Question

Symmetries of the cube. The intersection point of the diagonals is the center of symmetry of the cube (Figure 78). There are nine planes of symmetry: 6 diagonal planes (such as \(\mathrm{D} B F H\) ) and three planes passing through the midpoints of each quadruple of parallel edges. The cube has nine axes of symmetry of the 2nd order: six lines connecting the midpoints of opposite edges (e.g. of \(A D\) and \(F G\) ), and three lines connecting the centers of each pair of the opposite faces. The latter lines are in fact axes of symmetry of the 4 th order. In addition, the cube has four axes of symmetry of the 3rd order, namely its diagonals (e.g. \(A G\) ). Indeed, the diagonal \(A G\), obviously, makes congruent angles with the edges \(A B, A D\), and \(A E\), and these angles make the same (right) angles with each other. If we connect the points \(B, D\), and \(E\), then we obtain a regular triangular pyramid, \(A B D E\), for which the diagonal \(A G\) is the altitude. When the cube is rotated about this diagonal through the angle of \(120^{\circ}\), the pyramid returns to its original position. The same is true about the pyramid \(G H F C\), centrally symmetric to the pyramid \(A B D E\). Thus, as the

Short answer

The cube has 9 planes, 9 second-order axes, and 4 third-order axes of symmetry.

Step-by-step solution

Understanding the Cube Symmetries

A cube has various points of symmetry: planes, lines (axes), and points (centers). The intersection point of the diagonals of a cube is considered the center of symmetry.

Finding the Planes of Symmetry

There are 9 planes of symmetry in a cube. This includes 6 diagonal planes (for instance, plane DBFH) and 3 planes passing through the midpoints of each set of quadruple parallel edges.

Identifying Axes of 2nd Order Symmetry

A cube has 9 axes of symmetry of the 2nd order. These axes include 6 lines that connect the midpoints of opposite edges such as AD and FG, and 3 lines that connect the centers of each pair of opposite faces, which are also known as axes of the 4th order.

Recognizing Axes of 3rd Order Symmetry

The cube features 4 axes of symmetry of the 3rd order, which are its space diagonals, such as AG. The diagonal makes equal angles with the edges AB, AD, and AE. It can also rotate the cube by 120° to return to its initial configuration.

Key concepts

Planes of Symmetry

A cube, as a three-dimensional figure, has several planes of symmetry. Each plane divides the cube into two mirror-image halves. Let's explore these planes a bit further.

The cube possesses a total of nine symmetry planes:

• Six Diagonal Planes: These planes cut through the center of the cube and pass between pairs of opposite edges. For example, the plane DBFH passes through diagonal pairs and splits the cube evenly.
• Three Planes Through Midpoints: There are three planes that pass through the midpoints of each set of quadruple parallel edges. These planes run parallel to the cube's faces, effectively dividing the cube into symmetrical left and right halves, front and back halves, or top and bottom halves, depending on the orientation.

Each plane divides the cube into parts that are perfectly identical mirrored images, maintaining the cube's symmetrical properties.

Axes of Symmetry

Axes of symmetry are lines that you can rotate an object around, and the object looks the same at specific orientation angles. The cube has several of these rotational axes of symmetry.

For the cube, there are different types of symmetry axes:

• Axes of the 2nd Order: A cube features nine axes of the 2nd order symmetry, including six axes that connect the midpoints of opposite edges. These allow a 180° rotation symmetry.
• Axes of the 4th Order: Some of these second-order axes also serve as fourth-order ones. These include axes that connect the centers of each pair of opposite faces. These allow rotations of 90°/270° around them to maintain symmetry.
• Axes of the 3rd Order: There are four space diagonals (like AG), which serve as the third-order symmetry axes. When the cube is rotated 120° around one of these diagonals, it appears unchanged.

Understanding these axes of symmetry helps you visualize how the cube can rotate in space while still looking exactly the same.

Geometric Transformations

Geometric transformations involve changing the position, size, and orientation of a shape in space while maintaining its integrity. In the case of the cube, these transformations preserve its structural symmetry.

Some notable transformations for a cube include:

• Rotation: You can rotate a cube around any of its symmetry axes. For example, rotating around a diagonal (third-order axis) by 120° preserves symmetry.
• Reflection: Each symmetry plane of the cube acts as a mirror, providing a reflection transformation.
• Translation: This is less commonly mentioned, but it's worth noting that if you shift or translate the cube in space without rotation, its symmetry remains unchanged.

Geometric transformations allow for the visualization of how the cube can change its position or orientation while still looking the same, due to its symmetric properties.