The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon

Reproduced  from  http://www.cs.uwaterloo.ca/journals/JIS/sommars/newtriangle.html

         

Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5

The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon

Steven E. Sommars
Lucent Technologies
Indian Hill, IL
Email address: sommars@enteract.com
and
Tim Sommars
Wheaton North High School
Wheaton, IL
Email address: ozzy50@juno.com

(Affliliation for identification only, the paper is not officially endorsed by either organization.)

Abstract: We consider the number of triangles formed by the intersecting diagonals of a regular polygon. Basic geometry provides a slight overcount, which is corrected by applying a result of Poonen and Rubinstein [1]. The number of triangles is 1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956 for polygons with 3 through 12 sides.

Introduction

If we connect all vertices of a regular N-sided polygon we obtain a figure with = N (N - 1) / 2 lines. For N=8, the figure is:

Careful counting shows that there are 632 triangles in this eight sided figure.

Derivation

All triangles are formed by the intersection of three diagonals at three different points. There are five arrangements of three diagonals to consider. We classify them based on the number of distinct diagonal endpoints. We will directly count the number of triangles with 3, 4 and 5 endpoints (top three figures). We will count the number of potential triangles with 6 endpoints, then correct for the false triangles. In each of the following five figures, a sample triangle is highlighted.

Three, Four and Five Diagonal Endpoints

3 diagonal endpoints. There are 56 such triangles in the figure at left.   The number of triangles formed by diagonals with a total of three endpoints is simply.

4 diagonal endpoints. There are 280 such triangles in the figure at left.   There are combinations of the four diagonal endpoints. For each set of four endpoints, there are four triangle configurations. Thus there are triangles formed.

5 diagonal endpoints There are 280 such triangles in the figure at left. For each of the N vertices of the polygon, there are four other diagonal endpoints which can be placed on the N-1 remaining locations. Thus there are triangles formed. This is equal to .

Six diagonal endpoints

The number of potential triangles formed by 6 line segments is , since there are 6 segment endpoints to be chosen from a pool of N. Often potential triangles are not created by three overlapping line segments because the line segments intersect at a single point. counts both of the following two situations.

6 diagonal endpoints, resulting in triangle. There are 16 such triangles in the figure at left.

6 diagonal endpoints, false triangle. There are 9 interior intersection points in the figure at left where such false triangles can be formed.

We use a result of [1] to count these false triangles. As in that paper, for a regular N-sided polygon, let a_m(N) denote the number of interior points other than the center where m diagonals intersect. Surprisingly, only the values m = 2, 3, 4, 5, 6 or 7 may occur. The requisite formulae from [1] are reproduced here:

 

where if , 0 otherwise.

If there are K line segments that intersect at one common point, where K>2, there are false triangles corresponding to that point. Thus the correction term for false triangles is

where the last term represents the contribution of the center point for even N. The correction is 0 for odd N. The number of triangles formed by line segments with six endpoints on the polygon is then:

 

Result

The table below summaries the results for . These values were checked through use of a computer program performing an exhaustive search.

N	Triangles with 3 diagonal endpoints	Triangles with 4 diagonal endpoints	Triangles with 5 diagonal endpoints	Triangles with 6 diagonal endpoints	Total Number of Triangles
3	1	0	0	0	1
4	4	4	0	0	8
5	10	20	5	0	35
6	20	60	30	0	110
7	35	140	105	7	287
8	56	280	280	16	632
9	84	504	630	84	1302
10	120	840	1260	180	2400
11	165	1320	2310	462	4257
12	220	1980	3960	796	6956
13	286	2860	6435	1716	11297
14	364	4004	10010	2856	17234
15	455	5460	15015	5005	25935
16	560	7280	21840	7744	37424
17	680	9520	30940	12376	53516
18	816	12240	42840	17508	73404
19	969	15504	58140	27132	101745
20	1140	19380	77520	38160	136200

The sequence formed by the total number of triangles was studied by the late Victor Meally in the 1960's, although it appears he did not find our formula for the N-th term. This is sequence A006600 in the On-Line Encyclopedia of Integer Sequences.

To summarize the final result, the number of triangles generated by intersecting diagonals of an N-regular polygon is:

References

[1] Bjorn Poonen, Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. on Disc. Math. Vol. 11, No. 1 (Feb 1998), pp. 133-156. Note that Theorem 1 has a typographical error: in the second line, 232 should be replaced by 262.

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Received Feb 24, 1998; published in Journal of Integer Sequences April 26, 1998.

The following are reproduced  from  http://www.research.att.com/~njas/sequences/A006600

A006600

Triangles in regular n-gon. (Formerly M4513)

+0 12

1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297, 17234, 25935, 37424, 53516, 73404, 101745, 136200, 181279, 236258, 306383, 389264, 495650, 620048, 772785, 951384, 1167453, 1410350, 1716191, 2058848, 2463384, 2924000, 3462305, 4067028, 4776219, 5568786, 6479551 (list; graph; listen)

	OFFSET	3,2
	COMMENT	Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?
	LINKS	T. D. Noe, Table of n, a(n) for n=3..1000 Sascha Kurz, m-gons in regular n-gons B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156. B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998). B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv version, which has fewer typos than the SIAM version. B. Poonen and M. Rubinstein, Mathematica programs for these sequences D. Radcliffe, Counting triangles in a regular polygon T. Sillke, Number of triangles for convex n-gon S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5. Sequences formed by drawing all diagonals in regular polygon
	EXAMPLE	a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.
	MATHEMATICA	del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; Tri[n_]:=n(n-1)(n-2)(n^3+18n^2-43n+60)/720 - del[2, n](n-2)(n-7)n/8 - del[4, n](3n/4) - del[6, n](18n-106)n/3 + del[12, n]*33n + del[18, n]*36n + del[24, n]*24n - del[30, n]*96n - del[42, n]*72n - del[60, n]*264n - del[84, n]*96n - del[90, n]*48n - del[120, n]*96n - del[210, n]*48n; Table[Tri[n], {n, 3, 1000}] - T. D. Noe (noe(AT)sspectra.com), Dec 21 2006
	CROSSREFS	Often confused with A005732. Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file. Adjacent sequences: A006597 A006598 A006599 this_sequence A006601 A006602 A006603 Sequence in context: A136016 A100907 A058102 this_sequence A005732 A040977 A036598
	KEYWORD	nonn,easy,nice
	AUTHOR	njas
	EXTENSIONS	a(3)...a(8) computed by Victor Meally (personal communication); later terms and recurrence from S. Sommars and T. Sommars.