Crystallography · Lattices & Point Groups series · Part 1 of 3
Crystallography I: Lattices & Point Groups
14 lattices, 32 point groups, one trainer, zero gatekeeping.
The 14 Bravais lattices, Miller indices and the Weiss zone law, and all 32 crystallographic point groups, in true rotatable 3D: drag anything on this page and turn it over in your hands.
Part 1 of 3 in the Crystallography series. Next: Crystallography II, Stereographic Projections · Crystallography III, Space Groups · see also Interfaces & the Coincidence Site Lattice.
Everything else in crystallography is built on three ideas: a lattice is a periodic scaffold of points, a crystal structure is that scaffold with a repeating motif hung on every point, and the scaffold's own symmetry, its point group, constrains what shapes and physical properties the structure is allowed to have. This lab covers chapter 1 of the source syllabus in full: the 14 distinct ways a 3D lattice can repeat itself (the Bravais lattices), the index notation used to name directions and planes inside it (Miller indices, and the Weiss zone law that connects them), and the complete catalogue of 32 point-group symmetries a real crystal can have.
Every scene below is a real, live 3D model, built the same way a modeling package would build it, not a flat drawing. Drag to orbit, scroll or pinch to zoom, and everything else on the page, cell shape, indices, symmetry elements, updates from the same verified geometry underneath.
The 14 Bravais lattices
Drag to orbit · scroll/pinch to zoom.
Conventional vs. primitive
The conventional cell is chosen for convenience, usually to show off the lattice's full symmetry as simply as possible, and it is allowed to contain more than one lattice point. The primitive cell is the smallest possible repeat unit, exactly one lattice point per cell, always. The two coincide for the primitive (P) lattices; for body- (I), face- (F), and base-centered (C) lattices, the primitive cell is smaller and, for F in particular, looks nothing like the cube or prism you'd draw by hand. The ratio of conventional to primitive cell volume, shown live in the readout, is exactly the number of lattice points the conventional cell contains: 1 for P, 2 for I and C, 4 for F, matched here to machine precision rather than the usual corner-counting-fractions argument.
Miller indices & the Weiss zone law
Symmetry-equivalent planes
Why the zone law is exact regardless of cell shape
A direction [uvw] lies "in the zone" of a plane (hkl), meaning it lies within that plane (through the origin) and every plane in a physical zone axis shares this direction, exactly when hu + kv + lw = 0. That sum is pure index arithmetic: it never touches the cell's actual lengths or angles, which is why the same rule works unchanged from cubic to triclinic. Everything else on this page, the actual 3D direction of an arbitrary [uvw] or the true orientation of an (hkl) plane, does depend on the real cell shape (through the reciprocal lattice for planes), which is why the picture visibly reshapes itself when you switch crystal systems even though the zone-law verdict for a given index set never does.
Point group identifier
All 32 crystallographic point groups are built here from 1–3 generator matrices and closed under multiplication, the same way a group-theory text derives them, not hand-listed.
Reading the general-position orbit
Pick any point with no special coordinates (not on an axis, not on a mirror), apply every operation in the group to it, and count the distinct images: that count is always exactly the order of the group, because a generic point has no leftover symmetry of its own to reduce the count. It's a genuinely load-bearing check, not just a pretty picture: count the dots for any group below and it will match the order printed above the readout, every time.
Several planes together: interplanar angles, d-spacings & zone axes
[100], [010], [001] are the unit cell's own edge directions, not screen x/y/z. The sliders set the edge lengths a, b, c along them and the angles between them. Every (h k l) and [u v w] on this page is indexed in that same cell system.
Up to 4 planes through the cell center, each in its own identity color (4 is the accessibility cap: every 5-color extension fails colorblind separation in all-pairs mode).
| Plane | dhkl |
|---|
| Pair | Pole angle | Dihedral | Zone axis |
|---|
What "the distance between planes" means
Two non-parallel planes through a common point have no well-defined distance between them; the only meaningful metric for a pair is the interplanar angle, shown above as both the sign-aware pole angle (0--180 degrees, what diffraction and stereogram work use) and the undirected dihedral angle (0--90 degrees). "Distance between planes" is well-defined only within a single family: the d-spacing dhkl = 1/|ghkl| is the perpendicular distance between successive lattice planes of that family, computed exactly in any crystal system via the reciprocal lattice. Toggle the "Family" checkbox for any plane above to see its successive lattice planes drawn through the cell.
Reading the family stacks
When you show a plane's family, the lab draws every lattice plane of that family that crosses the unit cell, spaced exactly d apart. The first, index-defining member (n = +1, the plane cutting the cell edges at intercepts 1/h, 1/k, 1/l) is drawn brighter. Two families shown at once visibly interleave: (100) and (200) together show the (200) stack running exactly twice as fine, because every (100) plane is also a (200) member but not vice versa. The reciprocal-space dot (g) along each family's normal collapses the whole stack to the single reciprocal-lattice point g = ha* + kb* + lc*, at a distance proportional to 1/d from the origin.
Simplifications made in this lab
Two deliberate choices are worth being explicit about. First, every point group here is drawn with its principal axis fixed along z, including the monoclinic groups; International Tables for Crystallography conventionally uses the b-axis as the monoclinic unique axis instead, a labeling convention only, it doesn't change which operations are in the group or how many there are, only which Cartesian direction they're drawn along. Second, the "symmetry-equivalent planes" panel in Mode 2 (index permutation under the point group) is mathematically valid only for the three orthogonal systems, cubic, tetragonal, and orthorhombic, where the crystallographic axes are mutually perpendicular and Miller indices double as ordinary coordinates for symmetry purposes. For hexagonal, monoclinic, and triclinic cells the axes aren't orthogonal, so that shortcut would silently give wrong equivalents; the panel disables itself with an explicit note for those systems rather than guess.
References
- Hahn, T. (ed.) International Tables for Crystallography, Volume A: Space-Group Symmetry. International Union of Crystallography, 5th ed., 2005.
- Giacovazzo, C. et al. Fundamentals of Crystallography. Oxford University Press, 3rd ed., 2011. (Standard cell-to-Cartesian construction used throughout.)
- Bhadeshia, H. K. D. H. Worked Examples in the Geometry of Crystals, 2nd ed. Institute of Materials, 2001. (Topic outline for chapter 1 of this series; equations and worked examples above are derived independently from the standard literature, not reproduced from this text.)
BibTeX
@misc{tripathy2026latticespointgroupslab,
author = {Tripathy, Manisha},
title = {Crystallography I: Lattices & Point Groups},
year = {2026},
howpublished = {\url{https://untetheredatom.com/crystallography/lattices-point-groups-lab}},
note = {Interactive teaching resource}
}