Pyritohedral symmetry
The mineral iron pyrite, also known as “fool’s gold”, usually forms cube-shaped crystals, but it can also form crystals like these. At first glance, these crystals may appear to have dodecahedral symmetry, but this is not the case, because their pentagonal faces are irregular pentagons. This irregular dodecahedron shape is called a pyritohedron, and the type of symmetry it has is called pyritohedral symmetry.
The picture below (see here for an animated version) shows all the axes of rotational symmetry of a pyritohedron. Notice that eight of the vertices are highlighted in yellow, and six of the edges are highlighted in red. If one joins each pair of opposite yellow vertices with a straight line, this gives four axes of symmetry, and we may rotate either clockwise or anticlockwise by 120° about each of these axes. This gives a total of eight rotations, each of which has order 3, which means that performing one of these rotations three times in succession gives the same effect as doing nothing. Another kind of rotational symmetry arises from joining the midpoints of each pair of opposite parallel red edges with a straight line. These lines give three axes of 180° rotational symmetry. Each axis gives one new rotation, because rotating 180° clockwise is the same as rotating 180° anticlockwise. Each of these three rotations has order 2.
The operation of rotating by 0° (about any axis) also counts as a rotation, even though it has no effect. This is called the identity rotation, and it has order 1. This rotation, together with the 11 rotations from the previous paragraph (eight of order 3 and three of order 2) forms what mathematicians call a group. Essentially, this means that one can undo the effect of any rotation by rotating about the same axis in the opposite direction, and also that (much less obviously) if one performs a sequence of rotations, the overall effect is the same as performing a single rotation. Groups measure symmetry in the same way as numbers measure quantity, and this particular group is known as A4. The “A” stands for alternating and the “4” is something to do with the way that the four axes connecting yellow points are permuted among themselves by the rotations.
The group A4 also shows up as the group of rotations of a regular tetrahedron, shown above. In this case, there are three axes of order 2 rotational symmetry, each of which connects the midpoints of opposite edges, and four axes of order 3 rotational symmetry, each of which connects a vertex to the midpoint of the opposite face. However, unlike the regular tetrahedron, the pyritohedron also has symmetries that do not come from rotations: for example, there is a symmetry about the middle of the solid that sends each point to its polar opposite. Taking these extra symmetries into account, the pyritohedron has a group of 24 symmetries, which mathematicians denote by A4×C2.
There are also examples of crystals in nature that exhibit the full symmetries of a regular dodecahedron, such as the Holmium–Magnesium–Zinc quasicrystal shown above. Crystals like this can be rotated in 60 different ways rather than just 12. This gives a larger group of rotational symmetries, called A5. The “5” in this case has something to do with the way one can fit five cubes inside a regular dodecahedron, shown below in red, orange, green, blue, and grey. These five cubes are permuted among themselves by the act of rotating the dodecahedron.
A cube can be rotated in 24 different ways, but it is possible to cut this symmetry in half by dividing each face of the cube into two rectangles, and then only allowing symmetries that respect this partition. This new shape has pyritohedral symmetry, because the six lines through the faces of the cube correspond to the six red lines in the pyritohedron, and the eight corners of the cube correspond to the eight yellow points. The animation below by Tom Ruen shows a cube with this kind of pyritohedral symmetry morphing into a pyritohedron and then into a rhombic dodecahedron, which is a certain solid that has 12 rhombus-shaped faces and the same symmetry group as a cube. During the morph, each of the six lines on the faces of the cube is shrunk to a point, which results in a shape with twice as much rotational symmetry.
Picture credits and relevant links
The photograph of the single pyritohedron crystal is by Rob Lavinsky of iRocks.com. The other photograph of the pyritohedron-shaped crystals is by Didier Descouens and appears on the Wikipedia page on pyrite (FeS2).
The graphics of the pyritohedron and of the compound of five cubes are by Tilman Piesk. The first of these appears on the Wikipedia page on the dodecahedron.
The graphic of the tetrahedron is by Aldoaldoz.
The photograph of the dodecahedral crystal is from the Ames National Laboratory, which is part of the US Department of Energy. The photo appears on Wikipedia’s entry on the Holmium–Magnesium–Zinc quasicrystal.
The animation is by Tom Ruen.
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