In young's geometry, suppose that axiom 2 is changed to read as follows: There are exactly two points on every line. How many points and lines would the geometry have? What if every line had exactly four points? Generalize you result for the case where each line contain exactly n points (n being some positive integer).
Just to be clear Young's Geometry axioms:
- There exists at least one line.
- There are exactly three points on every line.
- Not all points are on the same line.
- There is exactly one line on any two distinct points.
- For each line l and each point P not on l, there exists exactly one line on P which is not on any point of l.
I think this would be an acceptable drawing if each line as exactly 2 points:
I was wondering if the above drawing is correct? Also does anyone have a drawing for young's geometry with exactly 3 points for a line? I don't have a drawing in my textbook.
I have looked at this further. I have concluded that the drawing is correct. I have also found a drawing for young's geometry with exactly 3 points. Now I am stuck on how to prove the number of lines in the 'exactly 4 points on a line' case. I do know that there must 16 points for this case.
I have solved this problem. For the curious minds, the 4 points on a line case has exactly 16 points and 20 lines.