Lecture 6 • Autumn 2026

Point Groups in High Symmetry &
Stereographic Projections

Delving into the symmetry of dual polyhedra, a systematic flowchart for point group identification, and the visual power of stereographic projections.

AS

Dr. Abhijeet L. Sangle

Assistant Professor (Gr- I)

alsangle@iitb.ac.in

+91 22 2159 6742

High Symmetry Groups

The Octahedron & The Cube

The Octahedron and the Cube are duals of one another. The object formed by joining the centres of the faces of a cube is an octahedron, and vice versa.
Rule: Duals share the same point groups.

Octahedron ($O_h$)

Order = 48
Drag to rotate

Symmetry Operations:

Click below to visualize the symmetry elements in the 3D viewer:

$O_h = \{E, 8C_3, 6C_2, 6C_4, 3C_2, i, 6S_4, 8S_6, 3\sigma_h, 6\sigma_d\}$

Cube ($O_h$)

Order = 48
Drag to rotate

Symmetry Operations:

Notice how the locations of axes swap compared to the Octahedron!

Exactly the same elements as $O_h$

Pure Rotational Group: $O$

If we remove all operations involving reflection or inversion from $O_h$ (i.e., $i, S_4, S_6, \sigma_h, \sigma_d$), we are left with the pure rotational subgroup $O$.

$O = \{E, 8C_3, 6C_2, 6C_4, 3C_2\}$

Order = 24

Icosahedron & Dodecahedron ($I_h$)

Another pair of Platonic duals sharing the highest discrete point group symmetry.

Icosahedron

Order = 120
Drag to rotate

Symmetry Axes:

Visualize major rotational axes:

$E$
$i$
$12C_5, 12C_5^2$
$12S_{10}, 12S_{10}^3$
$20C_3$
$20S_6$
$15C_2$
$15\sigma$

Dodecahedron

Order = 120
Drag to rotate

Symmetry Axes:

Notice the dual relationship: Face and vertex axes swap!

Pure Rotational Subgroup ($I$)

$I = \{E, 12C_5, 12C_5^2, 20C_3, 15C_2\}$

Determining Point Groups

A systematic flowchart approach to identify the point group of any molecule or object.

NO YES NO YES NO YES YES YES YES NO NO YES YES NO NO YES YES NO YES NO NO NO YES NO YES NO YES NO YES YES NO YES NO
START
$C_n$?
Low symmetry
point group
$i$?
$C_i$
$\sigma$?
$C_s$
$C_1$
6 $C_5$, or 3 $C_4$, or 4 $C_3$?
n $C_2$ perpendicular to $C_n$?
High symmetry
point group
6 $C_5$?
$i$?
$I_h$
$I$
3 $C_4$?
$i$?
$O_h$
$O$
4 $C_3$?
$\sigma_h$?
$T_h$
$\sigma_d$?
$T_d$
$T$
Rotational group
$\sigma_h$?
$C_{nh}$
$\sigma_v$?
$C_{nv}$
$S_{2n}$?
$S_{2n}$
$C_n$
Dihedral group
$\sigma_h$?
$D_{nh}$
$\sigma_d$?
$D_{nd}$
$D_n$

Interactive Flowchart: Real World Cacti

Follow the flowchart steps interactively to identify the point group for these two star cacti. Click the correct answers to reveal the path.

Monocolour Cactus

Monocolor Star Cactus

    Step 1: Is a $C_n$ axis present?

    Incorrect. Hint: Look at the top-down symmetry.

    Bicolour Cactus

    Bicolor Star Cactus

      Step 1: Is a $C_n$ axis present?

      Incorrect. Hint: Does the color alternating pattern allow full rotation?

      Visualizing 3D in 2D

      Stereographic Projections

      Definition & Construction

      A 2-D representation of a 3-D object located at the centre of a sphere. This projection preserves angular relationships between faces or vectors, making it invaluable in crystallography.

      How to Construct:

      • Consider a point $P$ on the surface of the sphere.
      • If $P$ is in the Northern Hemisphere, draw a line to the South Pole. The intersection with the equatorial plane is the projection (marked as a • filled circle).
      • If $Q$ is in the Southern Hemisphere, draw a line to the North Pole. The intersection is marked as a larger (○ open circle).
      N
      S
      P P' Q Q'

      All 32 Crystallographic Point Groups

      Stereographic projections showing symmetry elements (cyan) and symmetrically equivalent points (white). Solid dots are in the Northern Hemisphere, open circles in the Southern Hemisphere.

      Interactive Projection Builder

      Select a point group and generate its stereographic projection step-by-step to see how symmetry operations propagate a general point.

      Select a group

      Example: $C_{3h}$ Point Group

      Fill the multiplication table by clicking on the ? cells. The stereogram will demonstrate $Op_2 \circ Op_1$ (applying Op 1 from the row first, then Op 2 from the column).

      Stereographic Visualization

      Click a cell in the table to see how the two symmetry operations combine.

      Op 1 (Row) \ Op 2 (Col) $E$ $C_3$ $C_3^2$ $\sigma_h$ $S_3$ $S_3^5$
      $E$
      $C_3$
      $C_3^2$
      $\sigma_h$
      $S_3$
      $S_3^5$

      Translations & Crystallography

      In crystallography, there is one more essential symmetry operation: Translation. Let's see how it forms lattices.

      Lattice

      An infinite array of points in space that has the exact same environment at every point.

      Motif / Basis

      The repeating physical element (atoms, molecules, ions) attached to each lattice point.

      Primitive Unit Cell

      The smallest possible space which repeats itself and contains exactly one lattice point in it.

      Generating Lattices via Translation

      All lattice points can be generated simply by translation along basis vectors ($T_1, T_2, T_3$), without needing any other symmetry operation.
      Vector $\mathbf{T} = n_1\mathbf{T_1} + n_2\mathbf{T_2} + n_3\mathbf{T_3}$

      1D Lattice

      T1 T1 T1

      Translation along one dimension only.

      2D Lattice

      T1 T2

      Translation along two distinct dimensions.

      3D Lattice

      Requires $T_1$, $T_2$, and $T_3$ (out of plane) to form full crystal structures.