Problem 60981. Mesh the pentagon (with the minimum number of triangles)

Created by Nicolas Douillet

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Problem statement
An pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set V, corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.
V = [1           0            0;
     cos(2*pi/5) sin(2*pi/5)  0;
     cos(4*pi/5) sin(4*pi/5)  0;
     cos(4*pi/5) sin(-4*pi/5) 0;
     cos(2*pi/5) sin(-2*pi/5) 0];
A triangulated mesh T (stands for triangles here) -or a triangulation- is simply a N x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where N is the number of triangles. 
Your task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, F. The row order of the triangles in the list doesn't matter.
Example
The first triangle here can be [1, 2, 3] if counterclockwise oriented.
Tip
Beware to avoid self intersecting triangles.
Forbidden functions / expressions
regexp
assignin
str2num
echo
See also
Mesh processing
Mesh generation toolbox

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25 Solutions
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Last Solution submitted on Sep 13, 2025

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Nicolas Douillet on 23 Jul 2025

To visualize and check your result you can do for instance :

trisurf(T,V(:,1),V(:,2),V(:,3));
colormap([0 1 1]);
axis equal, alpha(0.5);

Pauli Huusari on 11 Aug 2025

Hi, I was wondering should the following mesh pass the test:
T = [1, 2, 3; 5, 1, 3; 4, 5, 3]

There should not be intersecting triangles, and the triangle vertices should be enumerated counter-clockwise.

Nicolas Douillet on 13 Aug 2025

Hi Pauli. You are right of course. Actually it corresponds to the test case #4,

T_correct4 = [3 4 5; 3 5 1; 3 1 2];

but while improving the tests, I forgot to put back the circular shift possibilities. I am going to do it now. Thanks again.

Nicolas Douillet on 13 Aug 2025

There was a little bug in the tests. I just fixed them. Thanks again for this feedback Pauli.

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Problem 60981. Mesh the pentagon (with the minimum number of triangles)

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Problem 60981. Mesh the pentagon (with the minimum number of triangles)

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Last 200 Solutions

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Problem Recent Solvers18

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