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Determining Symmetry of Crystals: An Introduction

Last Updated: 29th Nov 2018

By Donald Peck & Erin Delventhal

Observing the symmetry of a crystal is often a way to distinguish one mineral from another. There are thirty-two crystal classes spread among seven crystal systems into which all crystalline minerals fit. Each crystal system and class is distinguished from the others by its own elements of symmetry, often called symmetry operations. There are six (6) elements of symmetry in crystals: a Center of Symmetry, an Axis of Symmetry, a Plane of Symmetry, an Axis of Rotatory Inversion, a Screw-axis of Symmetry, and a Glide-plane of Symmetry.

The mirror plane of symmetry and the axis of rotational symmetry are the easiest to see. Also, when viewing a crystallized mineral specimen they are the most important in determining the system to which a crystal belongs. The axis of rotatory inversion and the center of symmetry are somewhat less important for diagnostic use and often difficult to see in a real mineral specimen. In the case of most crystals, the lower termination does not exist because often the crystal is attached to the matrix. As we shall see in a moment, this severely limits our ability to characterize the full symmetry of the crystal.

Center of Symmetry



If an imaginary straight line can be passed through a crystal from any point such that the point of entry is similar to the point of exit on the surface of the ideal crystal, the crystal has a center of symmetry.


Axinite-(Fe)
Center of Symmetry
Axinite-(Fe)
Center of Symmetry
Axinite-(Fe)
Center of Symmetry

In the photo of Axinite-(Fe), above, an imaginary line from the top left corner to the bottom right corner passes through similar points at the intersection of three faces on the exterior of the crystal, as do other lines between similar points (as shown in the diagram). All of those lines pass through a common point at the center of the crystal, thus axinite-(Fe), a triclinic mineral, has a center of symmetry.

In general, a center of symmetry can not be observed when the crystal is attached to matrix, as the lower termination is not visible. Thus, any line through the center that starts at the upper termination has no place to go.

Crystals in only eleven of the thirty two crystal classes exhibit a center of symmetry (see table below).

Axis of Symmetry



When an imaginary line can be passed through a crystal such that the crystal can be rotated 360o about the line to fill the same space two, three, four, or six times, it has an axis of symmetry.


Galena
Axis of Symmetry
Galena
Axis of Symmetry
Galena
Axis of Symmetry


Galena has cubic symmetry. If we were to grasp the two ends of the axis through the centers of the top and bottom face of the galena cube (Upper diagram, blue symbols), and turn the cube through 90o, that cube would fill the same three dimensional space. Put another way, we would have the same view of the cube, even though from a different side. We could turn the cube through 90o two more times, for a total of four identical views of the cube. The axis we are turning is a 4-fold axis of rotation. Now look at the square face at the left-front. Another 4-fold axis passes through that face and the face on the rear-right side; and another through the top and bottom faces. There are three 4-fold axes of rotational symmetry in a cube.

Look at the near corner of the galena. Imagine spinning it around the corner. How many positions will provide the same view of the corner? (Center diagram, red symbols) An axis of rotation passes through that corner, across the body diagonal, and through the far rear corner. There are three identical faces that meet at that corner. Each 120o rotation places the cube in the same 3D-space. The axis is a 3-fold axis of symmetry. Can you find the other three? Since the cube has four body diagonals, it has four 3-fold axes of rotational symmetry.

Galena has six 2-fold axes of rotational symmetry. Try to find them. Notice the axis that runs through the midpoints of the cube edges (Lower diagram, yellow symbols). These are 2-fold axes of rotation. A cube has 12 edges, and since each axis bisects two of them, there are six 2-fold axes. Four from the top to bottom edges; and two through the ertical edges.

If you have difficulty visualizing the axes, cut a cube out of a large potato. Stick toothpicks in it to make the axes. Turn the axes and count the number of times you see the potato occupy the same space in one 360o rotation.

Celestine
Axis of Symmetry - Celestine
Celestine
Axis of Symmetry - Celestine
Celestine
Axis of Symmetry - Celestine

The image above is of celestine, an orthorhombic mineral. The lower termination is lost in the matrix, but if you could look directly at the top of the crystal, it would appear to be rectangular. With an actual crystal in hand, this termination would resemble the gable roof of a house. There is a 2-fold axis of symmetry that emerges from the center of the upper horizontal edge (again, green dot). Turning the axis 180o places the terminal view in exactly the same space. Without the lower termination, we cannot actually see it; but we can infer that probably it is there.




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{100}
Locality: Many locations
Haüy, 1801/1823 ('Plomb Sulfuré'), and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923 ('Bleiglanz').
Manipulating the Model of Galena, Above.

1. Orient the model with the following settings: Axes: ON, View: ALONG A-AXIS.

2. Note that the a-axis is facing you, b-axis is horizontal, c-axis vertical. Place the mouse pointer on the b-axis, hold the left-button and pull straight across to the left (rotating about the c-axis). Count the number of square faces that turn to face you, before the the a-axis again comes to the front. The c-axis is a 4-Fold Axis of Rotation.

3. Orient the model back to the view along the a-axis. Place the mouse pointer on the c-axis, hold down the left-button and pull straight down (rotating about the b-axis). Repeat until the a-axis return to its original position, pointing at you. How many times did you rotate the cube through 90o before it was in its original position? How many times did it fill exactly the same space? In both instances, the answer should be "4". The b-axis is a 4-Fold Axis of Rotation.

4. Use either the b-axis or c-axis views to rotate about the a-axis. Satisfy yourself that it, also, is a 4-Fold Axis of Rotation. There are three 4-Fold axes.

5. Observing a 3-fold or a 2-fold axis is difficult by Manipulating the model. Try turning the model so that you are looking directly at one corner of the cube Notice that there are three identical faces showing. If you could turn the cube clockwise, you would obtain three identical views in a 360o rotation. The body diagonal is a 3-fold axis. Click a-axis and then pull the b across until an edge is centered. Imagine turning the crystal around an axis that passes through the center of that edge. Can you see a 2-fold axis (edge-center to opposite edge-center)?



Plane of Symmetry



When an imaginary plane is passed through a crystal such that the portion of the crystal on one side of the plane is a reflection, or mirror image, of the portion on the other side of the plane, the plane is a plane of symmetry (often called a mirror plane).


Fluorite
Plane of Symmetry
Fluorite
Plane of Symmetry
Fluorite
Plane of Symmetry


The fluorite crystal, shown above, has nine planes of symmetry. Look at the diagram at the right. The blue plane divides the cube in half so that the rear half is a mirror inage of the front half. There is another mirror plane that divides the cube from front to back and two more that are vertical diagonals mirror planes (the top edges are seen as dotted lines on the top of the cube). So far, that is 4 mirror planes.

One of those vertical planes divides the cube into left and right mirror images. Its edge would trace a vertical line down the center of the front of the cube. There is also a mirror plane dividing the top half of the crystal from the bottom half. And the cube can be divided into two mirror images by passing a plane from the top left edge to the lower right edge. That can be repeated from the top right edge to the bottom left edge. Again, that is four mirror planes, but one was counted when we considered the top view. leaving us now with seven planes.

Inspecting the (left) side view of the crystal. we should see four more mirror planes. This time, the vertical and horizontal planes have already been counted, leaving the two diagonal planes. Thus the total number of planes of symmetry for fluorite (and any cube) is nine. Don't expect to find them all in a crystal on matrix. If you can see two sides and recognize the existence of seven of them, you will have nailed the Hexoctahedral crystal class.

Return to the rotating model of galena. Square up the model by clicking Along a-axis. Note the plane of the a-axis and c-axis - it is a mirror plane. Do the same with the a-axis and b-axis - another mirror plane. Also, imagine mirror planes that divide the square face from corner to far opposite corner - two more mirror planes. That is a total of four mirror planes seen in one side of the cube. Where are the remaining five planes?

Look back at the photo and diagram for celestine. Notice that the "roof ridge" line in the diagram looks like a mirror plane. It also appears that there is another mirror plane perpendicular to "the ridge" and coincident with the 2-fold axis. We can infer that these planes exist, but without seeing the lower termination, we can not be certain.

Natrolite
Plane of Symmetry
Natrolite
Plane of Symmetry
Natrolite
Plane of Symmetry
Clinohedrite
Plane of Symmetry
Clinohedrite
Plane of Symmetry
Clinohedrite
Plane of Symmetry


The photo on the upper left shows a radiating group of natrolite crystals. On close inspection you will note that the terminal pyramid is wider in one dimension than in the other, making the cross-section rectangular. How many vertical mirror planes can be passed through the crystal? Two (dotted lines in termination plan), one that includes the a/c-axes and the other the b/c-axes. Is there a horizontal mirror plane? There is no way to know, as we cannot see the lower termination. The c-axis (vertical axis) is a 2-fold axis and if you consult the large table below, you will find that the crystal is most probably, orthorhombic, holohedral class.

The photo immediately above is of clinohedrite. Notice that it also has a rectangular cross-section. Examining the cross-section and termination, you will discover that the left half is a mirror image of the right half, but the front half is quite different from the rear half. There is one mirror plane (front to rear) and no vertical 2-fold axis of symmetry. This is the symmetry of the holohedral monoclinic class.

Learning to recognize the monoclinic prismatic and orthorhombic dipyramidal classes is worth the effort. They contain slightly more than half of all mineral species.

Axis of Rotatory Inversion



If an axis turned through n degrees (where n = 120, 90, or again 120) and the crystal inverted through its center brings the crystal into coincidence with its original space, then the crystal has, respectively 3, 4, or 6 -fold axis of rotatory inversion.


This sounds worse than it is. Since most crystals are on matrix, observing a inversion axis is usually impossible. Therefore, a 4-fold axis is mistaken for a 2-fold, and a 6-fold for a 3-fold. A 2-fold axis of rotation is always a 2-fold inversion axis and is always perpendicular to a mirror plane, so it is generally not considered as an inversion axis. The most common mineral to exhibit an axis of inversion is chalcopyrite. The crystal appears to be a tetrahedron in the isometric crystal system, but in fact it is a tetragonal disphenoid, which has its (very slightly) longer axis as a 4-fold inversion axis.

Screw-axis of Symmetry & Glide Plane of Symmetry



If an axis of symmetry, when turned through an angle of 360/no (where n = 2, 3, 4, or 6), causes a translation along the line of the axis in a helical pattern; and making one rotation in n intervals, the crystal exhibits a screw axis of symmetry.


If a plane of symmetry is passed through a crystal, and the reflection of one half relative to the other is translated parallel to the plane a distance of one half a unit cell dimension, the crystal exhibits a glide plane.


Screw-axes and glide planes are not visible in a macroscopic, or even a microscopic, crystal. They only appear in the unit cell, and are visible to the x-ray crystallographer. They are included here because one or both are required by 25 of 32 crystal classes; and while we cannot observe them, we occasionally see their effects. The axes of rotation, mirror planes, inversion axes, and center of symmetry combined can describe only the Holohedral Class in each of the seven crystal systems. The Holohedral Class is the crystal class that exhibits the highest symmetry in its system.

Screw axes may rotate either clockwise or anticlockwise. Enantiomorphic symmetry is one consequence. Enantiomorphism is left or right handedness in crystals. Handedness is often observed in quartz crystals. In the rotating model below, notice the trapezoidal faces at the corners of the prism. Turn the crystal and observe the location of the face. It is a left-handed crystal, no matter how you turn it, the positive trapezoidal face always cuts the top left corner of the prism face below it. If it were a right-hand crystal, the right hand top corner would be clipped.

Left-handed Dauphiné twinned quartz.

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Left-handed Dauphiné twin (mimetic twin on [001])
Haüy, 1801, and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.


Hemimorphism is the lack of symmetry between the two ends of a crystallographic axis (usually the c-axis). Tourmaline is a major example. The faces of the upper termination of the c-axis are quite different from those at the lower end. If the c-axis is the axis of heminorphic asymmetry, there is not a horizontal mirror plane, Hemimorphite is another common example.

Hemimorphism in hemimorphite.

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{010}, {100}, {031}, {12-1}, modified
Lewis, 1899. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.


Some Useful Generalizations



1. Crystals in the isometric (cubic) crystal system all have four 3-fold axes of symmetry. If you find more than one 3-fold axes the crystal is isometric. If you find only one 3-fold axis the crystal could be isometric, trigonal, or hexagonal.

2. Two or three 4-fold axes indicate an isometric crystal. A single 4-fold axis could be either isometric or tetragonal.

3. A termination showing two vertical mirror planes at 90o to each other and a vertical 2-fold axis (c-axis) is from the orthorhombic crystal system. The orthorhombic system holds about 21% of all minerals and the vast majority are in the orthorhombic dipyramidal class (the only other class possible is the orthorhombic pyramidal, hemimorphic).

4. When viewing the termination of a crystal from the monoclinic prismatic class, a single vertical mirror plane usually can be seen (coincident with the a/c-axes). That there is not also a perpendicular plane (coincident with the b/c-axes) nor a 2-fold axis (coincident with the c-axis) eliminates the orthorhombic dipyramidal class. These two classes contain nearly 45% of minerals. Learning to make this differentiation is significantly helpful.

5. A crystal with no axes of symmetry and no mirror planes is triclinic. Be cautious, the axial angles of many triclinic minerals are so close to 90o that it is easy to make a mistake.



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{021}, modified
Locality: Nebida, Sardinia, Italy
Tacconi, 1911. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
Try to Make the Following Observations: Click [Axes On] and [Transparent].

1. Does this crystal have a center of symmetry?

2. What is the symmetry of the principal axis of rotation (the c-axis)?

3. How many vertical mirror planes does this crystal have?

4. Is there a set of 3 mirror planes that includes the a-axis in the plane?

5. Is there a horizonatal mirror plane?


Answers:

1. Yes. Line up the intersection of the axes with any point on the near side of the crystal; there is a similar point through the crystal. behind the center.

2. It is a 3-fold axis. Turn the crystal to look down the c-axis. Notice that everything is repeated 3 times around the axis. This is more difficult, but it is a 3-fold inversion axis.

3. Three vertical mirror planes. Click [Along the C Axis]. Mirror planes bisect the angles between the a-axes.

4. No. Cick [Along a-axis] Notice that the view left of the c-axis is not a mirror image of the view right of the axis.

5. No. Click [Along the a-axis] the upper half of the crystal is not a mirror image of the bottom half.


Symmetry Elements in Each Crystal System & Class



The following is an explanation of the codes used in the table, below.
Nbr 1 to 32: The number of the Crystal Class: the progression is from least symmetrical to most symmetrical.
Axis: A bar over a number indicates an inversion axis (e.g. 3)
Mirror Planes: Vertical planes are indicated because they are usually visible on the upper termination. Horizontal planes often cannot be seen because the lower termination is "in the matrix", Total planes may exceed the vertical + horizontal planes.
Asterisk (*): An asterisk indicates "usually easy to see". If enclosed in parentheses as (1*) then only one of the set is easy to see.
Representative Mineral: Click on it to go to that mineral gallery. This will provide a large number of photos that you may examine for symmetry. If you then click the name of the mineral at the top of the gallery page, you will open the Mineral Page and can drop down to the rotatable model. (Try it with Hemimorphite, Model #99). Close the tab to return here.

Symmetry Elements in Each Crystal System & Class

NbrSystemClassHermann-Mauguin SymbolType2-Fold Axis3-Fold Axis4-Fold Axis6-Fold AxisVert. PlanesHoriz PlanesTotal PlanesCenterRepresentative Mineral
1TriclinicPedial1Hemihedrite
2Pinacoidal1Holohedral1Axinite-(Fe)
3MonoclinicDomaticmHemimorphic1*1*Clinohedrite
4Sphenoidal2Enantiomorphic1none common
5Prismatic2/mHolohedral1*1*1*1Diopside
6OrthorhombicOrthorhombic pyramidal3Hemimorphic1*2*2*Hemimorphite
7Orthorhombic disphenoidal33 (1*)Epsomite
8Orthorhombic dipyramidal2/m 2/m 2/mHolohedral3 (1*)2*131Baryte
9TrigonalTrigonal pyramidal31*33Jarosite
10Trigonal rhombohedral31Dioptase
11Ditrigonal pyramidal3mHemimorphic1*3*3*Elbaite
12Trigonal trapezohedral32Enantiomorphic31*Quartz
13Hexagonal scalenohedral3 2/mHolohedral313*3*1Calcite
14HexagonalTrigonal dipyramidal6 = 3/m111none common
15Hexagonal pyramidal6 Hemimorphic1Cancrinite
16Hexagonal dipyramidal6/m111Vanadinite
17Ditrigonal dipyramidal6m2313*14Benitoite
18Dihexagonal pyramidalHemimorphic6mm16*6*Zincite
19Hexagonal trapezohedral622Enantiomorphic61β-Quartz
20Dihexagonal dipyramidal6/m 2/m 2/mHolohedral61*6*171Beryl
21TetragonalTetragonal disphenoidal41Cahnitel
22Tetragonal Pyramidal4Hemimorphic1none common
23Tetragonal dipyramidal4/m1111Wulfenite
24Tetragonal scalenohedral42m3122Chalcopyrite
25Ditetragonal pyramidal4mmHemimorphic44none common
26Tetragnl. trapezohedral42241Sincosite
27Ditetragnl. dipyramidal4/m 2/m 2/mHolohedral414151Apophyllite
28IsometricTetartoidal23Enantiomorphic34none common
29Diploidal2/m 33*4 2*13Pyrite
30Hextetrahedral43m3432*6Tetrahedrite
31Gyroidal43264none common
32Hexoctaheral4/m 3 2/mHolohedral64*3*4*191Fluorite



Articles in This Series

Links to the "Determinig . . ." Series: How To
    What Is a Mineral? The Definition of a Mineral
  1. Determining Color and Streak
  2. Determining Lustre: For Beginning Collectors
  3. Determining the Hardness of a Mineral
  4. Determining the Specific Gravity of a Mineral
  5. Determining Symmetry of Crystals: An Introduction
Links to the Crystallography Series
  1. Determining Symmetry of Crystals: An Introduction
  2. Miller Indices
  3. Hermann-Mauguin Symmetry Symbols
  4. Crystallography: The Triclinic System
    Crystallography: The Monoclinic System
    Crystallography: The Orthohombic System
    Crystallography: The Trigonal System
    Crystallography: The Hexagonal System
    Crystallography: The Tetragonal System
    Crystallography: The Isometric System

References


Mason, Brian and Berry, L.G. (1968) Elements of Mineralogy. W. H. Freeman and Company, San Francisco.
Smith, Jennie R. (1991) Understanding Crystallography. The Rochester Mineralogical Symposium.
Peck, Donald B. (2007) Mineral Identification: A Practical Guide for the Amateur Mineralogist. Mineralogical Record, Tucson, Arizona.
https://en.wikipedia.org/wiki/Crystal_system; Includes symmetry operations, point groups, and space groups





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Comments

Another excellent, clearly explained article! For me, crystals systems have always been difficult, perhaps because of my own difficulty with three-dimensional concepts. This has been so helpful, as have the instructions combined with toggling the crystal diagrams; I will refer back to this often, especially as I have mineral sessions with a neighbour's 14-year old son. He soaked up your earlier articles like a sponge! Kind regards, Becky

Becky Coulson
11th Mar 2018 6:38pm
Crystallography is one of them things that, if you don't engage it constantly, it will elude you. I had to learn it in mineralogy class back in '75. I've used it less and less since then, although I can make some judgements every now and then. Just no details. Someday I may sit down and try to relearn some of this, using my collection. Something else I haven't looked at in about 10 years.

Doug Daniels
12th Mar 2018 1:30am
Hi Doug,

Thanks for the comments. I think you and I studied crystallography at just about the same time. I think the elements of it stay with one, but there are parts that do get away (Herann-Mauguin Symbols being one of them).

Don

Donald B Peck
14th Mar 2018 12:06am
Don,

You refreshed my memories of a lecture couple of decades ago by the late Bill Moller about crystal symmetry. I think it was one of the most significant mineralogical talks I ever attended! He made large crystal models for the purpose and took the great pains you did to explain it well.
Today I am completely spoiled by the availability of x-ray diffraction but I appreciate it is all based on the symmetry ideas you so well explain. Thanks to you and Erin

John Attard
20th Mar 2018 5:14am
John,

Thank you for your kind message. I enjoy writing and am a lifelong teacher. Putting the two together with my love of minerals is almost unbeatable for me.

Don

Donald B Peck
20th Mar 2018 4:13pm

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