[QUOTE 4351964, member: 9609"]
But what I can't find out is WHY ?[/QUOTE]
As said already by another poster circle theorems are useful.
However in where group theory meets geometry and a few other things like graph theory there are plenty of demonstrations or proofs.
There are three regular tessellations in plane euclidean geometry because of the number of sides in relation to the number of vertices.
Call this [p,q] so p is the number of sides of the regular polygon and q is the vertices (where the points meet).
ICBST the angle sum of a polygon is 180(p-2) so each interior angle of the polygon is 180(p-2)/p
You have q polygons meeting at each vertex so
q. 180(p-2)/p = 360
divide by 180 and pull the p in the denominator across
q. (p-2) = 2p
pq - 2q = 2p
(p-2)(q-2) =4
It can be trivially seen that for a square which has 4 sides and 4 vertices meeting, i.e. [p,q] = [4,4]
that it works. It also works for [6,3] and [3,6].
Probably something wrong in there but you get the idea.
Endless fun beyond circle theorems. There are only 17 plane symmetry groups or 'wallpapers'.
https://caicedoteaching.files.wordpress.com/2012/05/nelson-newman-shipley.pdf
But what I can't find out is WHY ?[/QUOTE]
As said already by another poster circle theorems are useful.
However in where group theory meets geometry and a few other things like graph theory there are plenty of demonstrations or proofs.
There are three regular tessellations in plane euclidean geometry because of the number of sides in relation to the number of vertices.
Call this [p,q] so p is the number of sides of the regular polygon and q is the vertices (where the points meet).
ICBST the angle sum of a polygon is 180(p-2) so each interior angle of the polygon is 180(p-2)/p
You have q polygons meeting at each vertex so
q. 180(p-2)/p = 360
divide by 180 and pull the p in the denominator across
q. (p-2) = 2p
pq - 2q = 2p
(p-2)(q-2) =4
It can be trivially seen that for a square which has 4 sides and 4 vertices meeting, i.e. [p,q] = [4,4]
that it works. It also works for [6,3] and [3,6].
Probably something wrong in there but you get the idea.
Endless fun beyond circle theorems. There are only 17 plane symmetry groups or 'wallpapers'.
https://caicedoteaching.files.wordpress.com/2012/05/nelson-newman-shipley.pdf
