Let's break down the solution for: **Define the three types of rotation axes in a cube**—with step-by-step explanation. --- ### **Step 1: Understanding the Problem** A cube can be rotated about certain axes such that its orientation changes but its appearance remains unchanged. These are called **symmetry axes of rotation**. We are to **define the three types** of rotation axes in a cube. --- ### **Step 2: List the Three Types** The three types of rotational axes in a cube are: 1. **Four-fold axes (through the centers of opposite faces)** 2. **Three-fold axes (through opposite vertices)** 3. **Two-fold axes (through the midpoints of opposite edges)** --- ### **Step 3: Define Each Type** #### **1. Four-fold Rotation Axes (Face Axes)** - **Definition:** An axis passing through the centers of two opposite faces of the cube. - **Rotation:** The cube can be rotated by 90°, 180°, or 270° around this axis, and it will look the same. - **Number of axes:** **3** (one for each pair of opposite faces: x, y, and z directions) #### **2. Three-fold Rotation Axes (Body Diagonal Axes)** - **Definition:** An axis passing through two opposite vertices (corners) of the cube, i.e., along the body diagonals. - **Rotation:** The cube can be rotated by 120° or 240° around this axis, and it will look the same. - **Number of axes:** **4** (each body diagonal forms one axis) #### **3. Two-fold Rotation Axes (Edge Axes)** - **Definition:** An axis passing through the midpoints of two opposite edges of the cube. - **Rotation:** The cube can be rotated by 180° around this axis, and it will look the same. - **Number of axes:** **6** (for each pair of opposite edges) --- ### **Step 4: Summary Table** | Axis Type | Axis Passes Through | Rotation Angle(s) | Number of Axes | |---------------|-------------------------------------|-------------------|---------------| | Four-fold | Centers of opposite faces | 90°, 180°, 270° | 3 | | Three-fold | Opposite vertices (body diagonals) | 120°, 240° | 4 | | Two-fold | Midpoints of opposite edges | 180° | 6 | --- ### **Final Answer (Summary)** **The three types of rotation axes in a cube are:** 1. **Four-fold axes:** Pass through the centers of opposite faces. The cube can be rotated by 90°, 180°, or 270° around these axes (3 such axes). 2. **Three-fold axes:** Pass through opposite vertices (body diagonals). The cube can be rotated by 120° or 240° around these axes (4 such axes). 3. **Two-fold axes:** Pass through the midpoints of opposite edges. The cube can be rotated by 180° around these axes (6 such axes). --- **In summary:** - **Three types of rotation axes in a cube:** four-fold (face), three-fold (body diagonal), two-fold (edge).