2018年4月23日月曜日

The Sylvester-Gallai Theorem

The deformation is the mathematical concept.

The Sylvester-Gallai Theorem is that a finite set of points in the plane have the proper line through any two of them, and it passes through a third point of the set. Moreover, it must be on the same line.

P is the the noncollinear finite set. S(P) is the set of connecting lines in P.

p∈P, s∈S(P)


p and s are perpendicular, so they are not the ordinary line. (s*,p*) is the smallest distance, and s* is in the ordinary line.

l∩P={p}


l must be the ordinary line. This is the contradiction.

l∩S(P)


There is the cyclically intersection points (p,x1).

x1,・・・xk is on l. Moreover it crosses S. This must be ordinary.


You also see the extra dimensions.



i lines are determined by P.



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