wendy wrote:Seems you go up a dimension every power of 4 completed. So your next post ought make you a line citizen,
Marek14 wrote:Although cube has six faces, they are squares (therefore the 4), and there are three squares meeting at every vertex of the cube (therefore the 3).
If we reduce the angle even more, we get into hyperbolic realm. The edge of triangle grows as the angle shrinks. The limit is {3,oo} covering with infinite triangles that have 0-degree angles.
iNVERTED wrote:Marek14 wrote:Although cube has six faces, they are squares (therefore the 4), and there are three squares meeting at every vertex of the cube (therefore the 3).
Gotcha.If we reduce the angle even more, we get into hyperbolic realm. The edge of triangle grows as the angle shrinks. The limit is {3,oo} covering with infinite triangles that have 0-degree angles.
What exactly is a hyperbolic realm? It's impossible to have a {3,7} thing... you just can't fit seven triangles round a point.
Marek14 wrote:2. There is a relation between angle sum and area. However, in hyperbolic realm, the area grows as the angle sum SHRINKS. One weird result is that even when you shrink the angle sum all the way to zero, putting the vertices at infinity, the area will stay finite - which means that there is an UPPER LIMIT of area of hyperbolic triangle which can be never surpassed.
http://www.superliminal.com/geometry/geometry.htm
iNVERTED wrote:Ok, I understand all that now, but...Marek14 wrote:2. There is a relation between angle sum and area. However, in hyperbolic realm, the area grows as the angle sum SHRINKS. One weird result is that even when you shrink the angle sum all the way to zero, putting the vertices at infinity, the area will stay finite - which means that there is an UPPER LIMIT of area of hyperbolic triangle which can be never surpassed.
The upper limit is a constant? Or does it depend on the curvature of the realm?
As the angle sum of triangle goes to zero, its sides grow to infinity, but the area stays within the limit.
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