Close-Packing of Spheres

Stack oranges in a crate and you will build one of the two most efficient arrangements of spheres possible in three-dimensional space. You do this without knowing any solid-state chemistry. Grocers have known this for centuries. Crystallographers only worked out why it is optimal in the twentieth century. They also only then discovered how many distinct ways of doing it exist. The same geometric problem governs many structures. That problem is how to pack identical hard spheres to fill as much space as possible. It governs the structures of most metals. It also governs the arrangement of anions in countless ionic solids. It even governs the packing of atoms in noble-gas crystals at low temperature. Close-packing matters because it is the geometric skeleton of inorganic structural chemistry. An enormous fraction of that chemistry is built on it. Once you know a lattice of anions is close-packed, you immediately know several things. You know its coordination geometry and its packing efficiency. Crucially, you also know where the empty spaces are. Those empty spaces are the tetrahedral and octahedral holes. Cations sit in these holes in ionic crystals. Their size and number follow directly from the close-packing geometry. This is why close-packing is the natural gateway between two ideas. One is the purely geometric idea of Ionic Radii and Their Determination. The other is the structural chemistry of real ionic solids. The big picture is deceptively simple. Spheres of one size, packed as tightly as possible, always achieve the same maximum density. That density is 74.05%. This holds regardless of whether the stacking sequence is hexagonal, cubic, or a random mixture of the two. What differs between the arrangements is not how much space is filled. Instead, it is the symmetry of the resulting lattice — hexagonal versus cubic. This difference has real consequences. It affects physical properties, phase transitions, and which cations can be accommodated in the resulting holes. This article builds close-packing from a single sphere layer upward. First, we examine how one layer of spheres packs into a plane. Then we see how a second layer sits in the dimples of the first. We also show how the third layer's position creates a fork in the road. That fork leads to either hexagonal close-packing (hcp) or cubic close-packing (ccp). We derive the coordination number (12) from first geometric principles. We also derive the packing fraction (74.05%) the same way. We work through three fully quantitative examples. Finally, we set up key vocabulary — hcp, ccp, holes. These terms are what the structures of real ionic and metallic solids are built from.

Key takeaways

A single close-packed layer of identical spheres has each sphere touching six neighbours; stacking such layers on top of each other, offset into the hollows, gives two related close-packed structures: hcp (ABAB…) and ccp (ABCABC…).
Both hcp and ccp give each sphere exactly 12 nearest neighbours — the maximum coordination number for identical touching spheres.
Both structures achieve the identical maximum packing fraction of 74.05% ( $\pi/(3\sqrt2)$ ) — this is the densest possible packing of identical spheres.
ccp is identical to the face-centred cubic (fcc) lattice; hcp has its own primitive hexagonal cell with a fixed axial ratio $c/a = \sqrt{8/3} \approx 1.633$ for ideal spheres.
Both structures leave two kinds of interstitial voids per sphere pattern — octahedral holes (1 per sphere) and tetrahedral holes (2 per sphere) — which will host cations in ionic structures.
Key idea: stacking identical close-packed layers in different sequences (ABAB… vs ABCABC…) gives structures with the same density and coordination number but different symmetry, and it is the pattern of gaps between spheres — not the spheres themselves — that determines where the cations of a real ionic crystal will go.

Where this lives today

Close-packing geometry is not a historical curiosity; it is the working vocabulary of modern structural chemistry and materials science. Metallic elements across the periodic table — magnesium, zinc, titanium, cobalt (hcp); copper, silver, gold, aluminium, nickel, platinum (ccp/fcc) — adopt these structures because non-directional metallic bonding favours maximising the number of nearest neighbours, and close-packing is the geometric solution to that problem; the density and mechanical properties (slip planes, ductility) of structural metals like aluminium alloys and titanium aerospace alloys trace directly back to whether the metal is ccp (many slip systems, generally more ductile) or hcp (fewer slip systems, often more brittle at room temperature). In ionic solids, the anion sublattice in structures like rock-salt (NaCl-type, foreshadowed here and treated in depth under Rock Salt (NaCl) Structure), fluorite (CaF₂-type), and the spinel and corundum structures used in ceramics, magnets, and catalysts is very often describable as (approximately) close-packed, with cations distributed among the tetrahedral and octahedral holes. This hole-filling framework, built directly on the close-packing geometry developed here, is how chemists predict and rationalise the stoichiometry and coordination environments of an enormous swath of solid-state inorganic compounds — the subject taken up formally under Tetrahedral and Octahedral Holes. Beyond classical solids, close-packing ideas underpin colloidal crystal engineering (photonic materials built from close-packed arrays of nanometre- to micrometre-scale silica or polymer spheres, exploited for structural colour and optical band-gap materials), the physics of granular materials and random close packing (relevant to powder metallurgy, pharmaceutical tablet compression, and even the geophysics of sediment compaction), and the packing of atoms in noble-gas cryocrystals (solid argon and neon adopt ccp structures at low temperature, since van der Waals forces are essentially non-directional, just like metallic bonding). Even virus capsid geometry and the packing of certain biological macromolecular assemblies borrow close-packing logic, since maximising contacts between identical subunits while minimising enclosed volume is a universal optimisation problem that these Barlow-era geometric arguments solved once and for all.

Ask the tutor: How do the sizes of the tetrahedral and octahedral holes in a close-packed lattice determine which cations can occupy them?

The physical picture

Begin with a single layer of identical spheres resting on a flat surface, each touching six others, arranged so that the layer has no gaps larger than necessary. This is the unique two-dimensional close-packed layer: every sphere sits at the centre of a hexagon of six neighbours, and the layer has hexagonal symmetry. Within this layer, spheres cover a fraction

\eta_{2D} = \frac{\pi}{2\sqrt3} \approx 0.9069

of the plane — you cannot pack circles of equal radius into a plane more efficiently than this (a result formally proven by Thue, 1890, and rigorously by Fejes Tóth in the 20th century). Now look at the hollows on top of this layer. There are two sets of triangular dimples, pointing in opposite orientations — call them the B-type and C-type hollows relative to the original A layer. A second layer of spheres can drop into either set of hollows (it cannot occupy both, since B and C hollows are too close together for two full layers of spheres). Suppose the second layer occupies the B-hollows: we now have an AB stack. The critical geometric decision arises when we place the third layer. Looking down through the AB stack, you find that some of the original A-layer hollows are now directly exposed again (the C-hollows), while others sit directly above A-layer spheres. There are exactly two choices:

Place the third layer directly above the first (A again): the sequence is ABABAB…, giving hexagonal close-packing (hcp).
Place the third layer over the remaining set of hollows, not used by either A or B (a new position, C): the sequence is ABCABC…, giving cubic close-packing (ccp).

Both choices are locally identical — each sphere in either structure has 12 nearest neighbours arranged identically in its immediate vicinity (6 in its own layer, 3 above, 3 below). The stacking sequence only affects the arrangement of second-nearest and further neighbours, which is why the two structures have the same density but different overall symmetry: hcp is described by a hexagonal unit cell, ccp by a face-centred cubic (fcc) unit cell.

Figure

A single close-packed layer: each sphere touches six neighbours, with triangular B- and C-type hollows marked

Figure

Stacking sequences: hcp (ABAB…) vs ccp (ABCABC…), each giving 12-coordination

Every sphere in either lattice touches 12 others: six in its own layer, three in the layer above nestled in alternating hollows, and three in the layer below. This is the maximum coordination number achievable for identical touching spheres and is a hallmark of close-packing — contrast it with the simple cubic lattice (coordination number 6) or body-centred cubic (coordination number 8), neither of which is close-packed.

The core ideas

The two defining numbers of close-packing are the coordination number and the packing fraction (packing efficiency). Both follow from pure geometry, independent of what the spheres represent (atoms, ions, or macroscopic oranges).

CN_{\text{close-packed}} = 12

where: $CN$ is the number of nearest-neighbour spheres in direct contact with any given sphere — 6 in-plane, 3 above, 3 below, identical for hcp and ccp. Reading: twelve is not a coincidence of these two structures; it is the geometric ceiling. No arrangement of identical, non-overlapping spheres in three dimensions can exceed 12 nearest neighbours while keeping all twelve in simultaneous contact with the central sphere (the "kissing number" problem, resolved by Newton and Gregory's 1694 argument and proven rigorously much later).

\eta = \frac{V_{\text{spheres}}}{V_{\text{cell}}} = \frac{\pi}{3\sqrt2} \approx 0.7405

where: $\eta$ is the packing fraction, $V_{\text{spheres}}$ is the total volume of spheres in one repeat unit, and $V_{\text{cell}}$ is the volume of that repeat unit (unit cell). Reading: 74.05% of space is occupied by spheres and 25.95% is void — this void is not scattered randomly but organised into two specific, differently-shaped and differently-sized holes (tetrahedral and octahedral), whose geometry is fixed by the packing and will determine where cations can be inserted in real ionic lattices.

\frac{c}{a} = \sqrt{\frac{8}{3}} \approx 1.6330 \quad (\text{ideal hcp axial ratio})

where: $c$ is the height of the hexagonal unit cell (twice the interlayer spacing) and $a$ is the hexagonal in-plane lattice parameter, equal to the sphere diameter $2r$ . Reading: real hcp metals deviate slightly from 1.633 (e.g. Mg: 1.624, Zn: 1.856) because real atoms are not perfectly hard, undirectional spheres; the ideal ratio is the geometric benchmark against which real structures are compared.

Derivation ACoordination number = 12, from layer geometry

Goal. Show that any sphere in a close-packed arrangement (hcp or ccp) touches exactly 12 neighbours, by direct counting layer by layer.

Step 1 — Neighbours within the same layer. In the 2D close-packed layer, every sphere sits at the centre of a regular hexagon formed by six neighbouring spheres, each at distance $2r$ (touching). This is forced by the hexagonal symmetry of the single-layer close packing: three-fold rotational symmetry about each hollow, combined with the requirement that spheres touch their nearest neighbours, generates exactly six equidistant contacts per sphere. Any attempt to add a seventh in-plane neighbour at the same distance $2r$ would require an angular spacing under 60°, which is geometrically impossible for coplanar circles of radius $r$ centred at distance $2r$ from a common point.

Step 2 — Neighbours in the layer above. Consider a sphere in layer A. The layer above (B, in hcp) sits in the hollows directly above alternating triangular gaps of the A layer. Each A-sphere is adjacent to six such triangular hollows around it, but only three of these six hollows are occupied by B-spheres (the layer chooses one hollow orientation consistently); the B-spheres occupying these three hollows are each in contact with the central A-sphere, since a sphere dropped into a triangular dimple of three touching spheres touches all three of them. This gives exactly 3 contacts from the layer above.

Step 3 — Neighbours in the layer below. By the mirror-symmetric argument, the layer below (also occupying hollows of the A-layer, on the opposite face) contributes another 3 contacts.

Step 4 — Total. Summing:

CN = 6\ (\text{same layer}) + 3\ (\text{above}) + 3\ (\text{below}) = 12

This count is identical whether the third layer above continues the ABAB (hcp) or ABCA (ccp) sequence, because the local environment — one layer above, one below, both offset into hollows — is the same in both cases. The stacking choice only affects the identity/position of the second layer above and below, which lie beyond the first coordination shell.

Limiting case check. If instead the layers were stacked directly on top of one another with no lateral offset (simple layers stacked as in simple cubic or simple hexagonal, not close-packed), each sphere would gain only 1 contact above and 1 below, giving $CN = 6$ , matching the known coordination number of simple cubic packing — confirming that the "offset into hollows" step is what produces the extra 6 contacts characteristic of close-packing.

Go deeper · Derivation B · 1 minPacking fraction = 74.05%, using the fcc (ccp) unit cell

Derivation BPacking fraction = 74.05%, using the fcc (ccp) unit cell

Goal. Compute the fraction of space occupied by identical touching spheres in the ccp (face-centred cubic) unit cell, and show it equals $\pi/(3\sqrt2)$ .

Step 1 — Geometry of the fcc cell. The fcc unit cell is a cube of edge length $a$ with spheres at each of the 8 corners and at the centre of each of the 6 faces. Spheres touch along the face diagonal (not the cell edge), because along the edge there is a gap, but along the face diagonal, corner–face-centre–corner spheres are mutually tangent.

Step 2 — Relate $a$ to sphere radius $r$ . The face diagonal has length $a\sqrt2$ and spans two sphere radii plus a full sphere diameter (corner sphere, radius $r$ ; face-centre sphere, diameter $2r$ ; opposite corner sphere, radius $r$ ):

a\sqrt2 = 4r

a = \frac{4r}{\sqrt2} = 2\sqrt2\, r

Step 3 — Count effective spheres per unit cell. Each of the 8 corner spheres is shared among 8 adjacent cells, contributing $1/8$ each; each of the 6 face-centred spheres is shared between 2 cells, contributing $1/2$ each:

n_{\text{eff}} = 8\times\frac18 + 6\times\frac12 = 1 + 3 = 4

So the ccp/fcc unit cell contains 4 complete spheres' worth of volume.

Step 4 — Volume of spheres versus cell. The total sphere volume in the cell is

V_{\text{spheres}} = 4\times\frac{4}{3}\pi r^3 = \frac{16}{3}\pi r^3

The cell volume is

V_{\text{cell}} = a^3 = (2\sqrt2\,r)^3 = 16\sqrt2\, r^3

Step 5 — Packing fraction.

\eta = \frac{V_{\text{spheres}}}{V_{\text{cell}}} = \frac{\frac{16}{3}\pi r^3}{16\sqrt2\, r^3} = \frac{\pi}{3\sqrt2}

Numerically,

\eta = \frac{3.14159}{3\times1.41421} = \frac{3.14159}{4.24264} = 0.74048

so $\eta \approx 74.05\%$ .

Limiting case check. Compare with simple cubic packing, where spheres touch along the cell edge ( $a = 2r$ ) and there is only 1 sphere per cell ( $8\times1/8$ ): $\eta_{sc} = \dfrac{\frac43\pi r^3}{(2r)^3} = \dfrac{\pi}{6} \approx 0.5236$ , i.e. 52.36%. This is markedly lower than 74.05%, confirming that offsetting layers into hollows (close-packing) genuinely increases density relative to naive stacking, exactly as the coordination-number derivation predicted (12 versus 6 neighbours).

Worked examples

Example 1 (Foundational) — Counting atoms and computing density in an hcp unit cell of magnesium. Magnesium crystallises in the hcp structure with metallic radius $r = 160$ pm and axial ratio close to ideal, $c/a = 1.624$ . Take the hexagonal cell (the primitive hexagonal prism containing 2 atoms) and compute the theoretical density. For hcp, the in-plane parameter equals the touching-sphere distance:

a = 2r = 2\times160\ \text{pm} = 320\ \text{pm}

Using the near-ideal ratio,

c = 1.624 \times a = 1.624 \times 320\ \text{pm} = 519.7\ \text{pm}

The hexagonal primitive cell (a rhombic prism with 120° angle, not the full hexagonal prism) contains 2 lattice points/atoms, and its volume is

V_{\text{cell}} = a^2 c \sin(120^\circ) = (320)^2 \times 519.7 \times 0.8660\ \text{pm}^3

V_{\text{cell}} = 102400 \times 519.7 \times 0.8660\ \text{pm}^3 = 4.6082\times10^7\ \text{pm}^3

Converting to cm³ ( $1\ \text{pm} = 10^{-10}\ \text{cm}$ , so $1\ \text{pm}^3 = 10^{-30}\ \text{cm}^3$ ):

V_{\text{cell}} = 4.6082\times10^{7}\times10^{-30}\ \text{cm}^3 = 4.608\times10^{-23}\ \text{cm}^3

Mass of 2 Mg atoms (molar mass 24.31 g/mol, Avogadro's number $6.022\times10^{23}\ \text{mol}^{-1}$ ):

m = \frac{2\times24.31}{6.022\times10^{23}}\ \text{g} = 8.073\times10^{-23}\ \text{g}

Density:

\rho = \frac{m}{V_{\text{cell}}} = \frac{8.073\times10^{-23}}{4.608\times10^{-23}}\ \text{g/cm}^3 = 1.752\ \text{g/cm}^3

This is close to the experimental density of magnesium metal (1.738 g/cm³); the small discrepancy comes from the real axial ratio (1.624, not exactly ideal 1.633) and from metallic bonding effects not captured by hard-sphere geometry. Limiting-case check: if we instead used the ideal axial ratio $c/a = 1.633$ exactly, we would recover the packing fraction 74.05% precisely; the small deviation of real Mg (1.624 vs 1.633) is why real hcp metals are not perfect hard-sphere close-packings but are extremely close to them.

Example 2 (Applied) — Coordination and hole geometry consequence: predicting the stoichiometry accessible in a close-packed oxide lattice. Consider a close-packed array of O²⁻ ions (ionic radius 140 pm) with $N$ oxide ions. Close-packing theory states that such an array generates $N$ octahedral holes and $2N$ tetrahedral holes (this result, foreshadowed here, is derived rigorously when discussing Tetrahedral and Octahedral Holes). Suppose a metal oxide $M_xO_y$ has all its cations in octahedral holes, with a fraction $f$ of the octahedral holes filled. The stoichiometry is fixed by charge balance and hole occupancy:

\frac{n(M)}{n(O)} = \frac{f\times N}{N} = f

For rock-salt-type MgO, all octahedral holes are filled ( $f=1$ ), giving $M:O = 1:1$ , consistent with the known formula MgO. For a hypothetical spinel-like oxide where only half the octahedral holes are filled by a divalent cation while all tetrahedral holes are empty, $f = 1/2$ , giving stoichiometry $M_{0.5}O$ , i.e. $M_2O$ ... To make this concrete: suppose $N = 6.022\times10^{23}$ oxide ions (1 mole) close-packed, generating $6.022\times10^{23}$ octahedral holes and $1.2044\times10^{24}$ tetrahedral holes. If a trivalent cation $M^{3+}$ occupies only two-thirds of the octahedral holes (charge neutrality with O²⁻ requires this, since $2\times3 = 3\times2$ balances when $2/3$ of octahedral sites per O are filled by a 3+ ion — this is exactly the corundum, Al₂O₃, structure motif), the mole ratio is:

n(M) = \frac23\times6.022\times10^{23} = 4.015\times10^{23}\ \text{mol}^{-1}\text{-ions}

\frac{n(M)}{n(O)} = \frac{4.015\times10^{23}}{6.022\times10^{23}} = 0.667 = \frac23

which reproduces the M₂O₃ stoichiometry ( $M:O = 2:3$ ) directly from close-packing hole-filling fractions, without needing any separate empirical input — a first glimpse of how close-packing geometry dictates real oxide formulas. Limiting-case check: if $f\to1$ (all octahedral holes filled, no tetrahedral occupancy), stoichiometry $\to MO$ (1:1), correctly reproducing rock-salt oxides like MgO, CaO, FeO — the extreme opposite of the two-thirds-filled corundum case just computed.

Example 3 (Exam/competition level) — Distinguishing hcp from ccp using packing fraction and density data alone, then confirming via a geometric consistency check. A newly discovered close-packed elemental metal M has measured density $\rho = 8.908\ \text{g/cm}^3$ , molar mass $58.69\ \text{g/mol}$ (this is nickel), and atomic radius (from ionic-radii-style crystallographic data) $r = 124.6$ pm. Determine whether the data are consistent with a close-packed structure, and if the crystal system is reported as cubic, confirm that the density matches an fcc (ccp) cell rather than being spuriously consistent with a non-close-packed cubic cell.

Step A — Compute the fcc cell edge from $r$ .

a = 2\sqrt2\, r = 2\times1.41421\times124.6\ \text{pm} = 352.4\ \text{pm}

Step B — Compute the fcc cell volume.

V_{\text{cell}} = a^3 = (352.4\ \text{pm})^3 = 4.376\times10^{7}\ \text{pm}^3 = 4.376\times10^{-23}\ \text{cm}^3

Step C — Predict density assuming ccp (Z = 4 atoms/cell).

\rho_{\text{calc}} = \frac{Z\times M}{N_A\times V_{\text{cell}}} = \frac{4\times58.69}{6.022\times10^{23}\times4.376\times10^{-23}}\ \text{g/cm}^3

\rho_{\text{calc}} = \frac{234.76}{26.35}\ \text{g/cm}^3 = 8.909\ \text{g/cm}^3

This matches the measured 8.908 g/cm³ to within experimental precision (nickel is indeed ccp/fcc).

Step D — Consistency check against a non-close-packed alternative. Suppose instead we (incorrectly) assumed a body-centred cubic cell ( $Z = 2$ , spheres touching along the body diagonal, $a\sqrt3 = 4r$ , so $a = 4r/\sqrt3 = 4\times124.6/1.7321 = 287.8$ pm):

V_{\text{cell,bcc}} = (287.8)^3\ \text{pm}^3 = 2.385\times10^{7}\ \text{pm}^3 = 2.385\times10^{-23}\ \text{cm}^3

\rho_{\text{calc,bcc}} = \frac{2\times58.69}{6.022\times10^{23}\times2.385\times10^{-23}} = \frac{117.38}{14.36} = 8.174\ \text{g/cm}^3

This bcc-based prediction (8.174 g/cm³) disagrees with the measured density (8.908 g/cm³) by nearly 10%, decisively ruling out bcc and confirming the ccp assignment made in Step C. This calculation illustrates how packing-fraction geometry (74.05% for ccp versus 68.02% for bcc) directly and measurably manifests in bulk density — a genuinely diagnostic, checkable structural fingerprint.

Limiting cases, scope & common traps

Situation	What close-packing predicts	Common error
Identical hard spheres, no directional bonding	CN = 12, $\eta$ = 74.05% for both hcp and ccp	Assuming hcp is "denser" than ccp — they are equally dense; only symmetry differs
Metals with strong directional (covalent) character (e.g. Sn, Ga)	May adopt non-close-packed structures instead	Assuming all metals are close-packed — many are bcc or have distorted structures
Ionic solids with large size mismatch between anion and cation	Anions close-pack; cations occupy holes, distorting the lattice	Treating the ionic lattice as literally two close-packed sublattices of touching ions of different size
Real hcp crystals (Mg, Zn, Ti)	Axial ratio should be near 1.633	Assuming exact 1.633 always holds — many hcp metals deviate substantially (Zn: 1.856; ideal hard-sphere touching is not always achieved because bonding is not purely non-directional)
Random stacking sequences (e.g. ABACABAB…)	Still 74.05% packing and CN = 12 locally	Assuming only strictly periodic ABAB or ABCABC sequences are "close-packed" — random close-packed sequences (polytypes) are equally valid and occur in real materials like SiC and ZnS
Comparing bcc to close-packed structures	bcc has $\eta = 68.02\%$ , CN = 8 (or 8+6 next-nearest)	Assuming bcc is close-packed because it looks dense in a simple cubic drawing — it is not one of the two true close-packed structures

Common trapMisconception: "hcp and ccp differ in density"

A very common mistake is to assume that because ccp (cubic close-packing, equivalent to fcc) "sounds" more symmetric, it must pack space more efficiently than hcp. In fact both structures share the identical packing fraction of 74.05% and the identical coordination number of 12 — the only difference is the long-range stacking sequence (ABAB… versus ABCABC…), which changes the space group and the macroscopic symmetry (hexagonal vs cubic) but not the local density or nearest-neighbour count. Real differences in physical properties between hcp and ccp forms of the same element (e.g. different hcp vs fcc allotropes of cobalt) arise from subtle differences in the second-nearest-neighbour shell and its effect on electronic band structure, not from any difference in packing efficiency.

The chemist behind it

Origins · 1 minWilliam Barlow and how this was discovered

William Barlow

1845–1934 · Victorian crystallography, prior to X-ray diffraction

William Barlow was an English amateur crystallographer — by profession a businessman, self-taught in geometry and mineralogy — who in the 1880s and 1890s worked out, purely from geometric reasoning about how identical spheres could stack, the possible symmetric arrangements now recognised as hexagonal and cubic close-packing. Remarkably, this was accomplished before X-ray diffraction existed (Laue's experiment was 1912), meaning Barlow had no direct experimental way to confirm his structural predictions; he reasoned entirely from packing geometry, space-group theory (building on the mathematical classification of the 230 crystallographic space groups completed independently by Barlow, Fedorov, and Schoenflies around 1891–1894), and physical intuition about how atoms of different sizes should pack together in ionic crystals. Barlow correctly predicted structural motifs — including the rock-salt structure and features of close-packed metal structures — years before diffraction methods could test them; when X-ray crystallography arrived in the 1910s, many of his geometric predictions were confirmed essentially unchanged. His work exemplifies how far pure geometric and symmetry reasoning can go in predicting real atomic-scale structure, a lesson that remains at the heart of how introductory inorganic chemistry teaches close-packing today: get the geometry right, and the chemistry follows.

History

- 1611 — Johannes Kepler conjectures that face-centred cubic (equivalently, cubic close-packing) is the densest possible arrangement of identical spheres (the Kepler conjecture).

- 1694 — Isaac Newton and David Gregory debate the maximum number of unit spheres that can touch a central sphere (the "kissing number" problem, resolved as 12 in 3D much later).

- 1883–1897 — William Barlow works out the possible close-packed structures (hcp, ccp) and related ionic structures from pure geometric/symmetry reasoning, before X-ray diffraction existed.

- 1912 — Max von Laue and colleagues demonstrate X-ray diffraction by crystals, providing the first direct experimental probe of the atomic arrangements Barlow had predicted.

- 1998 — Thomas Hales completes a computer-assisted proof of the Kepler conjecture, rigorously confirming that 74.05% is indeed the maximum possible packing density for identical spheres in three dimensions.

Why do hcp and ccp have exactly the same packing fraction if they look so different?

Both structures are built from identical two-dimensional close-packed layers; the only difference is whether the third layer repeats the position of the first (hcp) or introduces a genuinely new offset position (ccp). Because packing fraction is a purely local property — it depends on how tightly each sphere is surrounded by its nearest neighbours, which is identical (12 neighbours, same local geometry) in both cases — the long-range stacking sequence has no effect on it. The visual difference between hcp and ccp only becomes apparent when you look several layers deep, at the second-nearest-neighbour shell, where the two structures genuinely diverge in symmetry (hexagonal versus cubic).

Is ccp the same thing as fcc?

Yes. "Cubic close-packing" (ccp) is the structural/stacking-sequence name (ABCABC…) used in chemistry to emphasise the relationship to hcp, while "face-centred cubic" (fcc) is the crystallographic name for the same lattice, emphasising its unit cell (a cube with atoms at corners and face centres). They describe the identical atomic arrangement; chemists tend to say ccp when discussing packing and stacking sequences, while crystallographers and materials scientists more often say fcc when discussing unit cells and space groups.

Why doesn't every metal adopt a close-packed structure?

Close-packing maximises nearest-neighbour contacts, which is energetically favourable for non-directional metallic bonding — but not all metals bond in a purely non-directional way. Elements like iron (bcc at room temperature), tungsten, and the alkali metals adopt body-centred cubic structures instead, partly because subtle electronic effects (partially filled d-orbitals, Fermi surface effects) can make a slightly less dense structure electronically more stable. Some metals (gallium, tin, bismuth) show significant covalent/directional bonding character and adopt distorted, non-close-packed structures altogether. Close-packing is the geometric default for "hard sphere" bonding, but real atoms are not hard spheres.

Do real ionic crystals actually have touching anions, as assumed in close-packing?

Not always exactly, but often approximately. In many ionic structures — like rock-salt oxides and halides — the anions (typically larger and more polarisable) are close to touching each other, forming an approximately close-packed sublattice, with the smaller cations sitting in the interstitial holes without necessarily touching the anions at the ideal close-packed anion–anion distance. The close-packing model works best when the radius ratio (cation radius / anion radius) is small to moderate; when cations become comparably large, the anion sublattice is forced open and the close-packing approximation breaks down, requiring the more general radius-ratio rules developed alongside Ionic Radii and Their Determination.

What are the "holes" mentioned but not fully explained here?

Between close-packed spheres there are two types of interstitial voids: tetrahedral holes, each surrounded by 4 spheres in a tetrahedral arrangement, and octahedral holes, each surrounded by 6 spheres in an octahedral arrangement. For every $N$ spheres in a close-packed array, there are exactly $N$ octahedral holes and $2N$ tetrahedral holes — a counting result that follows from close-packing geometry but is developed rigorously, along with the maximum cation size each hole type can accommodate, under Tetrahedral and Octahedral Holes.

Why is 74.05% considered a "law of nature" rather than just an approximation?

The value $\pi/(3\sqrt2) \approx 0.7405$ is not an empirical average or approximation — it is the exact, provably maximum density achievable by any arrangement whatsoever of identical non-overlapping spheres in three-dimensional space, a statement known as the Kepler conjecture. Kepler proposed it in 1611 based on physical intuition about cannonball and fruit stacking, but a fully rigorous mathematical proof (using extensive case-by-case computer verification of geometric configurations) was not completed until Thomas Hales's work culminating around 1998–2005. No packing of identical spheres, however cleverly arranged, can ever exceed 74.05% density — hcp and ccp are simply the two simplest structures that achieve this proven maximum.

How does close-packing connect to the next topic of tetrahedral and octahedral holes?

Close-packing tells you exactly where every sphere sits, which automatically fixes the size, shape, and number of every gap between spheres. Once the close-packed anion lattice is established, the tetrahedral and octahedral holes are not adjustable — their positions, sizes (as a fraction of the anion radius), and counts per anion are geometrically forced consequences of the close-packing arrangement itself. This is why close-packing is treated first: every subsequent statement about which cations fit into which holes, and in what numbers, is a direct geometric corollary of the packing geometry developed here.