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. Excellent free tools already exist for practicing the classic pencil-and-Wulff-net construction (DoITPoMS's Stereographic Projection TLP in particular), so rather than duplicate that, this lab leans on what a browser can do that paper can't: show the projection actually happening in 3D, generate a point group's full stereogram automatically instead of plotting it by hand, and handle the hexagonal system's extra index cleanly.
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 scene below to orbit it, and use "view from above" to see it collapse into the familiar flat diagram.
The projection, live
Drag to orbit · scroll/pinch to zoom.
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.
Point-group stereograms
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.
Hexagonal indices & interplanar angles
4-index converter
Interplanar angle
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.
Simplifications made in this lab
The flat 2D distance between two points on a stereogram is not their true angular separation, only the projection's angle-true (conformal) property at each individual point is guaranteed, not a global distance relationship, which is exactly why real Wulff-net work rotates points onto a meridian using the net's printed grid before measuring, rather than reaching for a ruler. Mode 3's angle calculator sidesteps this entirely by computing angles from the actual 3D normal vectors, never from flat projected positions, so nothing on this page relies on ruler-measuring a projection.
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. (The established interactive Wulff-net tool this lab deliberately doesn't duplicate; see it for hands-on 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}
}