A333944 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.

2, 6, 203, 1723816861

Offset: 1

Author

Joel David Hamkins, Apr 11 2020

Keywords

• nonn
• hard
• more

Comments

The fourth value was computed by Teofil Camarasu, taking 6 days of computer time performing exact algebraic computations (see link to Mathematics Stack Exchange).

Note that circles are formed only by specifying an already existing segment as radius (so we have collapsible compasses only).

Examples

			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.
		

References

• Joel David Hamkins, Lectures on the Philosophy of Mathematics, MIT Press, 2020, chapter 4. (in press)

Links

• Joel David Hamkins, Mathematics Stack Exchange question, What is the next number on the constructibility sequence? And what is the asymptotic growth?, including answers by Teofil Camarasu and Pace Nielsen.
• Joel David Hamkins, Illustrating the construction sequence: Begin with 2 points, stage 1, 6 points at stage 2, 203 points at stage 3.