Final answer:
A shape can have line symmetry if it can be divided into identical mirror-image halves, and rotational symmetry if it can look the same at specific rotation angles. Shapes can exhibit both forms of symmetry simultaneously, such as a cube's symmetrical rotations about its axes. Symmetry is important in various fields, reflecting natural and mathematical harmony.
Step-by-step explanation:
A shape can indeed have line symmetry, rotational symmetry, or both. Line symmetry, also known as reflection or mirror symmetry, occurs when a shape or design can be divided into two identical halves that are mirror images of each other. This is commonly seen in everyday life, for instance, in the bilateral symmetry of a butterfly's wings. In mathematics and geometry, bilateral symmetry refers to an object being divisible into two symmetrical parts across a unique plane, also known as a mirror plane.
Rotational symmetry occurs when a shape can be rotated about a central point and still look the same. For example, a cube exhibits rotational symmetry as it can be turned 90° around axes perpendicular to its faces and appear unchanged. This kind of symmetry is not limited to simple geometric figures but is also fundamental in fields such as physics and chemistry, where symmetry plays a major part in contemporary research.
In summary, the concept of symmetry is not only a fascinating aspect of mathematics but also has a profound impact on understanding the natural world and the fundamental properties of matter, as seen in the study of sub-atomic particles using particle accelerators like the Large Hadron Collider at CERN.