Use of a computer to find the number of regular pentagons that can simultaneously touch a given one

Abstract

Around an initial regular pentagon one describes a contour L on which one introduces a measure m. One investigates the difference S(M)=1/7m(L)−m(L∩M) where M is a pentagon touching the initial one and congruent to it. The geometric part of the investigation reduces the proof of the inequality S(M)<0 for all M to the proof of the negativity of two effectively computable functions F(u,v) and G(v) in the compact domain of the variation of the arguments. By the method of demonstrative computations, one calculates on a computer the values of these functions at the nodes of a rectangular net of the domain of the variation of the arguments by taking into account the monotonicity and one estimates the computational error. The results of the computation show that we have the inequality S(M)<0, from where it follows that the desired number is equal to six.