The image you uploaded depicts a symmetric pattern made up of blue triangles arranged in a star-like formation. The question asks how many times the image can be rotated and remain unchanged before returning to its original position.

This is a problem of rotational symmetry. We need to determine how many degrees the image can be rotated while looking exactly the same. By observing the pattern, we can deduce the order of rotational symmetry.

Steps to determine rotational symmetry:

1. Count the identical triangular units forming the star-like shape.
2. Calculate how many degrees of rotation bring the image back to a matching configuration.

Upon inspection, the image appears to have 6-fold rotational symmetry, meaning the image looks the same after every $\frac{360^\circ}{6} = 60^\circ$ rotation.

Conclusion:

The image can be rotated by 60°, 120°, 180°, 240°, 300°, and finally 360° (which brings it back to the original position), making it unchanged at 6 positions before returning to the starting orientation.

Would you like more details on the symmetry, or do you have any other questions?

Here are some relative questions for deeper exploration:

1. What is rotational symmetry, and how is it defined mathematically?
2. How can rotational symmetry be related to group theory in mathematics?
3. How would you compute rotational symmetry for different geometric shapes?
4. What is the relationship between rotational symmetry and reflective symmetry?
5. How would the symmetry change if the pattern in the image had more or fewer arms?

Tip: Identifying symmetry in patterns can also help simplify complex calculations in geometry and physics.