This paper presents a brief historical review of G. I. Taylor’s solution of the point blast wave problem which was applied to the Trinity test of the first atomic bomb. Lie group symmetry techniques (also referred to throughout this paper as geometric techniques) are used to derive Taylor’s famous two-fifths law that relates the position of a blast wave to the time after the explosion and the total energy released. The theory of exterior differential systems is combined with the method of characteristics to demonstrate that the solution of the blast wave problem is directly related to the basic relationships that exist between the symmetry (or geometry) and the physics of wave propagation through the equations of motion. The point blast wave model is cast in terms of two exterior differential systems, and both systems are shown to be integrable with local solutions for the velocity, pressure, and density along curves in space and time behind the blast wave. This work is dedicated to the memory of Professor Roy Axford, who introduced many of his students to the topic of symmetry analysis of differential equations.