Take a square cut by its diagonals and mark the incentre of each of the four triangles. The incentre is the centre of the inscribed circle, equally far from all three sides. This long read follows its one number, √2 − 1 ≈ 0.414, into chemistry, into tilings, and into every dimension.
Quick recap. Take a cube of side 2 with its centre at the origin. Cut it from the centre into 6 square pyramids, one on each face. In each pyramid, put a ball inside that touches all five of its faces. That is the biggest ball that fits, called the insphere. Its radius is R = √2 − 1 ≈ 0.414 and its centre, the dot, is k = 2 − √2 ≈ 0.586 from the cube’s centre. The hands-on cube shows this in 3D.
A crystal is atoms stacked in a repeating pattern. The smallest block that repeats is the unit cell, and for many crystals it is a cube. So a cube with special points inside it is a natural way to describe one.
Put six equal balls around a centre, one on each of the three axes, in both directions, so each ball touches its four neighbours. They leave a small hole in the middle. How big is that hole, compared with a ball? Exactly √2 − 1. (With balls of radius √2 − 1, as in the cube, the hole has radius 3 − 2√2.)
Chemists met this number in the 1920s and 30s. In a salt-like crystal the atoms are really ions, atoms that have gained or lost electrons and so carry a charge. A small ion sits among big ions of the opposite charge. Pauling's radius-ratio rule says: if the small ion is at least 0.414 times the size of the big ones, it can be surrounded by six of them. This is octahedral coordination, meaning "six neighbours at the corners of an octahedron" (an eight-faced shape, like two pyramids glued at the base). Rock salt is the classic case. If the small ion is smaller than 0.414, it rattles in the hole, and the crystal usually picks 4 neighbours instead (tetrahedral coordination).
Try it. The big balls touch each other. Slide the small ball's size.
Figure 1. Six equal balls on the three axes, touching. The small red ball sits in the hole between them. The dashed ring marks the biggest ball that fits.
One centre, six neighbours straight along the axes. That is exactly the six dots of your cube, and it shows up all over chemistry: the gas SF₆ (sulfur with six fluorines), many metal complexes (a metal ion with six attached groups), and the TiO₆ unit in oxide crystals (a titanium atom with six oxygens).
Put a titanium atom (B) at the cube's centre, an oxygen atom (O) at the middle of each of the 6 faces, and a calcium or strontium atom (A) at each of the 8 corners. An atom on a face is shared by 2 cubes, so each cube owns half of each: 6 ÷ 2 = 3 oxygens. A corner is shared by 8 cubes, so 8 ÷ 8 = 1 A atom. Count them up: A, B and three O, written ABO₃. This is the perovskite structure: CaTiO₃ (the mineral itself), SrTiO₃, and the materials in the new perovskite solar cells.
In your cube, each pyramid is one Ti→O direction, with 4 A atoms around its base.
Corners plus the centre is body-centred cubic: iron at room temperature. If the centre atom differs from the corner atoms it is caesium chloride. The pyramid edges, from the centre to the corners, are the nearest-neighbour bonds. Corners plus the 6 face centres is face-centred cubic: copper, aluminium and gold. Notice that each face, with its diagonals, is the original drawing again. All corners, centres, face centres and edge midpoints, with two kinds of atom alternating, is rock salt (NaCl, table salt).
A symmetry is a turn or a mirror flip that leaves the shape looking unchanged. The cube has 48 (24 turns, 24 more when mirror flips count). All the crystals above share this full set, called Oh. The 48 mirror halves from the cube page are the smallest repeating piece under it, and a general point in such a crystal has 48 copies.
Stack cubes face to face. Across each shared face the two dots touch, exactly 2R apart. They form a pair centred on the face centre and pointing straight through the face. That looks like the split dumbbell in copper-type metals: a spot where two atoms share one site, pushed apart along a cube axis (written ⟨100⟩). But here only the face-centre sites split, so this is only a resemblance, not the same thing.
Use the viewer to see all of this. Choose a crystal, drag to rotate.
Figure 2. One cube cell, shown as your dots or as four classic crystals. The faint lines are the cube cut into six pyramids.
Repeat the square drawing in a grid. Now look at a spot where four squares meet. The eight nearby dots (two from each square) are all the same distance, 1.0824, from that corner, and they are joined by edges all the same length, 0.8284 (that is 2R). So they form a regular octagon: eight equal sides and eight equal angles.
Inside each square, the four dots make a small square. Squares and regular octagons together, every edge equal, is the truncated square tiling. Its name in the catalogue of Archimedean tilings (tilings by regular shapes where every corner looks the same) is 4.8.8: at each corner you meet one square and two octagons.
This is no accident. It works exactly because √2 − 1 = tan 22.5°, and 22.5° is half of 45°, the angle between neighbouring corners of a regular octagon seen from its centre. Circles of radius R at the dots touch 3 neighbours each and cover about 53.9% of the plane.
Figure 3. The drawing repeated in a grid. The dots of four squares form a regular octagon around each shared corner, and the four dots of one square form a small square.
Now stack cubes in all three directions. Inside each cell the 6 dots make an octahedron. Around every cube corner, the 24 dots from the 8 cells meeting there form a truncated cube: a cube with its 8 corners sliced off. It has 24 corners and 36 equal edges, 3 at each corner. Measured from the corner, the 24 points are all the ways of writing (±(√2−1), ±1, ±1) with the numbers in any order. That is the standard recipe for a truncated cube.
Octahedra and truncated cubes fill space with no gaps. This is a uniform honeycomb (a space-filling stack of regular-faced shapes), called the truncated cubic honeycomb, and every edge has length 2R. Each ball touches 5 others: 4 inside its own cube and 1 through the face. The balls fill about 22.3% of space. The biggest ball that fits in the cavity at a cube corner has radius about 2.56 R.
Figure 4. A block of 2 × 2 × 2 cells. The six dots of each cell make an octahedron. The 24 dots around the middle corner make a truncated cube.
We can do the same cut in 4, 5 or any number of dimensions. An n-dimensional cube cut from its centre gives 2n pyramids. Take a ball inside one pyramid, with its centre at distance c from the cube's centre along the axis. The pyramid's base is at distance 1 − c from it. Its slanted faces are at distance c/√2. The ball touches all of them when these are equal, which gives c = 2 − √2.
That does not depend on n. So in every dimension the ball has radius √2 − 1, sits at distance 2 − √2 from the centre, and neighbouring balls touch. The hole in the middle keeps the ratio √2 − 1. Stacked cubes give the vertices of the uniform truncated n-cubic honeycomb (the 2D and 3D coordinates were checked directly; the general-n form is the standard one, not checked here). Each ball touches 2n − 1 others: 3 in 2D, 5 in 3D, 7 in 4D.
On the cube page, 24 small pieces matched the cube's 24 rotations. Is that deep, or luck? Count in n dimensions. A piece is a dot plus one slanted face. There are 2n pyramids with 2(n − 1) slanted faces each, so 4n(n − 1) pieces. The cube has 2n·n! symmetries with mirrors, and half of those, 2n−1·n!, are turns.
Table 1. Pieces against symmetries, by dimension. Try other dimensions with the slider.
| dimension n | pieces | all symmetries | rotations only | balls touched |
|---|---|---|---|---|
| 2 | 8 | 8 | 4 | 3 |
| 3 | 24 | 48 | 24 | 5 |
| 4 | 48 | 384 | 192 | 7 |
| 5 | 80 | 3840 | 1920 | 9 |
Only in 2D do the pieces match all the symmetries (8 and 8). Only in 3D do they match the rotations (24 and 24). From 4D on, the pieces fall short of both. In general each piece splits into 2n−2·(n − 2)! equal smallest pieces: 1 in 2D, and 2 in 3D, which are the mirror halves. So the 3D match is a happy accident of small numbers. The number √2 − 1 is not an accident, but the 24 = 24 is.
Why one number keeps appearing. Every face of your pyramids lies on a mirror of the cube pattern: the cube faces themselves, and the diagonal planes, which are the diagonals of the flat drawing carried into 3D. Your dot is the point the same distance from all of those mirrors. Reflect it in any one mirror and the copy lands exactly 2 × (√2 − 1) away. So every edge of the pattern comes out the same length. That is what makes the octagons regular and the honeycomb uniform. Distance to a mirror doesn't depend on the dimension, so the number never changes.
This is a known method. It is the Wythoff construction, published by the Dutch mathematician Willem Wythoff in 1918 and developed by H. S. M. Coxeter. It builds every uniform tiling and honeycomb the same way: treat the symmetry planes as a kaleidoscope, and place one point at equal distance from the chosen mirrors. Your dot is that point, and your pyramid is one chamber of the kaleidoscope.
The same rule gives chemistry's whole table, not just 0.414. Cut each of three regular solids from its centre into pyramids, one per face, and put a dot equidistant from every face of each pyramid. These three solids work because every pyramid wall is a mirror of the solid. In general the identity needs mirror walls, or else an insphere plus equal dihedral angles. Without that it can fail. Neighbouring balls always touch, because neighbouring dots are mirror images across the shared wall. The hole left in the middle comes out exactly at chemistry's size limit for that arrangement:
Table 2. The same rule on three solids. Each hole-to-ball ratio is the classical radius-ratio limit for that neighbour shape.
| Shape cut from its centre | Dots form | Hole ÷ ball | Chemistry's limit |
|---|---|---|---|
| Tetrahedron (4 pieces) | 4 balls, a tetrahedron | 0.2247 | 0.225: four neighbours (zinc blende) |
| Cube (6 pieces) | 6 balls, an octahedron | 0.4142 | 0.414: six neighbours (rock salt) |
| Octahedron (8 pieces) | 8 balls, a cube | 0.7321 | 0.732: eight neighbours (caesium chloride) |
The limits are defined as “the hole left when the surrounding balls just touch”, and under those conditions the dot rule produces exactly that touching arrangement. So the thresholds are facts about symmetry, not about chemistry. Chemistry decides which arrangement real ions prefer; the kaleidoscope decides where the cut-offs fall. This explains the numbers, but it doesn't make the rule predict better: real ions aren't hard balls, and many crystals break the rule.
Mostly no. Every individual fact in sections 1 to 4 is known. The radius-ratio rule dates from the 1920s and 30s. The 4.8.8 tiling and the truncated cubic honeycomb are classical uniform tilings. Perovskite is a standard crystal. Section 3 is easy to derive, and anyone who studies these shapes would find it.
What is nice is the route. One rule, “put the dot the same distance from all the edges”, lands exactly on the regular octagon pattern, on the truncated cube pattern, and on chemistry’s 0.414 limit, in every dimension, and for one shared reason: tan 22.5° = √2 − 1.
The method behind the route is Wythoff’s 1918 construction (section 4). It is textbook that each size limit equals the coordination shape’s centre-to-corner distance over half its edge, minus one. Getting all three limits, and the uniform tilings, from one rule about dots equidistant from mirror walls is a tidy way to see it that we have not seen written down. Searches for prior work are summarised on the Ledger. This is a new view of known results, not a new result.
The results listed in the box near the top are different. They are not classical, and no prior description of them was found by web search. They are also not peer reviewed, so the Ledger labels how strong each one is.
See it all at once on the lattice page.