Counting the Number of Triangles in a Polygon

GrahamHills · 2014-01-07 22:05:53

Is there a formula to calculate the below problem

When a square has diagonals drawn across it the resultant triangles you can see equals 8. The four small ones down each side, plus the four larger triangles formed by the diagonals.

When a Pentagon has the diagonals drawn inside it the number of triangles you can see now is 35.

I am looking for a formula that will give me the number of triangles you will see in a hexagon based on the number of sides in relation to the number of triangles you will have.

anonimnystefy · 2014-01-07 22:55:16

Are you using regular polygons only?

Bob · 2014-01-07 23:45:08

hi Stefy,

Does regular matter, then? Concave/convex will though.

Come to think of it, when you make a regular hexagon the diagonals meet at a point. If you offset one point on the edge a new triangle appears at that point. Hmmm. This may not be so easy.

Bob

anonimnystefy · 2014-01-08 00:02:18

That's why I'm asking.

http://oeis.org/A006600

bobbym · 2014-01-08 00:46:00

Hi;

There is a formula on one of the links.

phrontister · 2014-01-08 02:27:56

Hi;

Edit: The following only works for regular polygons, and finds the "total number of triangles visible in regular n-gon with all diagonals drawn."

On stefy's link there appears the following Mathematica formula by T. D. Noe that produces a table comprising the number of triangles in the first 998 polygons (ie, from 3-sided to 1000-sided):

del[m_, n_] := If[Mod[n, m] == 0, 1, 0]; 
Tri[n_] := 
 n (n - 1) (n - 2) (n^3 + 18 n^2 - 43 n + 60)/720 - 
  del[2, n] (n - 2) (n - 7) n/8 - del[4, n] (3 n/4) - 
  del[6, n] (18 n - 106) n/3 + del[12, n]*33 n + del[18, n]*36 n + 
  del[24, n]*24 n - del[30, n]*96 n - del[42, n]*72 n - 
  del[60, n]*264 n - del[84, n]*96 n - del[90, n]*48 n - 
  del[120, n]*96 n - del[210, n]*48 n; 
Table[Tri[n], {n, 3, 1000}]

Changing the last line to the following will open a 'prompt' window asking for input of the number of sides of a single polygon, and the output will be the result for just that polygon, being the last entry in a table (not printed) comprising the number of triangles in all polygons up to and including the input one:

Last[Table[Tri[n], {n, 3, Input["Enter number of sides of a single polygon"]}]]

I have no idea how that program works - only that it does - nor its upper limit (or if there is one).

Last edited by phrontister (2014-01-08 09:17:32)

Bob · 2014-01-08 03:45:27

hi GrahamHills

Welcome to the forum.

That link is for regular polygons only.

See diagrams below.

Bob

phrontister · 2014-01-08 09:09:25

Hi Bob,

Ah, yes...I'd seen that important point discussed somewhere, with a piccy too like yours that clearly demonstrates the different result between regular and irregular, but overlooked mentioning that in my post.

anonimnystefy · 2014-01-08 11:46:25

Of course, if we look at irregular ones as well, there is obviously no formula. We might only be able to provide the range of possible numbers.

bobbym · 2014-01-08 20:57:04

Prof Scott wrote:

When I have 12 unanswered questions I stop asking anymore.

Forget about non-regular polygons. It will be hard enough to try to find an algebraic formula for the regular ones. There is one that is known but it is not elementary.

If this were another forum I would put a bounty on the problem of 100 points to the first person who could relate directly the number of sides ( of a regular polygon ) to the number of triangles using only elementary functions.

Wikipedia wrote:

In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and roots of polynomial equations through composition and combinations using the four elementary operations (+ - × ÷). By allowing these functions (and constants) to be complex numbers, trigonometric functions and their inverses become included in the elementary functions (see trigonometric functions and complex exponentials).

Bob · 2014-01-08 21:30:17

Hhmmm. 100 points eh? Interesting.

Bob

anonimnystefy · 2014-01-09 01:29:11

Can't we use Wilf's roots of unity idea here?

bobbym · 2014-01-09 02:03:09

I thought that was only for say every other coefficient or every third and so forth?

anonimnystefy · 2014-01-09 02:06:19

That's exactly what this is. Look at the del function a bit better. m and n have switched places!

bobbym · 2014-01-09 02:20:29

You would need to replace del with the general term. I do not see how to do that using the roots of unity. They would sum the gf not get the general term. I have already succeeded for the first few del's.

Math Is Fun Forum

#1 2014-01-07 22:05:53

Counting the Number of Triangles in a Polygon

#2 2014-01-07 22:55:16

Re: Counting the Number of Triangles in a Polygon

#3 2014-01-07 23:45:08

Re: Counting the Number of Triangles in a Polygon

#4 2014-01-08 00:02:18

Re: Counting the Number of Triangles in a Polygon

#5 2014-01-08 00:46:00

Re: Counting the Number of Triangles in a Polygon

#6 2014-01-08 02:27:56

Re: Counting the Number of Triangles in a Polygon

#7 2014-01-08 03:45:27

Re: Counting the Number of Triangles in a Polygon

#8 2014-01-08 09:09:25

Re: Counting the Number of Triangles in a Polygon

#9 2014-01-08 11:46:25

Re: Counting the Number of Triangles in a Polygon

#10 2014-01-08 20:57:04

Re: Counting the Number of Triangles in a Polygon

#11 2014-01-08 21:30:17

Re: Counting the Number of Triangles in a Polygon

#12 2014-01-09 01:29:11

Re: Counting the Number of Triangles in a Polygon

#13 2014-01-09 02:03:09

Re: Counting the Number of Triangles in a Polygon

#14 2014-01-09 02:06:19

Re: Counting the Number of Triangles in a Polygon

#15 2014-01-09 02:20:29

Re: Counting the Number of Triangles in a Polygon

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