Determining Symmetry of Crystals: An Introduction
Last Updated: 15th Jul 2020By Donald Peck & Erin Delventhal
Observing the symmetry of a crystal is often a way to distinguish one mineral from another. There are thirtytwo 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 Screwaxis of Symmetry, and a Glideplane 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.
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 portion or end/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).
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 90^{o}, 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 90^{o} two more times, for a total of four identical views of the cube. The axis we are turning is a 4fold axis of rotation. Now look at the square face at the leftfront. Another 4fold axis passes through that face and the face on the rearright side. There is a still another through the top and bottom faces. There are three 4fold 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 120^{o} rotation places the cube in the same 3Dspace. The axis is a 3fold axis of symmetry. Can you find the other three? Since the cube has four body diagonals, it has four 3fold axes of rotational symmetry.
Galena has six 2fold 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 2fold axes of rotation. A cube has 12 edges, and since each axis bisects two of them, there are six 2fold axes. Four from the top to bottom edges; and two through the vertical 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 360^{o} rotation.
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 (blue figure). With an actual crystal in hand, this termination would resemble the gable roof of a house. There is a 2fold axis of symmetry that emerges from the center of the upper horizontal edge (again, green dot). Turning the axis 180^{o} 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. The figure at right shows the shape of an ideal celestine crystal. Note that it has 3 2fold axes, each perpendicular to the other two. The long axis is the caxis. The baxis is intermediate in length, with the shortest being the aaxis.
Manipulating the Model of Galena, Above.
1. Orient the model with the following settings: Axes: ON, View: ALONG AAXIS.
2. Note that the aaxis is facing you, baxis is horizontal, caxis vertical. Place the mouse pointer on the baxis, hold the leftbutton and pull straight across to the left (rotating about the caxis). Count the number of square faces that turn to face you, before the the aaxis again comes to the front. The caxis is a 4Fold Axis of Rotation.
3. Orient the model back to the view along the aaxis. Place the mouse pointer on the caxis, hold down the leftbutton and pull straight down (rotating about the baxis). Repeat until the aaxis return to its original position, pointing at you. How many times did you rotate the cube through 90^{o} 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 baxis is a 4Fold Axis of Rotation.
4. Use either the baxis or caxis views to rotate about the aaxis. Satisfy yourself that it, also, is a 4Fold Axis of Rotation. There are three 4Fold axes.
5. Observing a 3fold or a 2fold 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 360^{o} rotation. The body diagonal is a 3fold axis. Click aaxis 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 2fold axis (edgecenter to opposite edgecenter)?
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 aaxis. Note the plane of the aaxis and caxis  it is a mirror plane. Do the same with the aaxis and baxis  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. The side in front of the "ridge line" is a mirror image of the side behind it. It also appears that there is another mirror plane perpendicular to "the ridge" and coincident with the 2fold axis. We can infer that these planes exist, but without seeing the lower portion of the crystal, which is lost in the matrix, we can not be certain. The image on the right shows an ideal representation. Celestine has three mutually perpendicular axes of rotation and three mutually perpendicular mirror planes. Each mirror plane is perpendicular to an axis and is the plane of the other two axes.
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 crosssection rectangular. How many vertical mirror planes can be passed through the crystal? Two (dotted lines in termination plan), one that includes the a/caxes and the other the b/caxes. Is there a horizontal mirror plane? There is no way to know, as we cannot see the lower termination. The caxis (vertical axis) is a 2fold 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 crosssection (see plan). Examining the crosssection 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. This can be only a monoclinic crystal. There is only one vertical mirror plane (front to rear), which tends to indicate a monoclinic crystal (it is possible but extremely rare for a monoclinic mineral not to have one vertical mirror plane, coincident with the aaxis and caxis). There is no vertical 2fold axis of symmetry, thus it cannot be an orthrhombic crystal. Orthorhombic crystals always have a 2fold caxis.
The monoclinic prismatic class and orthorhombic dipyramidal class contain nearly half of all mineral species. Learning to recognize the symmetry of these two systems, or even classes, is worth the effort.
This sounds worse than it is. Since most crystals are on matrix, observing an inversion axis is usually impossible. Therefore, a 4fold axis is often mistaken for a 2fold, and a 6fold for a 3fold. A 2fold axis of rotation is always a 2fold 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 4fold inversion axis.
The diagram at right shows the first two of eight moves in following a four fold inversion axis, that is through the first 90^{o} of a 360^{o} rotation about the caxis. After the 4fold axis is turned 90^{o}, all points on the crystal are inverted through the crystal's center of symmetry. Of course it is physically impossible to do this, so a single point is chosen (in this case the corner identified by the yellow dot) and the crystal is rotated around the axes as shown. When the operation, a (90^{o} rotation around the major axis followed by a 90^{o} rotation around the perpendicular axis) three more times, the original point returns to where it started,
Screwaxes 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 xray crystallographer. They are included here because one or both are found in several crystal classes; and while we cannot observe them, we occasionally see their effects.
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 this face. It is a lefthanded crystal, no matter how you turn it, the positive (upper) trapezoidal face always cuts the top left corner of the prism face below it. If it were a righthand crystal, the right hand top corner would be clipped.
Hemimorphism is the lack of symmetry between the two ends of a crystallographic axis (usually the caxis). Tourmaline is a major example. The faces of the upper termination of the caxis are quite different from those at the lower end. If the caxis is the axis of hemimorphic asymmetry, there is not a horizontal mirror plane. Hemimorphite is another common example.
1. Crystals in the isometric (cubic) crystal system all have four 3fold axes of symmetry. If you find more than one 3fold axis the crystal has to be isometric. If you find only one 3fold axis the crystal could be isometric, trigonal, or hexagonal.
2. Two or three 4fold axes indicate an isometric crystal. A single 4fold axis could be either isometric or tetragonal.
3. A termination showing two vertical mirror planes at 90^{o} to each other with a vertical 2fold axis (caxis) 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 class, which is hemimorphic).
4. When viewing a crystal in the monoclinic prismatic class from the positive end of the caxis, a single vertical mirror plane usually can be seen (coincident with the a/caxes). That there is not also a perpendicular plane (coincident with the b/caxes) nor a 2fold axis (coincident with the caxis) 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 90^{o} that it is easy to make a mistake.
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 caxis)?
3. How many vertical mirror planes does this crystal have?
4. Is there a set of 3 mirror planes that includes the aaxis 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 3fold axis. Turn the crystal to look down the caxis. Notice that everything is repeated 3 times around the axis. This is more difficult, but it is a 3fold inversion axis.
3. Three vertical mirror planes. Click [Along the C Axis]. Mirror planes bisect the angles between the aaxes.
4. No. Cick [Along aaxis] Notice that the view left of the caxis is not a mirror image of the view right of the axis.
5. No. Click [Along the aaxis] the upper half of the crystal is not a mirror image of the bottom half.
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 in the Isometric System may exceed the vertical + horizontal planes due to the presence of diagonal 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 #91). Close the tab to return here.
What Is a Mineral? The Definition of a Mineral
Determining Color and Streak
Determining Lustre: For Beginning Collectors
Determining the Hardness of a Mineral
Determining the Specific Gravity of a Mineral
Determining Symmetry of Crystals: An Introduction
Determining Fracture and Cleavage in Minerals
Links to the Crystallography SeriesCrystallography: 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
Mason, Brian and Berry, L.G. (1968) Elements of Mineralogy. W. H. Freeman and Company, San Francisco.
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
We are indebted to Dr. Peter Richards, Morphological Crystallographer, who offered many helpful suggestions.
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 on the surface of crystal such that the point of entry is similar to the point of exit, then the crystal has a 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 portion or end/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 360^{o} about the line to fill the same space two, three, four, or six times, it has an 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 90^{o}, 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 90^{o} two more times, for a total of four identical views of the cube. The axis we are turning is a 4fold axis of rotation. Now look at the square face at the leftfront. Another 4fold axis passes through that face and the face on the rearright side. There is a still another through the top and bottom faces. There are three 4fold 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 120^{o} rotation places the cube in the same 3Dspace. The axis is a 3fold axis of symmetry. Can you find the other three? Since the cube has four body diagonals, it has four 3fold axes of rotational symmetry.
Galena has six 2fold 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 2fold axes of rotation. A cube has 12 edges, and since each axis bisects two of them, there are six 2fold axes. Four from the top to bottom edges; and two through the vertical 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 360^{o} rotation.
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 (blue figure). With an actual crystal in hand, this termination would resemble the gable roof of a house. There is a 2fold axis of symmetry that emerges from the center of the upper horizontal edge (again, green dot). Turning the axis 180^{o} 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. The figure at right shows the shape of an ideal celestine crystal. Note that it has 3 2fold axes, each perpendicular to the other two. The long axis is the caxis. The baxis is intermediate in length, with the shortest being the aaxis.
1. Orient the model with the following settings: Axes: ON, View: ALONG AAXIS.
2. Note that the aaxis is facing you, baxis is horizontal, caxis vertical. Place the mouse pointer on the baxis, hold the leftbutton and pull straight across to the left (rotating about the caxis). Count the number of square faces that turn to face you, before the the aaxis again comes to the front. The caxis is a 4Fold Axis of Rotation.
3. Orient the model back to the view along the aaxis. Place the mouse pointer on the caxis, hold down the leftbutton and pull straight down (rotating about the baxis). Repeat until the aaxis return to its original position, pointing at you. How many times did you rotate the cube through 90^{o} 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 baxis is a 4Fold Axis of Rotation.
4. Use either the baxis or caxis views to rotate about the aaxis. Satisfy yourself that it, also, is a 4Fold Axis of Rotation. There are three 4Fold axes.
5. Observing a 3fold or a 2fold 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 360^{o} rotation. The body diagonal is a 3fold axis. Click aaxis 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 2fold axis (edgecenter to opposite edgecenter)?
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).
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 aaxis. Note the plane of the aaxis and caxis  it is a mirror plane. Do the same with the aaxis and baxis  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. The side in front of the "ridge line" is a mirror image of the side behind it. It also appears that there is another mirror plane perpendicular to "the ridge" and coincident with the 2fold axis. We can infer that these planes exist, but without seeing the lower portion of the crystal, which is lost in the matrix, we can not be certain. The image on the right shows an ideal representation. Celestine has three mutually perpendicular axes of rotation and three mutually perpendicular mirror planes. Each mirror plane is perpendicular to an axis and is the plane of the other two axes.
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 crosssection rectangular. How many vertical mirror planes can be passed through the crystal? Two (dotted lines in termination plan), one that includes the a/caxes and the other the b/caxes. Is there a horizontal mirror plane? There is no way to know, as we cannot see the lower termination. The caxis (vertical axis) is a 2fold 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 crosssection (see plan). Examining the crosssection 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. This can be only a monoclinic crystal. There is only one vertical mirror plane (front to rear), which tends to indicate a monoclinic crystal (it is possible but extremely rare for a monoclinic mineral not to have one vertical mirror plane, coincident with the aaxis and caxis). There is no vertical 2fold axis of symmetry, thus it cannot be an orthrhombic crystal. Orthorhombic crystals always have a 2fold caxis.
The monoclinic prismatic class and orthorhombic dipyramidal class contain nearly half of all mineral species. Learning to recognize the symmetry of these two systems, or even classes, is worth the effort.
Axis of Rotatory Inversion
If an axis turned through n degrees (where n = 120^{o}, 90^{o}, or 60^{o}) and the crystal inverted through its center brings the crystal into coincidence with its original space, then the crystal has, respectively a 3, 4, or 6 fold axis of rotatory inversion.
This sounds worse than it is. Since most crystals are on matrix, observing an inversion axis is usually impossible. Therefore, a 4fold axis is often mistaken for a 2fold, and a 6fold for a 3fold. A 2fold axis of rotation is always a 2fold 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 4fold inversion axis.
The diagram at right shows the first two of eight moves in following a four fold inversion axis, that is through the first 90^{o} of a 360^{o} rotation about the caxis. After the 4fold axis is turned 90^{o}, all points on the crystal are inverted through the crystal's center of symmetry. Of course it is physically impossible to do this, so a single point is chosen (in this case the corner identified by the yellow dot) and the crystal is rotated around the axes as shown. When the operation, a (90^{o} rotation around the major axis followed by a 90^{o} rotation around the perpendicular axis) three more times, the original point returns to where it started,
Screwaxis of Symmetry & Glide Plane of Symmetry
Screwaxes 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 xray crystallographer. They are included here because one or both are found in several crystal classes; and while we cannot observe them, we occasionally see their effects.
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 this face. It is a lefthanded crystal, no matter how you turn it, the positive (upper) trapezoidal face always cuts the top left corner of the prism face below it. If it were a righthand crystal, the right hand top corner would be clipped.
Hemimorphism is the lack of symmetry between the two ends of a crystallographic axis (usually the caxis). Tourmaline is a major example. The faces of the upper termination of the caxis are quite different from those at the lower end. If the caxis is the axis of hemimorphic asymmetry, there is not a horizontal mirror plane. Hemimorphite is another common example.
Some Useful Generalizations
1. Crystals in the isometric (cubic) crystal system all have four 3fold axes of symmetry. If you find more than one 3fold axis the crystal has to be isometric. If you find only one 3fold axis the crystal could be isometric, trigonal, or hexagonal.
2. Two or three 4fold axes indicate an isometric crystal. A single 4fold axis could be either isometric or tetragonal.
3. A termination showing two vertical mirror planes at 90^{o} to each other with a vertical 2fold axis (caxis) 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 class, which is hemimorphic).
4. When viewing a crystal in the monoclinic prismatic class from the positive end of the caxis, a single vertical mirror plane usually can be seen (coincident with the a/caxes). That there is not also a perpendicular plane (coincident with the b/caxes) nor a 2fold axis (coincident with the caxis) 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 90^{o} that it is easy to make a mistake.
1. Does this crystal have a center of symmetry?
2. What is the symmetry of the principal axis of rotation (the caxis)?
3. How many vertical mirror planes does this crystal have?
4. Is there a set of 3 mirror planes that includes the aaxis 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 3fold axis. Turn the crystal to look down the caxis. Notice that everything is repeated 3 times around the axis. This is more difficult, but it is a 3fold inversion axis.
3. Three vertical mirror planes. Click [Along the C Axis]. Mirror planes bisect the angles between the aaxes.
4. No. Cick [Along aaxis] Notice that the view left of the caxis is not a mirror image of the view right of the axis.
5. No. Click [Along the aaxis] 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 in the Isometric System may exceed the vertical + horizontal planes due to the presence of diagonal 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 #91). Close the tab to return here.
Symmetry Elements in Each Crystal System & Class  

Articles in This Series
Links to the "Determining . . ." Series: How ToWhat Is a Mineral? The Definition of a Mineral
Determining Color and Streak
Determining Lustre: For Beginning Collectors
Determining the Hardness of a Mineral
Determining the Specific Gravity of a Mineral
Determining Symmetry of Crystals: An Introduction
Determining Fracture and Cleavage in Minerals
Links to the Crystallography SeriesCrystallography: 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.
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
Acknowledgements
We are indebted to Dr. Peter Richards, Morphological Crystallographer, who offered many helpful suggestions.
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