Geometric solution of a six order equation by three-fold origami

Abstract

Robert J. Lang has proposed a theorem that when solving equations using multi-folds origami, general equations of order n can be solved using n-2 simultaneous folds. However, recently Jorge C. Lucero proved that arbitrary five order equations can be solved using two simultaneous folds. Combining this with the fact that one single fold can solve general quartic equations, the writer questions that whether the theorem may be altered into general equations of order n can be solved by n-3 simultaneous folds. Thus, in this paper, a method of geometric graphing -Lills method is used to try to solve six order equations with three simultaneous folds. By conducting case analysis using theoretical knowledge, it can be found that the six order equations can be possibly solved by three simultaneous folds. Besides, a comparison of solving equations with origami constructions and compass-and-straightedge is carried out. The result will encourage more research on using origami to solve higher order equations and inspire people to pay more attention to origami construction, which is more powerful, accurate, and efficient than the compass-and-straightedge people usually use.