---
title: "Axis of symmetry; isometric system"
subject: "Engineering Geology I"
chapter: 3
key: "semester_3/engineering_geology_i:axis-of-symmetry-isometric-system"
source: https://www.bctengineer.com/q/semester_3/engineering_geology_i/axis-of-symmetry-isometric-system
site: BCT Engineering
---

# Axis of symmetry; isometric system

Engineering Geology I · Chapter 3: Crystallography & Mineralogy

- Asked in: 2080 Bhadra, 3 marks
- Video: [Mineralogy, Crystallography: Point Symmetry, Rotational Axes](https://www.youtube.com/watch?v=2M6SNtLctTw) — Erin Adlakha
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_geology_i/axis-of-symmetry-isometric-system
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_geology_i/chapter-3

## Question

Define axis of symmetry with a figure. Describe the isometric crystal system.

## Short answer

An axis of symmetry is an imaginary line about which a crystal repeats its appearance 2, 3, 4, or 6 times in a full rotation. The isometric system has three equal axes at 90°, the highest symmetry, with forms like cube and octahedron; examples include galena and pyrite.

## Answer

#### Axis of symmetry
An **axis of symmetry** is an imaginary straight line through the centre of a crystal about which, when the crystal is rotated through 360°, the same appearance is repeated two, three, four or six times.

```text
 Diad   (2-fold): 180 deg, repeats 2 times
 Triad  (3-fold): 120 deg, repeats 3 times
 Tetrad (4-fold):  90 deg, repeats 4 times
 Hexad  (6-fold):  60 deg, repeats 6 times

   Cube:   tetrad axis through face centres
           |
        +--|--+
        |  |  |
        +--|--+
```

#### Isometric (cubic) system
- Three crystallographic axes of **equal length**, all at **90°**: $a = b = c$, $\alpha = \beta = \gamma = 90^\circ$.
- Most symmetrical system; normal class has 9 planes, 13 axes (3 tetrad, 4 triad, 6 diad) and 1 centre of symmetry.
- Forms: cube, octahedron, dodecahedron.
- Minerals: galena, pyrite, fluorite, halite, garnet, diamond.

## Check yourself

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

1. An axis of symmetry is an imaginary line through the crystal centre about which a 360 degree rotation:
   A. Repeats it 5 or 7 times
   B. Changes the crystal system
   C. Produces mirror images only once
   D. Repeats the same appearance 2, 3, 4 or 6 times
   Answer: D. Only diad, triad, tetrad and hexad axes occur. (Key point: Axis definition)

2. A tetrad axis repeats the form every:
   A. 90 degrees, 4 times in a full turn
   B. 120 degrees, 3 times
   C. 60 degrees, 6 times
   D. 180 degrees, 2 times
   Answer: A. 360/4 = 90. (Key point: Tetrad)

3. Isometric axes are:
   A. Unequal and at 90 degrees
   B. Equal in length and all at 90 degrees
   C. Equal with one at 120 degrees
   D. Unequal with none at 90 degrees
   Answer: B. a = b = c, alpha = beta = gamma = 90 degrees. (Key point: Isometric axes)

## 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
- [Orthorhombic system; isometric mineral examples](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Aorthorhombic-system-isometric-mineral-examples) — Engineering Geology I, chapter 3, asked 2×