Circles and Infinity
As we go outword from the inner circle, the gaps between the lines joining the centre to the circumference widens:
This essentially means that even though you can theoretically you can have an infinite number of lines on the inner circle, they are never enough to cover the outer circle(s). I was made to ponder on this fact while watching a BBC documentary titled “Dangerous Minds”.
Here is the MATLAB code (works in GNU Octave):
clear all;close all % radius for r=1:10 lines=zeros(361,2); cx=;cy=; for theta=0:360 x=r*cos(theta); y=r*sin(theta); cx=[cx;x]; cy=[cy;y]; end % plot the circle plot(cx,cy,'r.'); hold on % lines matrix lines=[lines cx cy]; % plot the lines for i=1:size(lines,1) plot([lines(i,1) lines(i,3)],[lines(i,2) lines(i,4)],'b-'); hold on end end
And here is another thing for you to think about, also mentioned in the documentary: A circle can be thought of as having infinite points on its circumference. The bigger circle looks like having more points on the circumference, right?
Now consider these two circles are concentric: so for every point on the outer circle, a line joining it to the centre intersects the inner circle at a point. So, is there really more points on the circumference of the bigger circle? Time to blame infinity.