc.) from the University of Nanjing in 1986 and was awarded a Sino-British Friendship Scholarship Scheme in 1987, subsequently gaining a Doctor of Philosophy (Ph.D.) from the University of Wales in 1992. He became a Full Professor at Shanghai Jiao Tong University in 1998 and was a recipient of the National Science Fund for Distinguished Young Scholars awarded by the National Natural Science Foundation of China in 2002. He was appointed a Distinguished Professor at the School of Ocean and Civil Engineering, Shanghai Jiao Tong University, China, in 2009. He spent two Michaelmas terms in 2012 and 2019 at the G.K. Batchelor Laboratory, the Department of Applied Mathematics and Theoretical Physics, the University of Cambridge, U.K. He was awarded a Senior Visitor by Emmanuel College Cambridge in 2019 and was elected a Visiting Fellow at Wolfson College Cambridge in 2019. His field of research is in environmental fluid mechanics with a focus on vortex and small-scale turbulence in the atmosphere and ocean. He is currently a Member of Editorial Board of Environmental Fluid Mechanics.