# Symmetry

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

## Intro

Symmetry is the concept object that looks identical same after some geometric transformation - usually reflection and rotation, sometimes scaling, translation, and some less obvious types.

Introductions to symmetry are often 2D.

Most concepts can be applied to 3D easily enough, but this can get more involved, so it helps if you understand 2D symmetries well first, then start with 3D symmetries that are simple enough to almost be 2D cases.

## Some concepts

• chirality refers to things that cannot be made to look the same with only rotation and translation - cases where you'ld need reflection.
• The common example is hands: Your left hand would need to be reflected before you can match it to the right (and vise versa).
• Similar things also happens in the study of molecules in chemistry and biology. You could say that chirality is one of the main concepts that point group symmetry deals with.

• symmetries (as a noun) refer to the amount of cases of symmetry there are for an object.
For example, an equilateral triangle has six symmetries: 3 reflection symmetries (from each point, straight through) and 3 rotation symmetries under which it looks the same.

• n-fold symmetry means that doing the operation 'n' times yields the original (also used when the object had no symmetry).
This mostly applies to rotations, where it is equivalent to saying that one full rotation sees n symmetries. For example, a plus symbol (+) has 4-fold rotational symmetry, an equilateral triangle 3-fold rotational symmetry.

• The point group of a molecule describes the set of symmetry operations that describes its overall symmetry.
For example, H2O has the operation set {E, C2, 2 σv} (the last meaning two different mirror planes)
Which (in Schoenflies notation) is the C2v point group

• Point symmetry: types of symmetry around a point. in 2D there are a few of these. In 3D, it's mostly just inversion.

• Pseudo-symmetry, approximate symmetry - there are a number of things that are almost symmetric, or superficially symmetric. For example, the human body looks fairly mirror-symmetric left to right (but certainly isn't on the inside).

## Typical operations

• E - identity. No change. Not really a symmetry, part of every point group. Only really there to refer to it - e.g. C1 is equivalent to E, and in many uses means 'no (particular) symmetry'.

• i - inversion. When a coordinate-wise inversion yields the same thing (in 2D: (x,y) to (-x,-y). In 3D: (x,y,z) to (-x,-y,-z))
happens in 2D mostly for already pretty symmetric/regular things
doesn't happen much in 3D
considered non-axial, since this mostly just has a center point, not an axis or plane
In crystallography this is called centro-symmetry

• σ - mirror on a plane. Is sometimes split into more qualified variants, often horizontal, vertical, and diagonal (when those directions have meaning, e.g. a main axis to compare to)
σh - perpendicular to the main axis. For examples, look at Cnh, Dnh, Cs
σv - along the main axis. For examples, look at C∞v and D∞h (verify)
σd - diagonal, indicating planes in-between (bisecting) two C2 axes (does that implicitly make them v direction or not?(verify))

• proper rotation - often denoted by a rotation set, often some variation in C
for n-fold proper rotation, "every (360/n) degrees you get the same thing"
typically refers to one axis at a time, unless mentioned otherwise.
The principal axis refers to the one with the highest n
often C or anything that involves it (D, T, O, I)

• improper rotation - point (sub)groups Sn and things that involve it (including Cnh, T, O, I)
for n-fold improper rotation: "same thing when you rotate by (360/n) and then reflect over the plane perpendicular to the rotation axis" (proper rotation is just the first part)
Sort of a strange operation, but it happens often enough to be useful
http://www.reciprocalnet.org/edumodules/symmetry/operations/improper.html
part of specific point groups,

To those looking for patterns in the (Schoenflies) point group names below, a few notes:

The category letters are more about the axes than about the types of operations involved. Yes, each specific point group has a fairly distinct set of operations (see e.g. this table), but, for example, there is C with mirror (Cnv, Cnh, C∞v), C without mirror, and there is D with mirror, and D without mirror.

As it happens, specific regular symmetries are often seen combined with simpler ones, often sets. In some cases, one symmetry implies another, but usually this is more a result of the shape being studied.

You may like to look at a lot of different examples, and then perhaps at some point group decision trees.

## Notation

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There are a number of notations.

The two most common seem to be Schoenflies notation and Hermann–Mauguin notation, the latter of which is better known as international notation . People often use whichever notation simpler or more succinct for a particular case. Which can depend on preference, an on the field you're in. The two just mentioned seem common in chemistry/biology.(verify) Physics has its own take, for example.

Schoenflies looks like D6, international looks like 422.

Wolfram Alpha can be useful in translation; see e.g. C2v point group and 422 point group.

## Symmetry types

### Some introduction

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

The most common parts of symmetry are rotation and reflection. Various symmetries may apply at the same time. For example, a square has four rotational and four reflective symmetries.

The cases are fairly easy to oversee in 2D. In 3D things become more complex, because added dimension adds more distinct ways in which those two can combine - more axes for things to happen on. For example, the axis/axes of rotation and the plane(s) of reflection may not always be in the same direction. Some of the implications tend to take some time and some examples to grasp.

Keep in mind that there is a difference between having a particular symmetry and being well defined by it. Various specific point groups have subgroups. For example, while C2h is a pretty definitive point group, C2 is a set common to it and many other specific point groups (e.g. most of the dihedral ones, the cubic ones).

You can summarize a point group by the types operations in the set, the amount of each, and (perhaps the best exercise) trying to list them exhaustively. For example, (staggered) methane has:

• One C3 (that rotates it along its long axis, which is its main axis)
• Three C2 (that flip it)
• One S6 (rotate and flip)
• Three reflections
• inversion

Its point group is D3d 

### Various point groups and examples

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

The list below is mostly of the various point groups. It might be a halfway decent introduction into things you see in molecular symmetry - I don't know. ...but naturally not definitive. And is biased to Schoenflies categories, because it uses its notation. For a more thorough understanding, do a lot more reading.

Non-axial / non-rotational (there is no axis of rotation involved)

• C1 - meaning no symmetry.
• the more complex, the more likely it is to be C1 (or at best approximately something else)
• Ci - point group with only inversion.
• Cs - one mirror plane, nothing else.
• example: CH2BrCl (plane through all but the hydrogens, which stick to the side)

Rotational

The other Csomething types all have exactly one axis of rotation

• Cn (nn) - one axis of rotation, nothing more
• Cnh (n*) - one axis of rotation, and a mirror plane perpendicular to that rotational axis
• known as (n-fold) prismatic symmetry
• Cnv (*nn) - one axis of rotation, and a mirror plane not perpendicular to that rotational axis (verify)
• known as (n-fold) pyramidal symmetry
• C∞v - infinite symmetry on an axis (and no mirror; compare to D∞h)
• happens with linear molecules, which have an axis of rotation in which any position means it's the same

• Sn, called improper axis rotation symmetry
• Example: CH4 has three S4 symmetries
• Note: S2 is equivalent to inversion (verify)

Examples:

• Cn in 2D is relatively straightforward. For example:
• Cn in 3D is still fairly simple, because it is constrained by that one axis we have:
• C2v examples: H2O
• C3v example: NH3
• C∞v example: HCl

Dihedral symmetry

All have one Cn axis (sort of the main axis), and n C2 axes.

The main axis is high-order (has a bunch of rotations (n-fold)), and the C2 axes are different ways to rotate it to flip back to front(verify).

• Dn (22n) - just those.
• Fairly rare, in part because having those while avoiding mirror symmetries requires some interesting detail
• Dnh (*22n) - basically meaning Dn, a back to front mirror symmetry, and n mirror symmetries along the rotation
• known as n-fold prismatic symmetry
• Cn, n C2, σh, n σv (also Sn?)

• Dnv - similar or identical to Dnd?
• D∞h - infinite symmetry on an axis, and a mirror (compare to C∞v)
• happens with linear molecules, which have an axis of rotation in which any position means it's the same
• Example: CO2

Examples:

• in 3D, Dn basically means there are no other axes of symmetry, and no mirror symmetries. For example:
• D3d: C3H6 (when staggered, which is its preferred conformation)
• D4d: S8

• D1 means a mirror symmetry, but no other symmetries. For example chairs, hands, pants, eyeglasses, the letters A, D, and E. There are relatively few things that have only mirror and no other sorts of symmetry(verify), and many of them are designed.

Cubic types:

These have more than one high-order axis.

• On - octahedral symmetry
• Tn - tetrahedral
• In - icosahedral

Screws are a potentially confusing example. If you take just the threading and pretend it's infinite, they have helical symmetry. If you take the screw as a real object, they may have Cn symmetry, depending both(verify) on how fast they wind around, and on whether they use double-start or triple-start thread thread (see Lead and pitch in screws

### Some further notes

• Biopolymers often see C, D, T, and O (verify)
• What we think of as rotational symmetry can be divided into radial symmetry (the type you're probably thinking of most of the time, and mentioned above) and helical symmetry, which means cases where a combination of rotation and translation (around and along the same axis) gives symmetry. It's relatively rare that helical symmetry is directly useful to you, in part because for real-world objects you have to ignore that they are finite. Given that, consider (the middle part of) screws and drill bits, spiral staircases (a discrete example)