Final answer:
The cube can perform specific rotations around its axes that keep its appearance the same, exemplifying symmetry through its C4 and C3 rotational axes. Other symmetrical shapes like the octahedron and icosahedron also exhibit interesting geometric properties.
Step-by-step explanation:
A cube has several special rotation attributes due to its symmetric structure. If we rotate a cube 90° around an axis perpendicular to any of its six faces, the appearance remains unchanged, showing that it possesses three mutually perpendicular four-fold rotational axes (C4 axes). Additionally, the cube can be rotated around axes that extend between opposite corners, requiring three 120° rotations to complete a full circle, known as three-fold or C3 axes.
An octahedron, another symmetrical shape with eight sides and six corners, can be visualized as a double square pyramid or laying on one of its faces within a close-packed lattice, illustrating different perspectives of geometric shapes. Similarly, assessing the structure and transformations of various polyhedra such as an icosahedron, with its 20 equilateral triangular faces and 12 corners, enriches our understanding of symmetry and geometry.