science

Tetrahedral symmetry

Tetrahedral symmetry is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tetrahedral symmetry rather than just read about it. In short: A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrah…

Key takeaways

Tetrahedral symmetry belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Tetrahedral symmetry to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tetrahedral symmetry from memory before moving on to harder problems.

Reference excerpt

A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4.

Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system.

Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at order 2 and 3 gyration points.

Chiral tetrahedral symmetry

T, 332, [3,3]+, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A4, the alternating group on 4 elements; in fact it is the group of even permutations of the four 3-fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23). The conjugacy classes of T are:

identity 4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321) 4 × rotation by 120° counterclockwise (ditto) 3 × rotation by 180° The rotations by 180°, together with the identity, form a normal subgroup of type Dih2, with quotient group of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d: the group G = A4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C6 or D3, but neither applies.

Subgroups of chiral tetrahedral symmetry

Achiral tetrahedral symmetry

Td, *332, [3,3] or 43m, of order 24 – achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 (4) axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion. See also the isometries of the regular tetrahedron. The conjugacy classes of Td are:

identity 8 × rotation by 120° (C3) 3 × rotation by 180° (C2) 6 × reflection in a plane through two rotation axes (Cs) 6 × rotoreflection by 90° (S4)

Subgroups of achiral tetrahedral symmetry

Pyritohedral symmetry

Th, 3*2, [4,3+] or m3, of order 24 – pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 (3) axes, and there is a central inversion symmetry. Th is isomorphic to T × Z2: every element of Th is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D2h (that of a cuboid), of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the direct product of the normal subgroup of T (see above) with Ci. The quotient group is the same as above: of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes. The conjugacy classes of Th include those of T, with the two classes of 4 combined, and each with inversion:

identity 8 × rotation by 120° (C3) 3 × rotation by 180° (C2) inversion (S2) 8 × rotoreflection by 60° (S6) 3 × reflection in a plane (Cs)

Subgroups of pyritohedral symmetry

Solids with chiral tetrahedral symmetry

The Icosahedron colored as a snub tetrahedron has chiral symmetry.

Solids with full tetrahedral symmetry

See also Octahedral symmetry Icosahedral symmetry Binary tetrahedral group

Citations

References

External links Weisstein, Eric W. "Tetrahedral group". MathWorld. Learning materials related to Symmetric group S4 at Wikiversity

Illustrations

Worked examples

Example 1 — a first encounter with Tetrahedral symmetry

Start with the simplest possible case. Write down what Tetrahedral symmetry claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tetrahedral symmetry before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tetrahedral symmetry ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tetrahedral symmetry

In research: Tetrahedral symmetry appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Tetrahedral symmetry in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Tetrahedral symmetry is common in secondary-school and first-year university syllabi. It links to neighbouring topics Finite groups, Rotational symmetry, Tetrahedra, so understanding it makes those chapters shorter.
In everyday life: Look for Tetrahedral symmetry outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Tetrahedral symmetry” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Tetrahedral symmetry in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tetrahedral symmetry means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Tetrahedral symmetry out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tetrahedral symmetry in simple terms?

Why does Tetrahedral symmetry matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tetrahedral symmetry?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tetrahedral symmetry.