---
title: "Isometric crystal system with symmetry elements"
subject: "Engineering Geology I"
chapter: 3
key: "semester_3/engineering_geology_i:isometric-crystal-system-with-symmetry"
source: https://www.bctengineer.com/q/semester_3/engineering_geology_i/isometric-crystal-system-with-symmetry
site: BCT Engineering
---

# Isometric crystal system with symmetry elements

Engineering Geology I · Chapter 3: Crystallography & Mineralogy

- Asked in: 2080 Baisakh, 2 marks; 2074 Asoj, 1.5 marks; 2072 Chaitra, 2 marks; 2066 Bhadra (old course), 2 marks; 2075 Chaitra, 2 marks (asked 5 times)
- Video: [Mineralogy - 2 | Crystal Structure, Forms, Miller Indices, Symmetry Elements | Geology Concepts](https://www.youtube.com/watch?v=2ZzVzPA3694) — Geology Concepts
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_geology_i/isometric-crystal-system-with-symmetry
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_geology_i/chapter-3

## Question

Describe the isometric crystal system with its symmetry elements.

## Short answer

In the isometric (cubic) system, three equal axes are mutually perpendicular (a=b=c, α=β=γ=90°). The normal class has 9 planes, 13 axes (3 tetrad, 4 triad, 6 diad), and a centre of symmetry; common forms are cube, octahedron, and dodecahedron with minerals like galena and garnet.

## Answer

The **isometric (cubic) system** is the most symmetrical crystal system. It has **three crystallographic axes of equal length, mutually perpendicular** (all at 90°).

$$
a = b = c, \qquad \alpha = \beta = \gamma = 90^\circ
$$

```text
        c
        |   b
        |  /
        | /
        |/____ a
   a = b = c, all at 90 deg
```

#### Symmetry elements (of the normal class, hexoctahedral)
- **Planes of symmetry**: 9 (3 axial planes + 6 diagonal planes).
- **Axes of symmetry**: 13
  - 3 tetrad (4-fold) axes through the centres of opposite faces,
  - 4 triad (3-fold) axes through opposite corners,
  - 6 diad (2-fold) axes through the mid-points of opposite edges.
- **Centre of symmetry**: 1.
- Total: 23 elements.

#### Common forms and minerals
- Cube, octahedron, dodecahedron.
- Minerals: galena, pyrite, fluorite, halite (rock salt), garnet, diamond, magnetite.

## Check yourself

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

1. The isometric system has axes that are:
   A. Three, equal in length and mutually perpendicular
   B. Three, unequal in length and perpendicular
   C. Three, equal in length with one not at 90 degrees
   D. Four, with three at 120 degrees
   Answer: A. a = b = c with all angles 90 degrees. (Key point: Axes)

2. Which gives the 13 axes of symmetry of the isometric normal class?
   A. 6 tetrad + 4 triad + 3 diad
   B. 3 tetrad + 4 triad + 6 diad
   C. 3 tetrad + 6 triad + 4 diad
   D. 4 tetrad + 3 triad + 6 diad
   Answer: B. Tetrad through face centres (3), triad through corners (4), diad through edge mid-points (6). (Key point: Axes of symmetry)

3. How many planes of symmetry does the isometric normal class have?
   A. 3 axial only
   B. 6 axial and 3 diagonal
   C. 9: 3 axial and 6 diagonal
   D. 13 axial and diagonal
   Answer: C. The cube has 3 axial + 6 diagonal planes. (Key point: Planes)

## Related in Engineering Geology I

- [Axis of symmetry; isometric system](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Aaxis-of-symmetry-isometric-system) — Engineering Geology I, chapter 3
- [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×
- [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
- [Symmetry elements of crystals](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Asymmetry-elements-of-crystals) — Engineering Geology I, chapter 3, asked 13×

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- [Elements of communication system; modulation is a must](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acomm-elements-modulation-justify) — Communication Systems, chapter 1
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