Zig-Zag Paths and Neusis Constructions of a Heptagon and a Nonagon

Abstract

SummaryAlthough it is impossible to construct a regular heptagon and a regular nonagon using a compass and unmarked straightedge, it is possible to construct them with a compass and marked straightedge using the neusis technique. We give a geometric proof of Johnson’s neusis construction of the regular heptagon, which he had proven using trigonometry. We do so using so-called central triangles and zig-zag paths in the polygons. We then give efficient neusis constructions of the regular heptagon and the regular nonagon. Additional informationNotes on contributorsDan Lawson Dan Lawson (dlawson@peralta.edu) received his MS in mathematics from San Jose State University in 1994. He teaches at Merritt College in Oakland, CA. He enjoys working on problems in recreational mathematics, including solving and creating sudoku puzzles.David Richeson David Richeson (richesod@dickinson.edu) is a professor of Mathematics and the John J. and Ann Curley Chair in Liberal Arts at Dickinson College. He is the author of Tales of Impossibility (Princeton University Press, 2019) and Euler’s Gem (Princeton University Press, 2008), and he is a past editor of Math Horizons.