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%I #21 Aug 20 2022 02:22:32
%S 2,6,203,1723816861
%N Constructibility sequence, the number of points constructed by straightedge and compass after successive stages of forming all possible lines, circles, and intersection points, beginning with 2 points.
%C The fourth value was computed by Teofil Camarasu, taking 6 days of computer time performing exact algebraic computations (see link to Mathematics Stack Exchange).
%C Note that circles are formed only by specifying an already existing segment as radius (so we have collapsible compasses only).
%D Joel David Hamkins, Lectures on the Philosophy of Mathematics, MIT Press, 2020, chapter 4. (in press)
%H Joel David Hamkins, Mathematics Stack Exchange question, <a href="https://math.stackexchange.com/q/3377988/413">What is the next number on the constructibility sequence? And what is the asymptotic growth?</a>, including answers by Teofil Camarasu and Pace Nielsen.
%H Joel David Hamkins, Illustrating the construction sequence: <a href="/A333944/a333944.jpg">Begin with 2 points, stage 1</a>, <a href="/A333944/a333944_1.jpg">6 points at stage 2</a>, <a href="/A333944/a333944_2.jpg">203 points at stage 3</a>.
%e Begin with 2 points A and B. Construct the line containing them and the two circles having AB as radius; these intersect at four additional points, making 6 points in all. Using these 6 points, one may construct 9 new lines and 14 new circles, leading to 203 intersection points in all.
%K nonn,hard,more
%O 1,1
%A _Joel David Hamkins_, Apr 11 2020