Fix the three side lengths of a triangle and the shape is fixed with them. No flexing is possible, because three sides determine the three angles completely, which is the content of the side-side-side congruence result at Euclid I.8. No other polygon has this property. A quadrilateral with four fixed side lengths can be pushed into a parallelogram and back again; a hexagon collapses more readily still.
The consequence is structural rather than symbolic. Bridge trusses, roof trusses, pylons, bicycle frames, scaffolding and space frames are all triangulated, and the diagonal brace added to a rectangular frame does nothing except convert one flexible quadrilateral into two rigid triangles. Buckminster Fuller's geodesic domes are triangulated spheres for the same reason. The engineering term is that a triangulated frame is statically determinate: the forces in it can be resolved by geometry alone, member by member.
Triangles also tile the plane, and unusually, any triangle tiles it, not merely the equilateral one. Take any triangle at all, rotate a copy through 180 degrees, and the two together form a parallelogram, which tiles trivially. The same holds for any quadrilateral. It stops being true at five sides. Among regular polygons only the triangle, square and hexagon tile alone, because the interior angle must divide 360 degrees exactly, and 60, 90 and 120 are the only regular interior angles that do.
Triangular numbers, 1, 3, 6, 10, 15 and so on, count the dots in a triangular array. The fourth of them, ten arranged in four rows, is the Pythagorean tetractys, which the ancient sources describe as an object of oath and reverence within the school.