| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mi>π</mi> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>- and <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> </semantics> </math> </inline-formula>-circles, <inline-formula> <math display="inline"> <semantics> <msub> <mi>π</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>π</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> </semantics> </math> </inline-formula>, respectively, are considered and turn afterwards over to the approximation of an <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-circle by a family of <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> </semantics> </math> </inline-formula>-circles with <i>q</i> converging to <i>p</i>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>→</mo> <mi>p</mi> </mrow> </semantics> </math> </inline-formula>. Then we ask whether or not there holds the continuity property <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>π</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo>→</mo> <msub> <mi>π</mi> <mi>p</mi> </msub> <mspace width="4.pt"></mspace> <mi>as</mi> <mspace width="4.pt"></mspace> <mi>q</mi> <mo>→</mo> <mi>p</mi> </mrow> </semantics> </math> </inline-formula>. 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.
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