This is based on the work of George Hart (www.georgehart.com). But I'm not using his templates :).
It is donde with 20 equal equilateral triangles, you need at least 5 colors so you don't get two triangles with the same color together.
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Step 1: Making the Triangles
So, we need 20 triangles. I used 5 square pieces of paper of different colors, I found that the configuration of the picture maximizes the size of the triangles with a side length of (square side)x((3-sqrt(3))/2).
One way to construct the triangles is first drawing an equilateral triangle with the same side length as the square and taking the distance from the top vertex to the center of the square with a compass.
Then mark that distance from each vertex of the square to the side on its right (clockwise).
Again with the compass take the length from the other vertex to the mark you just did.
Now you have to make the triangles with the length you have with base on the sides of the square as they are on the picture.
Now you cut all the triangles :D.
Step 2: Preparing the Triangles
Now we have to make a little cut to every triangle so we can ensamble the structure later.
I made the cut one third the side of the triangle but you can make it smaller. You ca also make them bigger but I don't know how much :P.
You can mark he thirds of the sides with a compass and then drawing the lines connecting each mark.
Now you have to cut the lines we just drew but only half of it. I drew marks on the middle of each line with not very rigorously but that wasn't a problem because the triangles were small enough. Anyway when you ensamble the triangles some times you have to cut a little more so they match.
You have to cut the halves in either clockwise or counter-clockwise orientation, otherwise you will have triangles that show the marks when you have the structure done.
Again, you have to do this with every triangle.
Step 3: Ensamble and Finish
To ensamble you have to start with 5 triangles and slide through the cuts of each of them so you get a ring of triangles.
Do this carefully, if you apply to much force, you can rip the paper and you will have to make a new triangle.
If you find the triangles won't slide cut a little more and try again.
Now, keep on adding triangles so you continue forming these rings with pentagons in their center. Keep an eye so you don't get 2 triangles of the same color touching each other.
Finally, you get it finished :D.
I hope you like it and find it a fun as I did :).