Chirality, Centrosymmetry, and the Stereographic Projection

Where the rung leaves us

The rung is nearly built. Guide 1 named the symmetry operations and the elements they act on — the rotation axis, the mirror plane, the centre of inversion, and the rotoinversion axis. Guide 2 proved through the crystallographic restriction theorem that a periodic lattice can host only 1-, 2-, 3-, 4-, and 6-fold rotation and never 5-fold. Guide 3 combined those operations into the exactly 32 point groups that sort into the seven crystal systems, and guide 4 gave you two ways to write each one down, the Hermann-Mauguin and Schoenflies notations. Everything is in place.

This last guide cashes the machinery in. It pulls out three things you can now decide about any crystal from its point group alone: whether the crystal is left- or right-handed (chirality), whether it carries a centre of inversion (centrosymmetry), and how to draw its whole symmetry on one flat diagram (the stereographic projection). These are not idle labels — handedness governs whether a crystal rotates polarised light, the missing centre is precisely what lets a crystal be piezoelectric, and the same absent centre is the reason diffraction can never fully read the symmetry it measures.

Chirality: crystals with a handedness

Hold up your two hands. They are mirror images, yet no matter how you turn one it will never sit exactly on top of the other — the thumbs point the wrong way. Any object with that property is chiral, or handed. A crystal is chiral in exactly one clean situation: when its point group contains only proper rotations and no improper operation at all — no mirror plane, no centre of inversion, no rotoinversion axis. Mirrors and inversions are the operations that would turn a shape into its own reflection; strip them all away and the crystal can no longer be superimposed on its mirror image. Exactly 11 of the 32 point groups qualify: 1, 2, 3, 4, 6, 222, 32, 422, 622, 23, and 432. This is what chirality means at the level of the crystal class.

A chiral crystal therefore comes in two distinct mirror-image forms, called enantiomorphs, the way gloves come in lefts and rights — and this is enantiomorphism. Quartz is the textbook case: it belongs to point group 32, and it grows as either left-handed or right-handed crystals whose small facets spiral around the vertical axis in opposite senses. The handedness reaches all the way down to the atoms, where the SiO4 tetrahedra wind along a 3-fold screw axis clockwise in one form and anticlockwise in the other. That atomic twist is what makes quartz optically active: left quartz rotates the plane of polarised light one way, right quartz rotates it exactly the opposite way.

Centrosymmetry, and why the missing centre matters

A crystal is centrosymmetric when it owns a centre of inversion — a single point such that every atom sitting at position r has an identical twin at -r, the whole structure looking unchanged when turned inside out through that point. Of the 32 point groups, exactly 11 have this centre and are centrosymmetric; the other 21 lack it and are non-centrosymmetric. That looks like a dry bookkeeping split, but centrosymmetry is one of the most physically loaded facts about a crystal, because a centre of inversion vetoes any property that points a direction.

Piezoelectricity is the sharpest example — squeeze the crystal and a voltage appears across it. Here is why it needs no centre. When you press a crystal, its ions shift a little. If the crystal has a centre of inversion, every ionic displacement has an inversion partner shifting the opposite way, so the tiny dipoles they create cancel exactly and no net charge separation survives: zero voltage. Remove the centre and the cancellation fails, so squeezing can genuinely pull the centres of positive and negative charge apart and produce a voltage. This follows from Neumann's principle — a physical property must have at least the symmetry of the crystal. Of the 21 non-centrosymmetric classes, 20 are piezoelectric (the sole exception is cubic 432, whose other symmetry still cancels the effect). Quartz, class 32, is one of them: the sliver of quartz oscillating in a quartz watch is piezoelectricity keeping time.

A subset of the non-centrosymmetric classes goes further still. Ten of them are polar: they keep a unique axis that no operation can reverse, so the crystal carries a built-in electric dipole even with nothing pressing on it (a spontaneous polarisation). Warm such a crystal and the polarisation shifts — that is pyroelectricity — and in some the polarisation can even be flipped by an applied field, which is ferroelectricity. The whole hierarchy is easiest to read as a single count.

The 32 point groups sorted by symmetry  (categories can overlap)
----------------------------------------------------------------
property                       count   note
centrosymmetric (has a centre)   11    = the 11 LAUE CLASSES
                                       (all diffraction can resolve)
non-centrosymmetric              21    no centre of inversion
   piezoelectric                 20    squeeze -> voltage
                                       (every non-centro class but 432)
   polar / pyroelectric          10    keeps a unique, unreversed axis
   enantiomorphic / chiral       11    rotations only -> left/right forms
                                       1 2 3 4 6 222 32 422 622 23 432

  11 centrosymmetric + 21 non-centrosymmetric = 32 point groups

The 11 Laue classes: what diffraction cannot see

Now to that missing centre from the point of view of an experiment. A diffraction pattern records the intensities of the reflections, |F(hkl)|^2, and in doing so it throws away the phase of the wave — the phase problem you met earlier in the ladder. One consequence is immediate and unavoidable: the reflection (hkl) and its opposite (-h -k -l) come out with equal intensity, because their structure factors are complex conjugates and so have the same magnitude. This is Friedel's law, and it means every diffraction pattern looks centrosymmetric — it has a false centre of inversion baked in — whether or not the crystal actually has one.

So diffraction cannot tell a non-centrosymmetric crystal apart from the centrosymmetric one you would get by simply adding a centre. Adding a centre of inversion to each of the 32 point groups collapses them onto just 11 distinct symmetries — the 11 Laue classes, which turn out to be precisely the 11 centrosymmetric point groups. An ordinary diffraction experiment can sort a crystal only into one of these 11 boxes. For instance, point groups 4, -4, and 4/m all show the Laue symmetry 4/m, while 422, 4mm, -42m, and 4/mmm all show 4/mmm. Determining which of several point groups sharing a Laue class you actually have needs extra evidence — a piezoelectric test, an optical measurement, or the trick in the next paragraph.

The stereographic projection: symmetry on a flat page

All of this — the rotation axes, the mirrors, the handedness, the presence or absence of a centre — needs a way to be drawn, since a point group lives in three dimensions and paper is flat. The stereographic projection is that map, and it is the same trick a geographer uses to flatten the globe. Put the crystal at the centre of an imaginary sphere; every face normal and every symmetry axis pierces the sphere at a point called a pole. To flatten the sphere onto its equatorial disc, take a pole in the upper hemisphere, draw a straight line from it down to the south pole of the sphere, and mark a filled dot where that line crosses the disc. A pole in the lower hemisphere is projected the other way, up to the north pole, and marked with an open circle. The rim of the disc — the equator itself — is called the primitive circle.

1. Draw the primitive circle and mark the symmetry axes with their standard glyphs — a filled lens for a 2-fold, a triangle for 3-fold, a square for 4-fold, a hexagon for 6-fold; draw mirror planes as bold lines or arcs.
2. Place one general pole — a filled dot — at a spot in the upper hemisphere that lies on no special axis, so it is free to be moved around.
3. Apply every operation of the point group to that pole; each operation carries it to an equivalent pole somewhere else on the diagram.
4. Watch what each operation does: a horizontal mirror or a centre of inversion sends an upper dot to a lower open circle, while a vertical mirror keeps it a dot on the other side.
5. Count the poles when you finish — their total equals the order of the group, which is the multiplicity of the general position.

The stereographic projection earns its place because it is angle-true (conformal) and maps circles on the sphere to circles on the page, so the angles between crystal faces can be read straight off the diagram — the reason it has served crystallographers and geologists for two centuries. It also displays this whole guide at a glance. A centrosymmetric group pairs every dot with a circle at the very same spot, betraying its centre; a polar group leaves one direction pointing out unmatched; and a chiral group, whose poles are all one handedness, gives a diagram that cannot be rotated to match its own mirror image. The 32 point groups are catalogued as exactly these stereograms in the International Tables — and in the next rung the same poles become the scaffold for the 230 space groups, where point symmetry finally marries the lattice's translations.