From six numbers to seven families
In guide 2 you learned that a unit cell is pinned down by just six numbers — three edge lengths a, b, c and the three angles between them (alpha, beta, gamma), together its lattice parameters. Those six numbers are the stamp's exact shape, and stamping it over and over builds the whole lattice. Now a natural worry: six free numbers can take infinitely many values, so surely there are infinitely many kinds of crystal?
The answer, one of the quiet triumphs of crystallography, is no. When you insist that the pattern repeat perfectly in every direction, symmetry does the sorting for you. Every possible unit-cell shape falls into exactly one of seven crystal systems — no more, no fewer. Seven boxes hold every crystal that has ever been measured or ever could be, from table salt to a snowflake to a diamond.
The seven systems, weakest symmetry to strongest
Line the seven up by how much symmetry they carry and they form a neat ladder. At the bottom sits triclinic, a squashed brick with no two edges equal and no angle special — the least symmetric shape a lattice can take (blue vitriol, CuSO4 crystals, live here). Climb up, adding one symmetry constraint at a time, and you pass through monoclinic (gypsum), orthorhombic (the sulfur in a match head), tetragonal (white tin), trigonal (quartz, calcite), and hexagonal (magnesium, graphite, a snowflake), until you reach cubic at the top — the perfect box of salt, copper, and diamond, the most symmetric of all.
CRYSTAL SYSTEM EDGES ANGLES MINIMUM SYMMETRY -------------------------------------------------------------------------- triclinic a != b != c al,be,ga all != 90 none (1 or -1 only) monoclinic a != b != c al=ga=90, be != 90 one 2-fold axis orthorhombic a != b != c al=be=ga=90 three 2-fold axes tetragonal a = b != c al=be=ga=90 one 4-fold axis trigonal (rhomb) a = b = c al=be=ga != 90 one 3-fold axis hexagonal a = b != c al=be=90, ga=120 one 6-fold axis cubic a = b = c al=be=ga=90 four 3-fold axes
Read any row and the logic runs from right to left. The tetragonal system, for instance, is DEFINED by owning a single four-fold rotation axis: turn the crystal by 90 degrees about that axis and it looks unchanged. That one demand instantly forces two of the edges to be equal (a = b) and squares up all three angles — the shape a = b != c with every angle 90 is what a four-fold axis leaves behind. The geometry is the shadow; the symmetry is what casts it.
Why the shape follows the symmetry
This ordering has a sharp, practical edge. Suppose you measure a crystal and its cell comes out with beta = 90.02 degrees — tantalisingly close to a right angle. Is it monoclinic (which allows beta != 90) or orthorhombic (which demands beta = 90)? The metric alone cannot tell you. Only its symmetry can: if the crystal carries just one two-fold axis it is monoclinic, and that 90.02 is a near-coincidence, not a right angle promoted by symmetry. Numbers can lie; symmetry cannot.
The cubic system carries the most beautiful example of this. Everyone remembers cubic as 'a = b = c, all angles 90' — the shape of a die. But the true defining mark of cubic is not its equal edges; it is the four three-fold rotation axes running along the body diagonals (the <111> directions). Those four triads are what make the cube's corners interchangeable, and they are what force the edges equal. A crystal with a = b = c but only a single three-fold axis is trigonal, not cubic. The cube's symmetry lives in its diagonals, not its edges.
- Find its symmetry elements first — the rotation axes, mirrors, and inversion centre — not its cell edges.
- Identify the highest-order rotation axis present: a six-fold means hexagonal, a four-fold tetragonal, a single three-fold trigonal.
- Check the cubic hallmark separately: four three-fold axes along the body diagonals means cubic, whatever the edge lengths happen to look like.
- With no axis above two-fold, count the two-folds and mirrors: three perpendicular ones give orthorhombic, one gives monoclinic, none gives triclinic.
- Only now read off the axial relations — they fall out automatically from the symmetry you found.
The forbidden fold: why crystals cannot be five-fold
Look back at the table and you will see rotation axes of order 1, 2, 3, 4, and 6 — but never 5, and never 7 or higher. This is no accident of which crystals happen to exist; it is a theorem. A crystallographic restriction proves that a lattice which repeats by translation can only host rotation axes of order 1, 2, 3, 4, or 6. Five-fold and eight-fold symmetry are mathematically impossible in a periodic crystal.
The everyday picture is a tiled floor. Squares tile perfectly, and so do triangles and hexagons — but try to cover a floor with regular pentagons and you cannot: gaps and overlaps appear no matter how you turn them. Five-fold symmetry simply will not fill a plane by repetition. The algebra says the same thing crisply: a repeating axis is allowed only when 2 times cos(360/n degrees) is a whole number. For n = 6 that is 2 times cos(60) = 1, for n = 4 it is 0, for n = 3 it is -1 — all whole numbers. For n = 5 it is 2 times cos(72) = 0.618, not a whole number, so five-fold is ruled out.
Seven systems, and the road ahead
The seven systems answer the question 'what shapes can a unit cell take?'. But recall from guide 3 that a cell can be more than an empty box — you can add extra lattice points at its body centre, its face centres, or its base. Combine the seven shapes with those centering options and you might expect 7 times 4 = 28 distinct lattices. You get far fewer, because many combinations are either redundant (a centred cell that is really just a smaller primitive one in disguise) or would destroy the very symmetry that defined the system. When the dust settles, exactly fourteen survive: the Bravais lattices, the subject of the next and final guide in this rung.
It is worth savouring how firm these numbers are. There are exactly 7 crystal systems, exactly 14 Bravais lattices, exactly 32 crystallographic point groups, and exactly 230 space groups. None of these is a 'so far' count that a new discovery might extend — each is a complete mathematical enumeration, proved to be exhaustive. When someone finds a wholly new material tomorrow, its symmetry is guaranteed in advance to be one of those 230, no matter how exotic it seems.
One honest footnote you will meet in the textbooks. Trigonal and hexagonal are counted as two separate systems here (a three-fold axis versus a six-fold axis), yet they share the same hexagonal-shaped cell, so some books instead group them into six 'crystal families'. Both counts are correct — they simply draw the line in slightly different places. It is the classic reminder that a clean number like 'seven' can quietly hide a real, defensible choice about where one category ends and the next begins.