Mysterious Circle Numbers.Does πp,q Approach πp When q Is Tending to p ?

Abstract

This paper aims to introduce a mathematical-philosophical type of question from the fascinating world of generalized circle numbers to the widest possible readership. We start with recalling well-known (in part from school) properties of the polygonal approximation of the common circle when approximating the famous circle number π by convergent sequences of upper and lower bounds being based upon the lengths of polygons. Next, we shortly refer to some results from the literature where suitably defined generalized circle numbers of l p - and l p , q -circles, π p and π p , q , respectively, are considered and turn afterwards over to the approximation of an l p -circle by a family of l p , q -circles with q converging to p, q → p . Then we ask whether or not there holds the continuity property π p , q → π p as q → p . The answer to this question leads us to the answer of the question stated in the paper’s title. Presenting here for illustration true paintings instead of strong technical or mathematical drawings intends both to stimulate opening heart and senses of the reader for recognizing generalized circles in his real life and to suggest the philosophical challenge of the consequences coming out from the demonstrated non-continuity property.