Matter
A crystal is a repeating arrangement of atoms, and the smallest block that repeats is often a cube. The cube cut into six pyramids fits that block naturally: its six dots are the six neighbours of an atom at the centre, and √2 − 1 is the size limit chemists found for an atom to be surrounded by six. This page follows the number into six-fold coordination, perovskites and the classic cubic crystals, and then into a framework from materials chemistry that has the same topology as the dots’ contact graph.
1Atoms, ions and the unit cell
The smallest block that repeats under translation is the unit cell. For many crystals it is a cube, so a cube with special points in it is a natural description. 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 of one charge sits among big ions of the opposite charge, and the question of how many big ones can touch the small one is mostly a question of geometry.
2The radius-ratio rule
Put six equal balls around a centre, one on each of the three axes in both directions, so that each ball touches its four neighbours. They leave a small hole in the middle. A slice through the centre shows it: four balls of radius 1 with centres at (±√2, 0) and (0, ±√2), each pair of neighbours 2 apart. The centre is √2 from each ball centre, so the hole has radius √2 − 1. A small ion fits touching all six only if its radius is at least √2 − 1 = 0.4142 times the big one’s.PROVEN
Chemists met this number in the 1920s and 30s. Pauling’s radius-ratio rule says that if the small ion is at least 0.414 times the size of the big ones, it can be surrounded by six of them, which is octahedral coordination: six neighbours at the corners of an octahedron (an eight-faced shape, like two pyramids glued at the base). If the small ion is smaller than that, it rattles in the hole and the crystal usually picks four neighbours instead (tetrahedral coordination).KNOWN Slide the small ball below.
3Three limits from three solids
The kaleidoscope page shows how one rule produces three limits. Cut a regular solid from its centre into pyramids, one per face, and put a dot equidistant from every face of each pyramid. Neighbouring dots are mirror images across the shared wall, so neighbouring balls touch, and the hole left among them is exactly the radius-ratio limit for that arrangement:
| solid cut from its centre | dots form | hole / ball | chemistry’s limit |
|---|---|---|---|
| tetrahedron (4 pieces) | tetrahedron | 0.2247 | 0.225: four neighbours (zinc blende) |
| cube (6 pieces) | octahedron | 0.4142 | 0.414: six neighbours (rock salt) |
| octahedron (8 pieces) | cube | 0.7321 | 0.732: eight neighbours (caesium chloride) |
The thresholds are defined as the hole left when the surrounding balls touch, and the dot rule produces that touching arrangement every time, so the thresholds are facts about symmetry. Chemistry decides which arrangement real ions prefer; the geometry decides where the cut-offs fall. This explains the numbers. It does not make the rule predict better: real ions are not hard balls, and many crystals break the rule.KNOWN (the rule and its thresholds); PROVEN (that the numbers are √6/2 − 1, √2 − 1, √3 − 1, recomputed for this page)
4Octahedral coordination
One centre, six neighbours straight along the axes: that is exactly the six dots of the cube, and it appears all over chemistry. The gas SF6 has a sulfur atom with six fluorines. Many metal complexes are a metal ion with six attached groups. The TiO6 unit in oxide crystals is a titanium atom with six oxygens.
5Perovskite, and the classic cubic crystals
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: one A, one B and three O, written ABO3. This is the perovskite structure: CaTiO3 (the mineral itself), SrTiO3, and the materials in the newer perovskite solar cells. In the drawing, each pyramid is one Ti–O direction with four A atoms around its base.KNOWN
The same cube describes the classic cubic crystals by changing which sites are filled. 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. Corners plus the 6 face centres is face-centred cubic (copper, aluminium, gold). Notice that the X on each face is the original drawing again. Corners, centre, face centres and edge midpoints, with two kinds of atom alternating, is rock salt (NaCl, table salt). Pick a structure below. The counter weights each site by the share of it the cell owns: a corner 1/8, an edge 1/4, a face 1/2, the centre 1.KNOWN
6Dot pairs: a resemblance, not a match
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.RESEMBLANCE ONLY Nothing here claims the dumbbell structure follows from the dots.
7A known framework with the dots’ topology
Treat the dots as a mechanism. Each cube’s six dots form an octahedron whose edges all have length 2R. An octahedron with its triangulated faces is rigid: it cannot change shape without stretching an edge. Neighbouring cubes are joined by one straight bar through the shared face, from a dot to its partner. So the structure is a stack of rigid octahedra joined corner to corner by straight bridges, which is the topology of Prussian blue (octahedra joined by linear bridges). Goodwin (Physical Review B 74, 134302, 2006) found that frameworks of this kind have an unusually large number of rigid unit modes, motions where the octahedra tilt without being distorted, a number that grows in proportion to the volume (an O(V) density), which is the signature of flat bands.KNOWN
In the counts of the mechanics report, the same structure has exactly 3 zero modes per cell in 3D at a generic wavevector, one for each independent rotation of an octahedron, and no state of self-stress.PROVEN (n ≥ 3) VERIFIED NUMERICALLY (n = 2 to 8) The next page uses those rotations to fold the whole lattice.
8Further chemistry: what the rule does in real crystals
Everything above uses the rule on one cube. A separate survey (research/2-chemistry.md, code in research/chemistry/) tiled the rule across space, tried it on other solids, and compared it with crystal data. Labels are as before.
8.1 Tiled, the drawing is calcium hexaboride
Put the cube’s corners on a cubic lattice and apply the rule in every cube. The dots are the boron atoms of CaB6 and the corners are the calcium atoms. Take the cell edge as a and put calcium at the origin. Boron is at (x, ½, ½) in the cell. The distance from a boron atom to the centre of its octahedron is ½ − x, so the six borons of one octahedron are √2(½ − x)a from each other, while the two borons facing each other across a cell face are 2xa apart. These are equal exactly when
x = (√2 − 1)/2 = 0.20711which is the position the rule gives (the dot is (2 − √2)/2 = 0.2929a from the centre). “Neighbouring balls touch” here means “all five boron–boron bonds are equal”.PROVEN VERIFIED NUMERICALLY The cell edge is then a = (1 + √2) times the boron–boron distance d. In real MB6 crystals the mean of a/d is 2.399, against 2.414 for the ideal. Each metal atom sits in a cage of 24 borons shaped like a truncated cube, at a distance 1.7788d from each (1.765d in real crystals), and the cage’s hole-to-ball ratio is 2.5576, the radius ratio of the truncated cube.VERIFIED NUMERICALLY (45 crystal structures from the Crystallography Open Database)
This is Pauling’s own model. Pauling and Weinbaum (Zeitschrift für Kristallographie 87, 181, 1934) put boron at 0.293(1) from the octahedron centre, which is x = 0.207. In a 1955 letter Pauling wrote that the boron–boron distances between octahedra are identical, to within 0.01 Å, to those inside an octahedron (Oregon State University Pauling archive). The vertex net of the truncated cubic honeycomb is the RCSR net cab, named for the CaB6 boron net.KNOWN The link to the drawing is our framing, not a discovery. Try it in the viewer above: choose CaB6 and turn on the six dots.
What do real crystals do? Of 45 database entries, 14 of the 31 MB6 entries carry x = 0.207 exactly (Ba, Ca, Ce, Er, Gd, Nd, Pr, Pu, Sr, Th, Y, Yb). They date from 1934 to 1954 or are handbook re-entries, so they are the equal-bond assumption copied forward, not measurements. The 17 modern refinements (Ca, Ce, Eu, La, Sm, Th) give x = 0.197 to 0.204, and the ratio of the bond between octahedra to the bond inside one averages 0.947 over those 17 entries (x ≠ 0.207), with a range of 0.920 to 0.977. (Averaged over all 31 entries, including the copied-forward ones, it is 0.970.) In modern refinements the bond between octahedra is about 5% shorter than the touching ideal.VERIFIED NUMERICALLY That fits the known bonding picture, in which each boron has one two-centre bond outward and shares multicentre bonds inside the cage (Longuet-Higgins and Roberts, 1954).KNOWN The ideal is a geometric zero, and the deviation from it measures the asymmetry of the bonding.
The perovskite reading is a framing. The calcium is at the A site and the empty octahedron centre is at the B site. The touching pair that straddles each face centre is the boron–boron bond between octahedra, whose midpoint lies exactly on the perovskite O site. No source was found that says this in words.FRAMING, NOT A RESULT A negative result: real perovskite oxygens move sideways (Glazer tilts), and the breathing distortions of RNiO3 and BaBiO3 move oxygen along the B–O–B axis as a whole without splitting it into a touching pair. No oxide in this check realised the split-oxygen picture. Hexaborides do.
8.2 The same rule on the fcc cell gives UB12
Replace the cube by the Voronoi cell of the face-centred cubic lattice, the rhombic dodecahedron: the region closer to one lattice point than to any other. Every face is equivalent and every wall is a mirror, and it has 12 rhombic pyramids. The dots form a cuboctahedron of 12 borons in each cell, and each dot touches 4 + 1 = 5 others, as in CaB6. Boron is at (½, x, x) with x = 1/6 and a = 3√2 d. This is the dodecaboride UB12 structure (space group Fm-3m, 52 atoms per cell). The metal sits in a cage of 24 borons shaped like a truncated octahedron, at √(5/2)d = 1.5811d, with hole-to-ball ratio 2.1623, the truncated octahedron’s radius ratio. Real crystals (Sc, U, Y, Yb, Zr) give 1.567, and a bond ratio of 0.956 on average over 14 entries, with a range of 0.931 to 1.012. That is a shortening of about 4% on average, not a uniform 5%: uranium (COD 1008115) has 1.012, a slight lengthening, and the other four metals lie between 0.931 and 0.967.VERIFIED NUMERICALLY
The body-centred cell (a truncated octahedron) fails: it has two kinds of pyramid, with radii 0.500 and 0.634, and no contacts across its square and hexagon walls.VERIFIED NUMERICALLY The UB12 structure itself is known (AFLOW prototype A12B_cF52_225_h_b; a dodecaboride review, arXiv 2010.16239).KNOWN That the equal-bond x is 1/6, and that both boride families come from one rule (“insphere of each pyramid of a Voronoi cell”), was not found in the literature after searches for augmented nets with borides and for Voronoi or insphere constructions. RCSR’s raw data file (rcsr.net/data/3dall.txt) can be reached, and its ubt entry (UB12, Fm-3m, boron at (1/6, 1/6, 0), five-coordinated) and its cab entry list these equal-edge geometries, so x = 1/6 is there implicitly.NO REFERENCE FOUND (web searches only)
That MB12 forms with smaller metals (Zr, Lu, Y, U) and MB6 with larger ones (Ca, Sr, Ba, La, Eu) is a known trend, and the cage sizes point the same way (M–B about 3.07 Å for MB6 and 2.79 Å for MB12). As a predictor this is qualitative only: no classifier was built.CONJECTURE
8.3 Hard-sphere packing peaks at the dots’ geometry
Treat calcium as a large sphere and boron as small ones, and maximise the packing fraction over the position x and the cell size for each size ratio γ (small radius over large radius). The CaB6 arrangement peaks at γ = 0.4142, x = 0.2071. There the large spheres form a touching simple cubic array and each small sphere touches five other small spheres. Calcium and boron do not touch: their centres are 0.7368a apart against a sum of radii of 0.7071a, a gap of 0.0297a, so calcium is held by its calcium neighbours alone. The fraction is
φ = (4π/3)[1/8 + 6((√2−1)/2)³] = (π/6)(30√2 − 41) = 0.746865(recomputed for this page). That is above fcc (0.7405) and below rock salt at the same γ (0.793), so hard spheres alone do not favour it.VERIFIED NUMERICALLY (the closed-form values) The upper bound and attainment are proved.PROVEN (attainment: exact all-pairs classification, internal review) This fits the description of CaB6-type nanocrystal superlattices as low-density phases held by kinetics or ligand shells (Shevchenko and others, Nature 439, 55, 2006).KNOWN The suggestion that observed superlattices cluster near an effective γ of 0.41 is a guess.CONJECTURE The UB12 optimum is not at its dot point (γ = 0.4625 gives φ = 0.606), so the coincidence is special to the cube.VERIFIED NUMERICALLY The packing curve for this lattice was plotted by Horst and Travesset (2016), so the peak can be read off it, and Travesset (2017) may give its location (not read). No explicit statement of the closed form, or of its identity with the octahedral limit √2 − 1, was found.PARTIALLY KNOWN
Attainment proved. PR #23 gives an exact all-pairs argument for the optimum described above, replacing the finite attainment check. Independent human review is pending. This two-size packing problem is distinct from the equal-ball continuous fcc motion.
8.4 Which radius-ratio limits the rule can reach
Claim. Cut a solid P from its centre into pyramids, one per facet, and put each dot at the point equidistant from the walls of its pyramid, with radius R equal to that distance. Suppose P is facet-transitive (all facets alike) and every wall (the plane through the centre and a ridge) is a mirror of P. Then (i) all dots are equivalent, at one distance t; (ii) a ball and its mirror image touch at the wall, so every pair of dots on adjacent pyramids is 2R apart, and the dots form an equal-edged polytope Q whose vertices are all alike, dual to P; (iii) the hole-to-ball ratio is (t − R)/R = t/(e/2) − 1 with e the edge of Q, which is the classical radius-ratio limit of Q; (iv) if P instead has an insphere and equal dihedral angles at every ridge (call this H2), the ratio is also 1/sin θ − 1, where 2θ is the angle between adjacent facet normals. Clause (iv) needs H2: mirror walls alone do not give it. The triangular bipyramid has mirror walls, and its true ratio is 0.5275 at every height, but 1/sin θ − 1 gives 1.236 and 0.291 at height 1 (its two kinds of adjacent normals) and 0.633 and 0.414 at height 0.5. The base facet never enters the ratio. It sets the scale, and it gives the contacts across faces when cubes are tiled. In the language of the mirrors page, the dot is a Wythoff point, and the walls being mirrors is what makes the balls touch.PROVEN (clauses (i) to (iii) under mirror walls or H2; clause (iv) under H2 only)
Checked by linear-programming insphere and brute-force contact counts:VERIFIED NUMERICALLY
| P, cut into pyramids | hole / ball | coordination shape, classical limit |
|---|---|---|
| triangle (2D) | 0.1547 | triangle, 3 neighbours: 0.155 |
| square (2D) | 0.4142 | square, 4: 0.414 |
| tetrahedron | 0.2247 | tetrahedron, 4 |
| cube | 0.4142 | octahedron, 6 |
| octahedron | 0.7321 | cube, 8 |
| triangular bipyramid, any height | 0.5275 | trigonal prism, 6 (self-tuning) |
| dodecahedron | 0.9021 | icosahedron, 12 |
| rhombic dodecahedron | 1.0000 | cuboctahedron, 12 (the 3D kissing arrangement) |
| tesseract (4D) | 0.4142 | 16-cell |
| 24-cell (4D) | 1.0000 | 24-cell (the 4D kissing arrangement) |
The survey also lists the n-gonal bipyramids, the icosahedron, the Catalan solids and the 4D simplex and cross-polytope; see the report for those rows. A bipyramid of any height gives the uniform prism, because mirror walls force all edges to 2R: the rule finds the equal-edge shape without being told. It fails, with unequal radii, on solids that are not facet-transitive, such as the cuboctahedron, the truncated octahedron and prisms.
Where it cannot reach: the square antiprism (8 neighbours, 0.645) is snub-type, its walls are not mirrors, and touching occurs only for one exact shape; half the contacts open otherwise.PROVEN The anion shapes for 7 neighbours, the tricapped trigonal prism (9, 0.732), the dodecahedral 8-neighbour shape (0.668) and the capped octahedron (0.592) are not vertex-transitive, and the rule’s dots always form one orbit, so it cannot produce them.PROVEN The radius ratios themselves are known (Pauling 1929).KNOWN The statement that the limits reachable this way are exactly those of the mirror-generated equal-edge polytopes is one for which no prior reference was found after searching radius ratio with Wythoff, kaleidoscope and uniform polyhedra. It is elementary and may well appear in a textbook.NO REFERENCE FOUND (web searches only)
8.5 Perovskites: a negative result
The survey used the 576 ABX3 compounds (313 perovskites) of Bartel and others (Science Advances 5, eaav0693, 2019). The octahedral cut μ = rB/rX ≥ √2 − 1 alone classifies 0.752 of them correctly. The best fitted single cut is 0.426 (0.769), within 3% of √2 − 1. The Goldschmidt tolerance window alone gives 0.736, and the two together 0.835. Bartel’s own factor τ gives 0.915.VERIFIED NUMERICALLY Oxides tolerate μ down to 0.29, whereas no halide perovskite is found below 0.43, matching an empirical halide bound near 0.44 (Li and others, Acta Crystallographica B 64, 702, 2008).KNOWN In the ideal cubic perovskite (tolerance factor 1), μ = √2 − 1 holds exactly when rA = rX: the B-site octahedral limit and the A-site cuboctahedral limit (ratio 1) coincide only there. That is the geometric content of Goldschmidt’s √2 and a restatement, not a new predictor.PROVEN The dot geometry derives the 0.414 bound but does not beat τ, and nothing more is claimed.
More broadly, George and others (Angewandte Chemie 59, 7569, 2020) found that Pauling’s first rule holds for about 66% of some 5000 oxide environments, and only 13% satisfy rules 2 to 5 together.KNOWN The kaleidoscope view explains one structural reason for failure: limits exist only for equal-edged, vertex-transitive shapes, while real environments are distorted. It does not explain the softness and size failures, which dominate. That the unreachable shapes are over-represented among the violations was not tested.CONJECTURE
8.6 What was not done, and the open question
RCSR’s raw data file was reached only late, to check the cab and ubt geometries; other net symbols come from OEIS A299266 and notes from the Yaghi group. No density functional calculation was run: the 4 to 5% shortening comes from crystallographic database entries only. Glazer tilts as rigid unit modes were not explored.
The open question: is “Wythoff points in Voronoi cells give equal-bond cluster frameworks” a generating rule for boron-rich and other cluster solids? The cube gives CaB6 and the rhombic dodecahedron gives UB12, both about 5% off the ideal in the same direction. Next steps would be to list the augmented nets from every face-transitive, mirror-walled cell as candidate frameworks; to explain the dodecahedron case, where the icosahedral B12 of α-boron would be the frustrated version that cannot tile; and to test with density functional theory whether the measured bond ratio (0.92 to 0.98) follows the metal’s electron count.