---
title: "Symmetry elements of crystals"
subject: "Engineering Geology I"
chapter: 3
key: "semester_3/engineering_geology_i:symmetry-elements-of-crystals"
source: https://www.bctengineer.com/q/semester_3/engineering_geology_i/symmetry-elements-of-crystals
site: BCT Engineering
---

# Symmetry elements of crystals

Engineering Geology I · Chapter 3: Crystallography & Mineralogy

- Asked in: 2081 Baisakh, 1 marks; 2079 Bhadra, 1 marks; 2078 Bhadra, 1.5 marks; 2078 Kartik, 1.5 marks; 2076 Chaitra, 1.5 marks; 2076 Asoj, 2 marks; 2075 Chaitra, 1.5 marks; 2075 Asoj, 2 marks; 2069 Chaitra, 1.5 marks; 2065 Shrawan (old course), 2 marks; 2062 Baisakh (old course), 2 marks; 2063 Baisakh (old course), 3 marks; 2059 Chaitra (old course), 2 marks (asked 13 times)
- Video: [Mineralogy: Lecture 4, Symmetry Elements](https://www.youtube.com/watch?v=tgLekElD47Q) — Kenneth Befus
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_geology_i/symmetry-elements-of-crystals
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_geology_i/chapter-3

## Question

Describe the symmetry elements of crystals.

## Short answer

Symmetry elements include planes of symmetry (mirror planes), axes of symmetry (rotation: diad, triad, tetrad, hexad), and centre of symmetry. A cube has 9 planes, 13 axes, and 1 centre, defining its isometric crystal system.

## Answer

**Symmetry** in a crystal is the repetition of faces, edges and angles when the crystal is rotated or reflected. The imaginary features about which this repetition occurs are the **elements of symmetry**.

#### 1. Plane of symmetry
An imaginary plane that divides a crystal into two equal halves, each the mirror image of the other.
- **Axial (principal) plane**: passes through two crystallographic axes.
- **Diagonal plane**: passes through one axis and bisects the angle between two others.
- A cube has 9 planes (3 axial, 6 diagonal).

#### 2. Axis of symmetry
An imaginary line through the centre about which the crystal, when rotated through 360°, shows the same appearance more than once.

| Axis | Repeats | Rotation angle |
|---|---|---|
| Diad (2-fold) | 2 times | 180° |
| Triad (3-fold) | 3 times | 120° |
| Tetrad (4-fold) | 4 times | 90° |
| Hexad (6-fold) | 6 times | 60° |

A cube has 3 tetrad, 4 triad and 6 diad axes, total 13 axes.

#### 3. Centre of symmetry
A point at the centre of the crystal such that any line through it meets identical faces, edges or corners at equal distances on opposite sides. A cube has 1 centre.

```text
  Cube (isometric): 9 P + 13 A + 1 C = 23 elements
   P = planes, A = axes, C = centre
```

The symmetry of a crystal decides which of the 7 crystal systems it belongs to.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are the three elements of crystal symmetry?
   A. Colour, streak and lustre
   B. Cleavage, fracture and tenacity
   C. Plane, edge and corner
   D. Plane, axis and centre of symmetry
   Answer: D. Symmetry is described by planes, axes and a centre. (Key point: Elements)

2. A diagonal plane of symmetry is one that:
   A. Passes through two crystallographic axes
   B. Passes through one axis and bisects the angle between two others
   C. Passes through no axes at all
   D. Passes through the centre only
   Answer: B. Axial planes pass through two axes; diagonal planes pass through one axis and bisect the other two. (Key point: Diagonal plane)

3. Which axis-angle pairing is correct?
   A. Diad 90 degrees, triad 120 degrees
   B. Tetrad 120 degrees, hexad 90 degrees
   C. Triad 120 degrees, tetrad 90 degrees, hexad 60 degrees
   D. Triad 60 degrees, hexad 120 degrees
   Answer: C. Angle = 360 / fold: diad 180, triad 120, tetrad 90, hexad 60. (Key point: Rotation angles)

4. How many symmetry elements does a cube (normal class) have?
   A. 13: 9 planes and 4 axes
   B. 19: 6 planes, 12 axes and 1 centre
   C. 9: 3 planes, 3 axes and 3 centres
   D. 23: 9 planes, 13 axes and 1 centre
   Answer: D. 9 + 13 + 1 = 23. (Key point: Cube total)

## Related in Engineering Geology I

- [Isometric crystal system with symmetry elements](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Aisometric-crystal-system-with-symmetry) — Engineering Geology I, chapter 3, asked 5×
- [Symmetry elements, crystal systems, rock-forming minerals](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Asymmetry-elements-crystal-systems-rockforming) — Engineering Geology I, chapter 3
- [Physical properties; orthorhombic symmetry elements](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Aphysical-properties-orthorhombic-symmetry-elements) — Engineering Geology I, chapter 3