Cubes, Conic Sections, and Crockett Johnson

Or:  “What do the straightedge lines and compass arcs do when two parabolas and a hyperbola double a cube, just stand watching?”

In the early 1970s Crockett Johnson, author of the children’s book Harold and the Purple Crayon, sent a geometric diagram to a friend noting that the diagram “answers the question in so many minds ‘What do the straightedge lines and compass arcs do when two parabolas and a hyperbola double a cube, just stand watching?’” [Crockett Johnson to Mickey Rosenau, n.d., Rosenau Collection, Smithsonian Institution].  The diagram and the answer to this question are addressed at the end of this paper.