Identifying 4-Fold Rotational Symmetry

An object exhibits $n$-fold rotational symmetry if it appears identical after being rotated by an angle of $ \frac{2\pi}{n} $ radians around a specific axis. For an object to have $4$-fold rotational symmetry, it must look the same after a rotation of $ \frac{2\pi}{4} = \frac{\pi}{2} $ radians, which is equivalent to $90^\circ$. The axis of rotation must be perpendicular to the plane of the screen.

Analyzing Object Symmetry

We examine each figure to determine if it maintains its appearance after a $90^\circ$ rotation about its center:

• Figure 1: This shape resembles a square with a diagonal line. Rotating it by $90^\circ$ changes the orientation of the diagonal line, making the object appear different. It does not have $4$-fold rotational symmetry.

  

• Figure 2: This figure is a square divided into four equal quadrants by two perpendicular lines intersecting at the center. Rotating this figure by $90^\circ$ around its center results in the exact same appearance. Therefore, it exhibits $4$-fold rotational symmetry.

  

• Figure 3: This shape is a rectangle with intersecting perpendicular lines. A $90^\circ$ rotation changes the orientation of the rectangle (from horizontal to vertical or vice versa), making it look different. It does not possess $4$-fold rotational symmetry; it has $2$-fold symmetry.

  

• Figure 4: This figure shows a circle with a diameter line. Rotating it by $90^\circ$ changes the position of the diameter line relative to the observer, so it does not look identical. It does not have $4$-fold rotational symmetry.

  

Based on the analysis, only the object in Figure 2 has $4$-fold rotational symmetry.