Crystallography · Lattices & Point Groups series · Part 2 of 3
Crystallography II: Stereographic Projections
An entire 3D crystal flattened onto a circle, and somehow no information died.
A stereogram is a 3D construction flattened onto paper. This lab keeps the 3D part: watch a direction get projected live, generate the standard stereogram for any of the 32 point groups, and work hexagonal indices, all in true rotatable 3D.
Part 2 of 3 in the Crystallography series. Previous: Crystallography I, Lattices & Point Groups · Next: Crystallography III, Space Groups · see also Interfaces & the Coincidence Site Lattice.
Covers chapters 2 and 3 of the source syllabus together (stereographic projections, then their extension to low-symmetry and hexagonal systems), deliberately as one lab rather than two: the construction itself doesn't change between crystal systems, only which indices and angles you feed into it. For pencil-and-Wulff-net practice, DoITPoMS's Stereographic Projection TLP is an excellent free tool. This lab has a Wulff net too, but its focus is what paper can't do: show the projection happening in 3D, draw a point group's full stereogram for you, and handle the hexagonal fourth index.
All three modes share one idea: a crystal direction is a point on a reference sphere, and the stereogram is what you get by flattening that sphere onto a plane from one pole. Drag any 3D scene below to turn it. Mode 1 shows the sphere and the flat stereogram side by side; in Modes 2 and 3, press 2D to see the flat diagram.
The projection, live
Try this: turn on Normal and Plane, then drag ρ past 90°. The pole drops into the lower hemisphere, but the plane and its blue curve stay where they are. Type (1 1 1) to put a real cubic plane on the sphere, add a second plane to find the zone axis where the two curves cross, or test yourself with Predict, then reveal.
The pole
Drag the 3D view to turn it; scroll or pinch to zoom. Click the flat map to move the pole there.
Show
Second plane
Predict, then reveal
Each question hides part of the picture. You answer by clicking the flat map, then the hidden part appears so you can check.
A plane is a dot or a curve
The pole is where the plane's normal meets the sphere. The plane itself cuts the sphere in a great circle, 90° from its pole everywhere.
n and −n: one plane
Both ends of the normal describe the same plane. The upper end is the one usually plotted.
Two planes, one zone axis
Two great circles cross at the zone axis, the line both planes contain. The poles of all planes in that zone lie on one great circle.
Upper hemisphere
A pole with z ≥ 0 is projected from the south pole, straight through the equatorial plane. It is plotted as a filled point.
Lower hemisphere
By convention a pole with z < 0 is projected from the north pole instead. It is plotted as a hollow point, so the two kinds stay distinct.
At the equator
Both constructions agree there: every direction on the equator lands on the unit circle, R = 1.
- Two great circles crossing at a zone axis is how you steer a specimen in the TEM: follow a band on the Kikuchi map navigator, or index the pattern you land on with the SAED zone-axis indexer.
- A pole figure is this Wulff net filled with the poles of thousands of grains: Texture: pole figures and ODFs.
- The angle between a slip plane's pole and the load decides which grain yields first: Schmid factor and slip systems.
More detail: poles, great circles, and why the projection point switches hemispheres
Every pole in the upper hemisphere is projected from the south pole, straight through the equatorial plane. A pole in the lower hemisphere would need a projection line that also passes back through the sphere before it could reach the plane if projected from the south pole too, so by convention it's instead projected from the north pole, and plotted with a hollow rather than filled symbol so the two families stay visually distinct on the same flat diagram. Both conventions agree exactly at the equator, where every direction (upper or lower construction) lands on the unit circle, R = 1.
A plane through the centre of the sphere cuts it in a great circle, and the stereographic projection maps that circle to a circular arc on the flat map (or a straight line through the centre when the plane stands upright). So a plane can be plotted two ways: as its pole, or as its great circle. The two carry the same information, since every point of the great circle is 90° from the pole. Stereograms of crystals usually plot poles, because one dot per plane stays readable when there are dozens of planes; Mode 2 lets you draw a whole point group both ways and compare. Great circles earn their place when you ask which planes share a direction: two planes' great circles cross at their zone axis, and the poles of every plane in that zone lie on the great circle whose pole is the zone axis. The same curves appear in electron diffraction: the centre line of a Kikuchi band follows its plane's great circle on a stereographic map of the beam directions (see the Kikuchi map navigator).
The Wulff net is the equal-angle grid drawn for this projection. Its curves running from top to bottom are great circles; the ones running across are small circles, 10° apart. To measure the angle between two poles by hand, turn the net until both poles sit on the same great circle of the net, then count the 10° steps along it. The readout gives the exact angle from the 3D normals, so you can check your count.
Point-group stereograms
Try this: pick a point group and read its symmetry symbols (oval, triangle, square, hexagon for 2-, 3-, 4- and 6-fold axes; solid curves for mirrors). Then click the flat map to move the starting pole and watch every copy follow. Click any pole to see which operation put it there, or switch Draw each plane as to great circles: with m-3m that is 24 curves, and the symmetry that the 48 poles show at a glance becomes hard to see.
Group and starting pole
Show
Test yourself
Two kinds of question: how many poles a group makes from a starting pole, and which point group made a stereogram.
How it is made
Apply every operation in the group to one starting pole and plot where each image lands.
General pole
A pole that does not sit on any symmetry element gives as many points as the group has operations: the multiplicity is the order.
Special pole
A pole on a rotation axis or mirror gives fewer images, because part of the group leaves it fixed. In m-3m: {100} gives 6, {111} gives 8, {110} gives 12.
Reading the symbols
Filled oval, triangle, square, hexagon: 2-, 3-, 4-, 6-fold rotation axes. A small open circle inside a symbol marks a centre of symmetry, so a triangle with a circle is 3̄. An open square with an oval inside is 4̄; an open hexagon with a triangle inside is 6̄. Solid curves are mirrors; a bold outer circle is a mirror in the equator.
Forms
The planes of one family make a crystal form. Some close into a solid (cube, octahedron); others are open (a pyramid, a prism) and need a second form to close the crystal.
Poles, not circles
Every pole could be drawn as its great circle, but with many planes the curves hide the pattern. Mirrors are the planes that are drawn as curves, because they are the symmetry itself.
- Work out which of the 32 point groups a crystal belongs to with the point-group identifier in Crystallography I: lattices and point groups.
- Why an EBSD orientation map needs only one small triangle of the stereogram (the standard triangle): in m-3m the whole sphere is 48 copies of it. Reading an IPF map.
- Symmetry at the microscope: every symmetry-equivalent family of planes shows up together as a set of bands on the Kikuchi map navigator.
More detail: reading a point-group stereogram
This is the diagram every crystallography text prints once per point group, generated here instead of traced by hand: apply every operation in the group to one starting pole and plot where each image lands. A fully general starting pole (not sitting on any symmetry element itself) gives as many points as the group has operations, its multiplicity is the order, the same load-bearing check used in Chapter 1's point-group identifier. A special pole, one lying on a rotation axis or mirror already, gives fewer images because part of the group leaves it fixed; picking [001]-type, [111]-type, or [011]-type from the dropdown shows exactly that reduction for the selected group.
The symmetry elements are read off the operations themselves. Each operation is a 3×3 matrix: a proper rotation (determinant +1) has an axis and an angle, and an improper one (determinant −1) is an inversion, a mirror (a 180° rotation combined with inversion), or a rotoinversion. Along each axis the page draws the highest-order symbol the group has there, following the usual textbook convention: 3̄ where a 3-fold axis comes with inversion, and 4̄ or 6̄ where the rotoinversion is of higher order than the plain rotation on that axis. The directions typed or listed here are Cartesian (x, y, z) in the group's own frame, which match Miller indices only in the cubic groups.
Hexagonal indices & interplanar angles
Try this: set plane 1 to (1 0 1) and plane 2 to (0 1 1), read the angle, then move the c/a slider or pick a real metal and watch the angle change. Turn on Direction with the same numbers: in a hexagonal cell [1 0 1] does not point along the normal of (1 0 1). Switch the system to cubic and it does.
Two planes
In the hexagonal cell the third box is i. It fills itself in as i = −(h + k); type a different value and you get a warning.
Show
4-index converter (hexagonal) switch between 3-number and 4-number notation
One route for every system
Build each plane’s normal from the reciprocal lattice, then take the angle between the two normals.
Why not plain coordinates
In hexagonal and monoclinic cells the axes are not perpendicular, so the indices cannot be used as Cartesian coordinates the way cubic ones can.
Switch to cubic
With the same indices the cubic calculation is a different geometry, so the angle can change. It is not a cosmetic change.
Same numbers, different arrows
Only in cubic crystals does [h k l] point along the normal of (h k l) for every set of indices. In a hexagonal cell the two usually differ; (0 0 1) and [0 0 1] are one exception.
c/a sets the angles
Angles between two prism planes, or between the basal plane and a prism plane, do not depend on c/a. Most angles involving a pyramidal plane do, so Mg, Ti and Zn give different angles for the same indices.
Why four indices
With i = −(h + k), the six prism planes of one family are all permutations of the same numbers, such as (1 0 −1 0) and (0 1 −1 0). In three indices they look unrelated.
- Indexing a diffraction pattern from an HCP metal uses the same 4-index notation and the same c/a-dependent angles: SAED indexing, g-vectors and real vs. fake spots.
- Which slip systems carry the load in hexagonal magnesium: the HCP option of the Schmid factor lab.
- On a hexagonal Kikuchi map, c/a moves the zone axes and the tilts between them: try the hcp presets on the Kikuchi map navigator.
More detail: one formula, every crystal system
The interplanar angle here is computed the same way for all six systems in the dropdown: build each plane's normal from the reciprocal lattice (which already accounts for however skewed the real cell is), then take the angle between the two normal vectors. For hexagonal or monoclinic cells that's the only correct route, since the crystallographic axes aren't mutually perpendicular, so the indices can't be treated as plain Cartesian coordinates the way cubic ones can. Switch the system dropdown to cubic with the same two index pairs still in the boxes and you're seeing a genuinely different geometric calculation land on a different angle, not a cosmetic change.
The zone axis of two planes is the cross product of their indices, [u v w] = (h₁ k₁ l₁) × (h₂ k₂ l₂). That rule (the Weiss zone law, hu + kv + lw = 0 for every plane in the zone) holds in every crystal system, because it is written in the lattice's own indices. The family of a plane is found by applying the symmetry of the lattice (6/mmm for hexagonal, m-3m for cubic, and so on) to its normal. The c/a values listed are room-temperature values for the pure metals.
One thing to know when reading a stereogram
On a stereogram, the distance between two poles on the flat map is not the angle between them. The same angle looks shorter near the centre and longer near the edge. To read an angle by hand, turn the Wulff net until both poles sit on one of its great circles, then count the 10° steps. This page never measures the flat map: every angle it shows comes from the 3D directions.
References
- Cullity, B. D. & Stock, S. R. Elements of X-Ray Diffraction. Prentice Hall, 3rd ed., 2001. (Standard stereographic-projection construction and Miller-Bravais conventions.)
- Hahn, T. (ed.) International Tables for Crystallography, Volume A. International Union of Crystallography, 5th ed., 2005.
- DoITPoMS, University of Cambridge. "Stereographic Projection" Teaching and Learning Package. (An interactive Wulff-net tool for pencil-and-net practice.)
- Bhadeshia, H. K. D. H. Worked Examples in the Geometry of Crystals, 2nd ed. Institute of Materials, 2001. (Topic outline for chapters 2-3 of this series; equations above are derived independently from the standard literature, not reproduced from this text.)
BibTeX
@misc{tripathy2026stereographicprojectionslab,
author = {Tripathy, Manisha},
title = {Stereographic Projections in 3D: the Construction, Point-Group Stereograms & Hexagonal Indices},
year = {2026},
howpublished = {\url{https://untetheredatom.com/crystallography/stereographic-projections-lab}},
note = {Interactive teaching resource}
}