Cubes, Conic Sections, and Crockett Johnson - Introduction

Or, Answering “yes” to the question:

“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.

Introduction

Although it was written over 2,000 years ago (c. 300 BCE), Euclid’s Elements, a compilation of definitions, postulates, and propositions, serves as the basis of high school geometry courses taught today.  Since constructions were utilized extensively in the Elements, we begin with a brief overview of three classical Greek construction problems that arose at least a century before Euclid.  We will also explain how the author of children’s books became interested in mathematical constructions and thereby came to pose a question about cubes and conic sections.