There are 104 triangles in the figure.
To organize our approach, we create a two-dimensional representation of the decorative object, labeling the triangles 1 to 24, noticing that the interior of the figure is a regular hexagon. First, we count 24 single triangles. Then we count triangles that are constructed from two single triangles, such as the triangle created by combining triangles 1 and 13 and that triangle's mirror image, triangles 2 and 14. We find a total of 12 triangles of this type. Two more triangles created by combining two single triangles are triangles 1 and 2 and its mirror image, triangles 13 and 14. We find 12 more triangles of this type. Therefore, we count a total of 24 triangles constructed from two single triangles.
Next we look for triangles that are created from three single triangles. An example of this type of triangle is triangles 1 and 13 and 24. Finding mirror images, we count a total of 12 such triangles, and we find no triangles that are exclusively in the regular hexagon of the figure.
Now we examine the figure for triangles made up of four smaller triangles, such as triangles 1 & 13 & 24 & 23. There are 12 of this type of triangle.
There are no large triangles made up of five smaller triangles.
We find 6 large triangles containing six smaller triangles, one of which is triangles 1 & 13 & 24 & 23 & 22 & 10.
No large triangles are formed by seven smaller triangles.
Large triangles containing eight smaller triangles can be created in two ways. We found 6 similar to triangles 1 & 13 & 24 & 23 & 22 & 21 & 20 & 8,
and 6 similar to triangles 1 & 2 & 13 & 14 & 24 & 15 & 23 & 16.
We found 12 large triangles that contain nine smaller triangles, one of which is triangle 1 & 13 & 24 & 23 & 22 & 10 & 9 & 21 & 20.
No large triangles can be made from ten, eleven, or even twelve smaller triangles. Indeed, the final large triangle that can be created in this figure is made of eighteen smaller triangles, and 2 of this type can be found in the figure. We have found a total of 104 triangles in the figure.