Lattices · symmetries · snowflakes

CrystalFrom a snowflake to a diamond

A crystal is atoms arranged in a repeating pattern, like bricks in an endless wall. Start with a snowflake, then look inside salt, diamond and quartz.

Keys 1–5 switch between the labs.

Chapter 1

Snowflake

No two are alike. Every snowflake starts as a tiny hexagon of ice — everything else is decided by the air it falls through.

Nº 0128Sep 29, 2026, 22:45preview

Fernlike dendrite

−15 °C · humidity 92%

Lab

Set the conditions in the cloud and watch the crystal change.

Conditions in the cloud

the preview updates live

Temperature −15°C

Humidity 92%

Shortcuts: G grow, P similar, S save SVG, ←→↑↓ move the cursor on the diagram (with Shift — faster).

Nakaya diagram

drag the cursor or use the arrow keys

The horizontal axis is temperature, the vertical one is humidity — a nominal measure of how supersaturated the air is with water vapor relative to ice. The boundaries between regions are approximate.

Letters from the sky

A short note on how we know what we know.

In the 1930s the Japanese physicist Ukichiro Nakaya of Hokkaido University became the first to grow snowflakes in a laboratory. In a freezing chamber the crystals grew on a single rabbit hair, while Nakaya varied the temperature and the amount of water vapor, photographing every form. He called snowflakes letters sent from the sky: their shape tells you what kind of air they fell through.

Thousands of observations produced the Nakaya diagram. Temperature selects the basic habit — a flat plate or an elongated column — and switches it several times between −1 and −30 °C. Humidity decides the complexity: in dry air the crystal grows slowly and stays compact, while at high supersaturation the corners outrun the edges and sprout branches.

The diagram on this page is simplified, and its boundaries are approximate. A real snowflake passes through different layers of the cloud on its way down, which is why habits often mix in nature. The generator draws idealized crystals with perfect sixfold symmetry.

Catalog

Snowflakes you have grown, newest first. There is room for 36 specimens.

The catalog is empty. Set the conditions and press “Grow a snowflake”.

Why exactly six arms? Look inside ice in the lab .

Chapter 2

Crystal lattices

The unit cell is the smallest “brick” which, repeated, builds the whole crystal. Rotate the model, count the atoms and see how tightly they can be packed.

fcc · copper

Structure

Coordination number12neighbors of each atom

Packing74.0%of space filled by spheres

Atoms per cell8 × ⅛ + 6 × ½ = 4corners are shared by 8 cells, faces by 2

Density from the lattice

Substance

How many atoms (formula units) per cell?

Enter the number of atoms you counted in the drawing of the cell.

ρ = Z · MN_A · V

Shortcuts: B cell / block, K spheres / sticks, N neighbors; on the model ←→↑↓ rotate, +− zoom, 0 resets the view. Click an atom to select it.

Diamond or graphite?

The same carbon, different bonds. In diamond each atom holds four neighbors in a rigid three-dimensional framework. In graphite atoms join in threes into flat sheets of hexagons, and the sheets hold on to each other very weakly.

	Diamond	Graphite
Neighbors	4, in 3D (tetrahedron)	3, in a flat layer
Hardness (Mohs)	10 — the hardest mineral	1–2 — rubs off on fingers
Electricity	insulator	conducts well along the layers
Heat	conducts exceptionally well, better than copper	well along the layers, poorly across
Density	3.51 g/cm³	2.26 g/cm³
Price	a gram of cut stone: usually thousands of dollars or more	a gram: pennies (pencil lead)

How to count it

An atom's share in a cell depends on how many cells share it: a cube corner belongs to 8 cells (⅛), an edge to 4 (¼), a face to 2 (½), and the center to just one.

Packing is the fraction of the cell's volume taken up by touching spheres of radius r:

f = Z · 4⁄3 πr³V

For simple cubic it comes out as π/6 ≈ 52%, for body-centered cubic √3π/8 ≈ 68%, and for face-centered cubic and hexagonal close-packed π/(3√2) ≈ 74%. You can't do better — Johannes Kepler conjectured this in 1611, and the proof (with the help of a computer) was presented by Thomas Hales only in 1998.

The model is simplified: atoms are spheres of fixed radius. For the salts we use the ionic radii of Na⁺ and Cl⁻ (Cs⁺ and Cl⁻), adjusted so they touch. Ice and quartz are built of molecules and tetrahedra, so for them we don't give a sphere packing.

Chapter 3

Crystal systems

All the crystals in the world fit into seven systems and fourteen lattices. Symmetry follows iron rules here — with one famous exception.

Halite · cubic

System

Axes
Symmetry

Mineral

Rotate about an axis and check

Symbols at the ends of the axes: lens — 2-fold axis (rotation by 180°), triangle — 3-fold (120°), square — 4-fold (90°), hexagon — 6-fold (60°).

14 Bravais lattices

Around 1850 Auguste Bravais showed that in space there are exactly 14 kinds of point lattices. Click a lattice to select its system.

Why is there no 5-fold axis?

A crystal is a pattern that repeats with no gaps and no overlaps. Try fitting regular polygons around one vertex: the angles must add up to exactly 360°.

Polygon

Interior angle108°(n − 2) · 180° / n

Sum at the vertex0°

For the same reason a crystal lattice can only have 2-, 3-, 4- and 6-fold axes. A rotation maps lattice points onto lattice points only when

2 · cos(360° / n) is an integer

Subdivisions5

Thick : thin rhombi—

Golden ratio φ1.618…(1 + √5) / 2

Quasicrystals

In 1974 Roger Penrose showed that two rhombi can cover the plane with no holes and without repeating. The pattern has local 5-fold symmetry but never settles into a periodic lattice. The ratio of thick to thin rhombi approaches the golden ratio.

On April 8, 1982 Dan Shechtman, studying a rapidly cooled alloy of aluminum and manganese at the U.S. National Bureau of Standards (NBS), saw in an electron microscope a diffraction pattern with 10-fold symmetry. According to the knowledge of the time, that was impossible. In his notebook he wrote three question marks next to it.

He was met with disbelief and ridicule; his team leader asked him to leave the group, and two-time Nobel laureate Linus Pauling insisted to the end of his life that these were just twinned ordinary crystals. The paper was only published in 1984.

Other laboratories confirmed the discovery, and in 1992 the very definition of a crystal was changed: what counts is a sharp diffraction pattern, not periodicity. In 2011 Shechtman received the Nobel Prize in Chemistry on his own. In 2009 a natural quasicrystal was found too — in fragments of a meteorite found in Chukotka, in Russia's Far East.

See the forbidden 10-fold symmetry with your own eyes in the lab .

Chapter 4

Growing crystals

A salt crystal grows from a solution in which more has been dissolved than the water wants to hold. This “excess” decides whether you get a smooth cube or a lacy dendrite.

cross-section · 160 × 160 grid

Supersaturation60%

Temperature40°C

Shape—

Size—

Steps0time in model steps

Crystals1

A simplified cellular model: the substance diffuses in the solution and attaches to the crystal more easily the more neighbors it already has. Supersaturation and temperature are in arbitrary units here — it's a model of the phenomenon, not a lab recipe. Click in the solution to drop in your own seed.

The seed

nucleation

A crystal doesn't appear “out of nothing”: first a tiny nucleus has to stick together. One that is too small falls apart, because its surface costs too much energy. That's why even a supersaturated solution can stand quietly for a long time — until we drop in a grain or a thread, or scratch the glass.

σ = c − c_sc_s

c — concentration, c_s — saturation concentration (solubility).

Cube or dendrite?

supersaturation

At low supersaturation the molecules have time to find the best spots in the lattice, so the faces grow layer by layer and stay flat. At high supersaturation the corners reach “fresh” solution faster than the centers of the faces: funnel-shaped skeletal crystals form, and then branching dendrites.

It's the same instability that turns a hexagonal plate of ice into a star — .

Slowly = bigger

Ostwald ripening

Fast cooling gives high supersaturation, and so lots of nuclei at once — and a crowd of tiny crystals. Slow cooling keeps supersaturation low: there are few nuclei, and each has time to grow.

Over time the tiny crystals dissolve and the big ones grow at their expense, because a small grain dissolves a little more easily than a large one. This is Ostwald ripening.

Growing crystals at home

Three tried-and-tested experiments. How much of each substance dissolves in water — roughly:

Substance (per 100 g of water)	20 °C	50 °C	100 °C
table salt NaCl	≈ 36 g	≈ 37 g	≈ 39 g
sugar (sucrose)	≈ 200 g	≈ 260 g	≈ 480 g
potassium alum (hydrated)	≈ 12–14 g	≈ 37 g	a great deal

The solubility of salt barely depends on temperature, so salt crystals are best grown by slow evaporation. Sugar and alum crystallize best as a hot solution slowly cools. The values are approximate; different tables give slightly different numbers.

Salt crystals

evaporation · a few days

Dissolve salt in a glass of warm water until it stops disappearing (about 36 g per 100 mL).
Pour the clear solution onto a flat saucer or into a jar with a hanging thread.
Put it in a warm, quiet place and wait while the water evaporates.

Under a magnifying glass you'll see little cubes and “stepped” funnels — skeletal crystals.

Rock candy

sugar · 1–2 weeks

Bring 1 cup of water to the boil and dissolve 2–3 cups of sugar in it (an adult at the stove).
Wet a stick and roll it in sugar — those are the seeds. Let it dry.
Pour the cooled syrup into a clean jar, hang the stick so it doesn't touch the sides, and cover with paper.

You can eat this candy.

An alum crystal on a thread

alum · 1–3 weeks

Dissolve as much alum as you can in hot water (about 40–50 g per 100 mL) and let it cool.
After a day pick the nicest little crystal from the bottom and tie it to a thread.
Hang it in fresh, filtered solution. Every few days move the crystal to new solution.

A transparent octahedron grows — like the fluorite in chapter 3.

Safety: hot solutions and sugar syrup scald — prepare them only with an adult. Never eat or lick alum; wash your hands afterwards and label the jar. Don't use containers that will later hold food.

Chapter 5

How we see atoms in a crystal

You can't see atoms through a magnifying glass. But X-ray waves reflected from successive layers of atoms reinforce each other only at certain angles — and from those angles we read the structure of the crystal.

Bragg's law

n · λ = 2 · d · sin θ

Angle θ15.9°

Wavelength λ154pm

Plane spacing d282pm

Presets

Path difference 2d·sinθ—

Reflection angles (n = 1, 2, 3…)—here the beams reinforce most strongly

Diffractometer

On the left, the sample — you can change it and click to add or remove atoms. On the right, its diffraction pattern, computed with a fast Fourier transform (FFT 256 × 256).

sample

diffraction pattern

Sample

Atom spacing12

Rotation0°

Missing atoms0%

Thermal vibrations0%

Pattern symmetry—

From Laue to DNA

1912Max von Laue realizes that a crystal could act as a diffraction grating for X-rays. Walter Friedrich and Paul Knipping shine X-rays through a copper sulfate crystal and see regular spots. Laue receives the Nobel Prize in Physics for it in 1914.
1913William Henry and William Lawrence Bragg — father and son — describe the reflections with the simple law nλ = 2d·sinθ and work out the structure of table salt: there are no NaCl molecules in it, just a checkerboard of ions. A shared Nobel Prize in Physics, 1915.
1952In Rosalind Franklin's laboratory at King’s College London, together with PhD student Raymond Gosling, she takes the X-ray Photo 51 of DNA fibers. Its characteristic cross betrays a helix.
1953James Watson and Francis Crick, who had seen this photo and Franklin's data without her knowledge, announce the double-helix model. Franklin died in 1958; the 1962 Nobel Prize went to Watson, Crick and Maurice Wilkins — the prize is not awarded posthumously.

See how the DNA molecule is built: the Helix lab.

What a diffraction pattern shows

Each spot is one family of parallel planes of atoms. The more closely the planes are spaced, the farther the spot is from the center — the diffraction pattern is the “reciprocal lattice”.

A square lattice gives a square pattern of spots, and a hexagonal one — a hexagonal pattern.
Rotating the sample rotates the pattern by the same angle.
Missing atoms and thermal vibrations don't move the spots; they only weaken them and add background noise.
Glass has no long-range order: instead of spots you see blurry rings.
A quasicrystal gives sharp spots with 10-fold symmetry, which no periodic crystal can have.

The model is two-dimensional: atoms are blurry dots, and the “beam” has soft, round edges. A real diffractometer measures the intensity of X-rays reflected from a three-dimensional crystal.

Crystals around us

Crystallography isn't just for mineral museums.

Jan Czochralski and the silicon in processors

Polish invention · 1916

According to a well-known anecdote, in 1916 in Berlin Jan Czochralski (born in 1885 in Kcynia, Poland) dipped his pen into a crucible of molten tin instead of the inkwell. He pulled out a thin thread of metal — a single crystal. He described the method in 1918.

Silicon single crystals are grown the same way today: a seed touches molten silicon and very slowly, rotating, moves upward. The result is a cylinder up to 30 cm across, later sliced into wafers from which processors are made. The vast majority of silicon for electronics is made with the Czochralski method.

After the war Czochralski was wrongly accused of collaborating with the occupiers and removed from his post at the Warsaw University of Technology. The university cleared his name only in 2011.

Quartz in a watch

piezoelectricity 32,768 Hz

Squeeze a quartz crystal and it produces a voltage, and a voltage deforms it. A tiny quartz tuning fork vibrates 32,768 times per second, i.e. 2¹⁵ — divide by two 15 times and you get exact seconds.

Look inside a watch →

Liquid crystals

LCD screens

A liquid whose elongated molecules line up in one direction like matches in a box. A voltage turns them and changes the polarization of light, and polarizing filters turn that into a bright or dark pixel. They were discovered by the botanist Friedrich Reinitzer in 1888.

The salt mine in Wieliczka

halite · cubic

Rock salt has been mined here since the 13th century. Chambers, chapels and sculptures were carved out of salt; the most famous is the Chapel of St. Kinga. The mine was one of the first 12 sites inscribed on the UNESCO World Heritage List in 1978.

Cave of the Crystals, Naica

gypsum · Mexico ≈ 11–12 m

In 2000 miners at the Naica mine broke into a cave full of gypsum crystals as long as a bus. They grew for hundreds of thousands of years in hot water that was only ever slightly supersaturated — ideal conditions for growing giants. Inside it is about 50 °C at nearly 100% humidity.

The Mohs hardness scale

scratch test · 1812

Friedrich Mohs ordered minerals so that each one scratches the ones before it. Check what scratches what:

Scratch with