Talk:Diagonal
|
|
Challenging Question: 2 for a triangle, 5 for a square, 12 for a pentagon...
We know a patter for the number of diagonals in an n-gon, but how about the number of regions determined by the number of diagonals??
To make sure you know what I mean, here are some examples:
- A circle determines 2 regions, the inside and the outside.
- A checkerboard determines 65 regions, the squares and the area outside the lines.
A regular hexagon has 25. A general hexagon, assumed to have no more than two diagonals intersecting at any point (other than a vertex) has 26. The general position is already done for us http://www.research.att.com/projects/OEIS?Anum=A027927 . -- Smjg 11:08, 26 May 2004 (UTC)
POLYGON DIAGONALS REGIONS DETERMINED BY DIAGONALS 2 "bi-angle" 0 1 3 Triangle 0 2 4 quadrilateral 2 5 (still 5 for a Square) 5 Pentagon 5 12 (still 12 for a regular pentagon) 6 Hexagon 9 26 (but only 25 for a regular hexagon) 7 Heptagon 14 51 8 Octagon 20 92 9 Enneagon 27 155 10 Decagon 35 247 11 Hendecagon 44 376 12 Dodecagon 54 551 13 Triskaidecagon 65 782 30 Icosagon 170 17876 Tricontagon 405 Tetracontagon 740 Pentacontagon 1175 Hexacontagon 1710 Heptacontagon 2345 Octacontagon 3080 Ennecontagon 3915 100 Hectagon 4850 Chiliagon 498,500 Myriagon 49,985,000 Googolgon Extra credit
The numbers in this table come from "the On-Line Encyclopedia of Integer Sequences" http://www.research.att.com/projects/OEIS?Anum=A027927 which lists "a(n) = number of plane regions after drawing (general position) convex n-gon and all diagonals".
Note that this is the "general position convex n-gon", not the "regular n-gon".
The formula is a(n)= 1 + binomial(n,4) + binomial(n-1,2).
(By "binomial()", I mean the binomial coefficient).
If there are any errors, *please* tell the people at "the On-Line Encyclopedia of Integer Sequences" so they can fix it. (Or tell me, and I'll forward it to them).
-- DavidCary 15:49, 26 Jun 2004 (UTC)
What are the number of plane regions after drawing the regular n-gon ?