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Chapter 2: Problem 124

Show that a cube has nine planes of symmetry.

Short Answer

Expert verified
A cube has 9 planes of symmetry: 3 face planes and 6 diagonal planes.

Step by step solution

01

Understanding the Problem

To find the planes of symmetry in a cube, we need to identify planes that divide the cube into two identical halves, where each half is a mirror image of the other. A cube is a three-dimensional shape with six equal square faces.
02

Identifying Face Planes of Symmetry

First, consider planes that pass through the center and are parallel to the faces of the cube. Each pair of opposite faces has one plane of symmetry. A cube has three such pairs of opposite faces: top/bottom, front/back, and left/right. Therefore, there are 3 face planes of symmetry.
03

Identifying Diagonal Planes of Symmetry

Next, consider planes that pass through the center and cut through the edges diagonally. Each plane is formed by diagonal lines extending across opposite faces. In each of the three dimensions, there are two such diagonal planes. Thus, there are 6 diagonal planes of symmetry (2 planes per dimension).
04

Counting Total Planes of Symmetry

By adding the face planes and the diagonal planes, the total number of symmetry planes is obtained. We have 3 face planes and 6 diagonal planes, resulting in a total of 9 planes of symmetry for the cube.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Planes of Symmetry
Planes of symmetry are imaginary planes that divide a three-dimensional shape into two identical and mirror-image halves. When talking about a cube, it is essential to identify such planes that meet these criteria. Each plane should ideally create a perfect mirror image on either side.
A cube is famous for its symmetrical properties, featuring multiple symmetry planes. To understand this better, imagine slicing the cube with a transparent, thin sheet of material such that both parts of the cube appear identical when viewed against the sheet.
  • **Face Planes**: These are the simplest to visualize. For a cube, these planes are parallel to the faces of the cube. Each of these planes cuts through the center of the cube and creates two equal halves with opposite faces. There are three such face planes, corresponding to opposite faces like top/bottom, front/back, and left/right.
  • **Diagonal Planes**: These are slightly more complex, as they cut across the cube along its diagonal lines. Each diagonal plane passes through the center of the cube and bisects it into identical mirror halves. There are six diagonal planes, with two planes of symmetry in each of the three dimensions.
Three-Dimensional Shapes
Three-dimensional shapes or 3D shapes are objects that have length, width, and height. Unlike two-dimensional shapes that lie flat on a plane, three-dimensional shapes occupy physical space and can be held in your hand or seen from various angles.
The cube is a classic example of a three-dimensional shape. It embodies symmetry and balance, making it a fundamental concept in geometry. A cube is a polyhedron, which denotes a solid figure with flat, polygonal faces.
  • **Characteristics**: A cube has six square faces, twelve edges, and eight vertices. Each face is congruent, meaning all faces are identical squares.
  • **Visualizing 3D Shapes**: Imagine a cube as a box where you can look at different sides, edges, and angles, showcasing its three-dimensionality.
  • **Real-Life Examples**: Everyday items like dice or a Rubik's cube represent three-dimensional shapes, providing practical examples and making the understanding of these concepts easier.
Geometric Properties
Geometric properties define the shape, size, and relative position of figures, as well as the properties of space. In terms of a cube, these properties encompass its edges, faces, vertices, and symmetry.
  • **Symmetry**: The most captivating property of a cube is its symmetry. The cube's nine planes of symmetry highlight its balance and uniformity.
  • **Edges and Faces**: With twelve straight edges and six flat faces, a cube boasts simplicity in its geometry while displaying complexity in its symmetrical properties.
  • **Vertices**: The points where the edges meet, or vertices, are a crucial aspect of the cube's structure. A cube showcases eight vertices as corner points.
  • **Angle and Measurement**: Each corner of a cube forms a right angle, maintaining consistency in geometric properties throughout.
Understanding these properties assists in recognizing the three-dimensional framework and the balanced nature of geometric shapes such as cubes. Mastering the concept of symmetry and other properties not only aids in solving geometric problems but also enhances spatial awareness.

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Most popular questions from this chapter

Compute volumes of regular tetrahedron and octahedron with the edge \(a\).

Prisms. Take any polygon \(A B C D E\) (Figure 39 ), and through its vertices, draw parallel lines not lying in its plane. Then on one of the lines, take any point \(\left(A^{\prime}\right)\) and draw through it the plane parallel to the plane \(A B C D E\), and also draw a plane through each pair of adjacent parallel lines. All these planes will cut out a polyhedron \(A B C D E A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) called a prism. The parallel planes \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) are intersected by the lateral planes along parallel lines (\$13), and therefore the quadrilaterals \(A A^{\prime} B^{\prime} B, B B^{\prime} C^{\prime} C\), etc. are parallelograms. On the other hand, in the polygons \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\), corresponding sides are congruent (as opposite sides of parallelograms), and corresponding angles are congruent (as angles with respectively parallel and similarly directed sides). Therefore these polygons are congruent. Thus, a prism can be defined as a polyhedron two of whose faces are congruent polygons with respectively parallel sides, and all other faces are parallelograms connecting the parallel sides. The faces \(\left(A B C D E\right.\) and \(\left.A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\right)\) lying in parallel planes are called bases of the prism. The perpendicular \(O O^{\prime}\) dropped from any point of one base to the plane of the other is called an altitude of the prism. The parallelograms \(A A^{\prime} B^{\prime} C, B B^{\prime} C^{\prime} C\), etc. are called lateral faces, and their sides \(A A^{\prime}, B B^{\prime}\), etc., connecting corresponding vertices of the bases, are called lateral edges of the prism. The segment \(A^{\prime} C\) shown in Figure 39 is one of the diagonals of the prism. A prism is called right if its lateral edges are perpendicular to the bases (and oblique if they are not). Lateral faces of a right prism are rectangles, and a lateral edge can be considered as the altitude. A right prism is called regular if its bases are regular polygons. Lateral faces of a regular prism are congruent rectangles. Prisms can be triangular, quadrangular, etc. depending on what the bases are: triangles, quadrilaterals, ete.

Prove that if all altitudes of a tetrahedron are concurrent, then each pair of opposite edges are perpendicular, and vice versa.

In a rectangular parallelepiped with a square base and the altitude \(h\), a cross section through two opposite lateral edges is drawn. Compute the total surface area of the parallelepiped, if the area of the cross section equals \(S\). e.

Show that each of the 12 rotations of a regular tetrahedron permutes the four vertices, and that to different rotations there correspond different permutations of the set of vertices.
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