Aluminal speciation in the crystal nucleus: a mass spectral interpretation

Abstract

Mass spectral line positions m/z are computed from a general formula for aluminium cations, Al_nO_p(OH)_q(OH₂)_r^{(3n−2p−q)+}, without treating n, p, q and r as independent variables but instead constraining parameters by tetrahedral or octahedral symmetry, in accordance with each of a number of structural possibilities. In one instance, a ring is distinguishable from a chain; in another, an orthorhombic sheet from a hexagonal prism. Amongst other polyhedra, five distinct icosahedral cases are discussed. Classical constructions are revisited in the light of aperiodic tiling. Successive Fibonacci numbers may describe the nucleation of a structure previously regarded as amorphous. Recent experimental observations are reviewed in the same context, and re-interpreted. A spectral value may be ascribed to a tetrahedral or an octahedral species (ambiguity is rare). The results may be a significant factor in an understanding of crystal nucleation.

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1. General significance

Aluminal species exist naturally in soil, river water, ice melt, plants and food. For many years they have been of interest to soil science,¹ especially in regions which have large human populations to feed. Soil acidification, which accompanies the aqueous dissolution of Al³⁺, may restrict plant growth and limit crop production.² Such species have no known physiological function, yet because of their ubiquity we ingest them uncritically. Despite their toxicity³ it was thought that only at abnormally high concentrations might the human suffer cellular damage in consequence. Recent findings however suggest that chronic exposure⁴ may bring an increased risk of the brain disease, Alzheimer's. In soil science hexameric ring species were first postulated over 50 years ago,⁵ though not confirmed incontrovertibly. Recent experimental advances⁶ now enable us to re-examine this and other possibilities.

Alumina, or aluminium hydroxide or oxide hydroxide, or aluminium oxide, or aluminal species generally in solution, may contain positive ions or ionic complexes that may be generally formulated as Al_nO_p(OH)_q(OH₂)_r^z+, where z = 3n − 2p − q and where {p, q, r} have often been treated as if they were independent variables and were independent of n (p, q, r are each ≥ 0). This formulation provides explicitly for minor variations in ligand, extent of hydration and ionic charge.

In a mass spectral interpretation, m = 27n + 16(p + q + r) + q + 2r. In what follows, I propose an analysis where {p, q, r} are not independently variable, but are constrained by the symmetry of the aluminal species. Limits are placed on the number of ligands which may surround any one Al³⁺ cation. In broad terms, the symmetry may be regarded as either octahedral, O_h, or tetrahedral, T_d; six or four ligands may surround the cation: s = 6 or 4. In this analysis, r = r(s, n, p, q). m/z is then computed for each instance of the formula so constrained.

I have distilled out the essence of the computational method and presented this in Tables 1–5. This enables the reader to make progress through the text without having to dwell on each step of the derivation. Each table contains a text reference column referring the interested reader to a section containing the full derivation. The detailed results of the computation are appended as ESI.† In this context, observations made by Urabe and his group^7,8 using Electrospray Ionisation (ESI) mass spectrometry have been reviewed independently and the results summarised in Table 6. Initially at least, negative ions are disregarded (except in the special case of Keggin or, more generally, Biliński: see below, Sections 2.4 and 2.6.6).

Table 1 Calculation of m/z values from model Al³⁺ structures

Special case	n	p	q	r	s	m	z	m/z	Text reference
Hexagonal ring									2.2
	6	0	2n = 12	ns − 2q = 12	6	582	3n − q = 6	m/z = 97
2-d tiling by hexagons - tessellated hexagon (serrated)									2.2.3
	6(1 + x)^2, where x is the integer part of half the greatest column-height ξmax	0	6(2 + 5x + 3x^2)	12(1 + x)		6(97 + 175x + 78x^2)	6(1 + x)	m/z = (97 + 175x + 78x^2)/(1 + x); e.g., for x = 0, 1, 2 or 3, m/z = 97, 175, 253 or 331
Chain of hexagons									2.2.8
	2(2ξ + 1), where a row has ξ adjacent hexagons	0	2(2ξ + 1)	4(ξ + 2)		350ξ + 232	2(ξ + 2)	m/z = (175ξ + 116)/(ξ + 2); e.g., for ξ = 1, 2, 3 or 4, m/z = 97, 116.5, 128.2 or 136, respectively
Chain (general)									2.3.1
	n	p	q	(s − 2)n + 2 − (p + q)	4 or 6	27n + 16p + 17q + 18r	3n − 2p − q	m/z
Keggin cage – may be a special case of Biliński, rD									2.4, 2.6.6
	4nL + 1, where the tetrahedral centre attaches 4 chains, assumed all to have length L and identical parametric values {nL, pL, qL}	4pL + y, where y is the number of O^2− ligands in the tetrahedron	4qL + (4 − y)	nL(sL − 2) − (K − 2) − (pL + qL), with 13 − (pL + qL) being the most likely case: K = 1 (apical bonding), n = 3, s = 6	sL = 6	27n + 16p + 17q + 18r	4c − (y + 1), where in each cationic chain c = 3nL − 2pL − qL	m/z = [4m(nL, pL, qL) + m(y)]/[4c − (y + 1)]
Prismatic species									2.5
including
Prism (general)									2.5.1
	[n(x)]j, where the prism is of length j ≥ 2	[p(x, h)]j	[q(x, h) − v[1 − (1/j)]]j + (j − 1)v	[r(x, h)]j		27n + 16p + 17q + 18r	3n − 2p − q + v[1 − (1/j)]	m/z
Sheet of octahedra									2.5.2
	x(2h + 1) − h, comprising 2h + 1 rows of octahedra	2h(x − 1)	2h(2x − 3)	4(1 + 2h + x)	6		3x + 7h − 2hx	Cannot equal an m/z above, e.g., for a 2-d tiling
DoubleSheet – a special case of the polygonal prism, where the polygon is a rectangle and j = 2									2.5.3
	2[x(2h + 1) − h]	h(6x − 7)	2h(2x − 3)	8(1 + 2h + x)	6		2(3x + 7h − 2hx)
Extended DoubleSheet – a more general case of the DoubleSheet, in which j ≥ 2									2.5.4
	2j[x(2h + 1) − h], where αvj hydroxyl groups are shared between adjacent DoubleSheets	j[h(6x − 7)] + h(j − 1) (x − 3)	j[3hx − 3h + (h/j)(x − 3)]	j[8(1 + 2h + x)]	6		[6x + 11h − 3hx − (h/j)(x − 3)]j − 2h(j − 1)(x − 3)
Polyhedra (non-prismatic)									Table 2

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Table 2 Calculation of Al³⁺ m/z values in non-prismatic polyhedra

Special case	n	p	q	r	s	m	z	m/z	Text reference
Polyhedron (non-prismatic)									2.6.1
including
Platonic solid (including the regular icosahedron)									2.6.2
	C + V = V	αD = 0, because D = 0	2βD + 2εE = 2εE, where E = F + V − 2	(s − (4εE/V))V	4 or 6	27n + 16p + 17q + 18r	3V − 2εE
of which the icosahedron is the special case, V = 2(1 + 5)									2.6.2
	V = 12		E = 30; q = 2V, for example (in the case where ε = 2/5)	r = V(6 − 10ε) = 2V, for example (where s = 6)	4 or 6		V(3 − 5ε), neutral if ε = 3/5; equal to V if ε = 2/5	m/z = 61 + 18(s − 4) (if ε = 2/5, for example)
Archimedean solid									2.6.3
	C + V = V	αD = 0	2εE, where E = F + V − χ (if convex, χ = 2)	(s − (4εE/V))V	4 or 6	27n + 16p + 17q + 18r	3V − 2εE
of which Goldberg or the truncated icosahedron is a special case, V = 60									2.6.3
	V = 60 (“buckminstalumina” – were it to exist)		F = 32, χ = 2; q = 2E = 3V = 180, for example (in the case where ε = 1)	r = 0, for example	4 or 6		3V(1 − ε), neutral if ε = 1
3-d space-filler (non-prismatic)									Table 3
	C + V	αD	2βD + 2εE	(s − (4εE/V))V	4 or 6	27n + 16p + 17q + 18r	3V − 2εE
Stochastic nucleation product									2.6.9
	cV = 2^3 or ck + 2(2^3 − ck) initially, in the classical case for example; in which ck = 2^k, where k = 0, 1, 2	aαD = αk2^k, in the example – see text	2bβD + 2eεE = βk2^k + 2eεE, in the same example	A cV(s − 6ε) component is to be expected, its r = 2^3(s − 6ε) or 2(2^3 − ck)(s − 6ε) initially	4 or 6	27n + 16p + 17q + 18r	3cV − 2(aα + bβ)D − 2eεE, in general (a = 2b, in the example)
N-dimensional species									2.6.10
	C(n) + V(n) = 2^n, on whose exterior V(n) = 2^3	αD(n) = a(n − 1)2^n−2, approx.	2β(D(n) + E(n)) = 2βn2^n−1	sV(n) − 4β(12 + V(n))	4 or 6	27n + 16p + 17q + 18r	z = 2^n3 − [α(n − 1) + 2βn]2^n−1, or 2^n−1[6 + α − (α + 2β)n], approximately. Charge-neutrality would arise when β = 3/n (if α = 0); or, more generally, when n = (6 + α)/(α + 2β); e.g., in the hexeract (α = 0, β = 1/2, s = 6)

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2. Special cases

2.1 Ring species

2.1.1 General. In crystal growth, the hexagonal shape is found world-wide. The ice crystal is an iconic example … though the general shape is far from unique. In an aluminal structure where the Al³⁺ cation may be surrounded by six ligands arranged octahedrally, it seems highly plausible that the ultimate crystal system in bulk might be determined as early as the event of the 3^rd cation attaching itself to a dimer in solution [whether in a straight line, or at a 60° angle].

Given that aluminium salt solutions are amenable to scientific study,⁶ there is now the opportunity to examine the possibility of linear, hexagonal or polyhedral crystal nucleation. This could be of fundamental importance. A bigger question still might be whether the salt solution is really nothing more than “a crystal waiting to happen” (i.e., comprises a pre-determined set of nuclei of given geometry) … or whether the introduction of a chemical reagent – or impurity such as a metal ion, or other electric field or perturbation – must first radically alter the structure of the solution to induce its own characteristic nucleus.

2.1.2 Octahedral symmetry. It may be assumed that perfect octahedral symmetry O_h is to be expected only in the special case of the monomer where the Al³⁺ cation is surrounded by six ligands indistinguishable one from another, e.g. Al(OH₂)₆³⁺ (s = 6, n = 1, p = q = 0, r = ns = 6).

2.1.3 Hexagonal ring. It may be further assumed that a perfect hexagonal ring is to be expected only in the special case of the hexamer comprising six imperfect but near-perfect octahedra, i.e., Al₆(OH)₁₂(OH₂)₁₂⁶⁺ (s = 6, n = 6, p = 0, q = 2n, r = ns − 2q = 12).

The imperfect octahedron (symmetry lower than O_h) may be conceptualised as a sphere of radius R that encloses all the atoms, regarded as if they were point-positions, within its geometric span d_geo. From Jung's theorem in 3 dimensions, R ≤ d_geo sqrt(3/8). Since all six octahedra are indistinguishable one from another, the six identical spheres may be positioned with each sphere touching two others in a plane so as to form a perfect hexagonal lamina or “wafer” 2R thin and of edge-length 2R with cross-sectional area 6R²sqrt(3).

2.1.4 Question of existence. Evidence is now sought for existence of the precise hexamer Al₆(OH)₁₂(OH₂)₁₂⁶⁺ (as distinct from other hexameric species such as chains).

2.1.5 Spectral value expected. In a mass spectrum containing the hexagonal ring, z = 3n − 2n = 6 and m = 27n + 16(2n + ns − 4n) + 2n + 2(ns − 4n) = n(18s − 11) = (6 × 97) = 582. m/z = 97.

2.1.6 Ambiguity. Unique spectral interpretation may fail: other aluminal species might exist with m/z = 97 (see below). If so, an experimental variation may be required, or even a novel experiment.

2.2 Hexagonal species

2.2.1 Symmetry. Wherever the species is hexagonal, octahedral symmetry (s = 6) is assumed.

2.2.2 A nuclear hexagon?. The simplest aluminal hexagon that might exist is the hexagonal ring described above. If the hexameric species Al₆(OH)₁₂(OH₂)₁₂⁶⁺ does exist, it could potentially act as a crystal nucleus. A crystal nucleated by this species might itself belong a hexagonal class (such as a hexagonal prism), although it need not. The crystal could be an aluminal species; though again, it need not be. Given that ice crystals, for example, also exhibit hexagonal shapes, it is not at all inconceivable that octahedral Al³⁺ species now found in clay⁹ might have played some earlier role in nucleating ice in a glacier; or that alumina in mud once nucleated ice in a comet.¹⁰

2.2.3 Tessellated hexagon. It is possible to describe a serrated hexagon which is itself tiled by regular hexagons. The hexagonal tiles are primitive, congruent with one another and may be arranged in parallel columns, sharing edges. The line drawn through the centre of each of the tiles furthest from the most central tile forms the edge of the largest outer hexagon. In this model, each vertex of each hexagonal tile would provide the site of an Al³⁺ cation, with cations shared between adjacent columns. See Fig. 1.

Fig. 1 2-d tiling by hexagons. The number of “annular rings” x is int(ξ_max/2), the integer part of half the number of regular hexagons in the longest chain. Though approximately hexagonal, the tiled area is not itself a regular hexagon, but serrated. Nuclear expansion in 2-d may be represented by the equation . One aluminal ring structure thought to exist, for example, takes the formula Al₅₄(OH)₁₄₄(OH₂)₃₆¹⁸⁺ (x = 2); for which m/z = 253. Nucleation of the primary hexagonal species Al₆(OH)₁₂(OH₂)₁₂⁶⁺ (x = 0) from a dimer such as Al₂(OH)₄(OH₂)₆²⁺ would appear first to require a non-linear trimer to form. This could be an essential precursor of, in 3-d, a hexagonal prismatic crystal or honeycomb. Evidence is required of the polymer geometry. The serrated hexagon may also be regarded, again approximately, as a symmetrically bi-truncated rhombus (also serrated). This particular rhombus is not however a golden rhombus.

2.2.3.1 Number of hexagonal tiles. If the number of hexagons in the highest column is an odd finite number, then the total number of hexagonal tiles, or extent of 2-dimensional growth G(x), is:
where for regular hexagonal shape x = int(ξ/2), the integer part of half the column-height, and where ξ > 2 (see below, Section 2.2.8). This is one instance of a symmetrically bi-truncated diamond shape or serrated rhombus X.X, where X = 1 + 2x. Because the sum of the first n natural numbers . A non-trivial example is x = 2, giving G(x) = 19.
2.2.3.2 Number of vertices. In this tessellated hexagon, the number n of Al³⁺ cations is given by , or 6(1 + x)². For x = 0, 1, 2…, n = 6, 24, 54….
2.2.3.3 Number of sides. Each side of each hexagonal tile is assumed to be occupied by two hydroxyl groups, with adjacent hexagons sharing two groups between them. None of the q hydroxyl groups is located elsewhere. So the total number of sides is . Or, the total number of OH⁻ ligands q = 6(2 + 5x + 3x²). Since no O²⁻ ligands need be postulated in the near 2-dimensional wafer (as distinct from in the 3-dimensional prism), p = 0.
2.2.3.4 Number of corners. There are corners, each occupied by two H₂O ligands: r = 12(1 + x).
2.2.3.5 Crystal nucleation and growth?. Given that nucleation and growth mechanisms for hexagonal crystal formation are of interest, it follows that besides seeking evidence for the nuclear hexagon Al₆(OH)₁₂(OH₂)₁₂⁶⁺ (x = 0), we should also seek a tessellated hexagon or hexille, such as Al₂₄(OH)₆₀(OH₂)₂₄¹²⁺ (x = 1) or Al₅₄(OH)₁₄₄(OH₂)₃₆¹⁸⁺ (x = 2); not necessarily ruling out Al₉₆(OH)₂₆₄(OH₂)₄₈²⁴⁺ (x = 3) or larger species. In this model of near 2-dimensional crystalline growth, or laminar growth, the edge-length of the crystal hexagon (x > 0) is 2Rxsqrt[(1 + cos(π/3)²) + sin(π/3)²], or 2Rxsqrt[2(1 + cos(π/3)]; that is, greater than that of the nuclear hexagon 2R by a factor of xsqrt(3). In a growth pattern such as this where hexagonal species might be expected to be relatively stable, non-hexagonal intermediate species could form by adding tiles from G(x) to G(x + 1), G(x) recurring, but these would be transient.
2.2.3.6 Spectral values. See Table 1. Any of these three values may be interpreted unambiguously (unlike the value of m/z = 97, which may or may not derive from the hexagon, G(0) = 1).

2.2.4 The serrated rhombus. The serrated rhombus is tessellated by hexagonal tiles. It is also a quadrille, i.e., tessellated by G(x − 1) diamond-shaped tiles or rhombi. For the serrated rhombus, unlike the serrated hexagon (where X odd > 2 is necessary), X > 1 is sufficient and may be odd or even. G(x) = X². It is natural to ask whether its diagonals exhibit the “golden ratio” τ, which has the property τ = (τ + 1)/τ. For this condition, its half-angle θ would have to equal cot⁻¹ τ, or tan⁻¹(2/(1 + sqrt(5))). But in the hexagonal tessellation or hexille, the array of regular hexagons is such that θ = π/6, or tan⁻¹(1/sqrt(3)); which is less than golden. It seems that the rhombus is tarnished!

2.2.5 Lower-symmetry hexagons?. By replacing each sphere by the more general ellipsoid, however, the hexagon could be elongated (as on the face of Fyodorov's elongated dodecahedron,¹¹ perhaps). It might also become narrower; and possibly thinner: ellipsoidal packing may exceed the Kepler maximum (see below). Elongation, by a factor of λ, would increase the half-angle θ above its π/6 value, and could be such as to produce the golden rhombus.

As a 1^st approximation, the linear ratio would be λ = tan(cot⁻¹ τ)/tan(π/6) = 2sqrt(3)/(1 + sqrt(5)). Despite its lower symmetry, the hexagon may have regained something of its lustre!

2.2.6 The golden ratio. While growth in the direction of cot⁻¹ τ may give rise to the “golden” rhombohedron, with all six faces congruent, this is not the sole manifestation of the golden ratio in crystal systems: τ is also one of the Cartesian coordinates defining each of the 12 vertices of an icosahedron (see below, Sections 2.2.7 and 2.6).

2.2.7 The regular icosahedron. With Penrose, it has been recognised that – contrast the hexille – rhombic tilings may be aperiodic: without translational repetition. Also, the classical Bravais lattice is transcended by a hypercubic lattice: in N-dimensional hyperspace, the 4-, 5-, 6-cube et seq are analogous with the cube (or 3-cube). Specifically, a regular icosahedral quasi-crystal¹² may be represented visually as the 3-d projection of a crystal from a 6-cubic or hexeractic lattice. Instead of rhombic tiles (in 2-d), in a 3-d tiling the tiles may be rhombohedra.

The only instance known to occur naturally is icosahedrite, the aluminium mineral Al₆₃Cu₂₄Fe₁₃, with 5-fold symmetry (rotation through 72°, or 2π/5 or cos⁻¹(τ/2)): not an “alumina” per se. In the regular pentagon, τ is the ratio of chord to edge length. Nucleation and growth mechanisms remain to be elucidated. More generally, quasi-crystalline growth remains possible, involving structural changes to an “amorphous” or chaotic precipitate. When modelling crystal growth, icosahedral symmetry need no longer be debarred.

2.2.8 Chain of hexagons. A chain of hexagons may be regarded as being equivalent to a single row of ξ adjacent hexagons, although its specific formula does not imply a straight line (or indeed any particular conformation): Al_2(2ξ+1)(OH)_2(5ξ+1)(OH₂)_4(ξ+2)^2(ξ+2)+. Potentially, any small species of this kind could play a role in the nucleation and growth of a crystal. Given that the spectral value m/z = 97 is ambiguous, in that this value is a set containing other aluminal species besides the single hexagon (ξ = 1), complementary evidence may be sought for species with ξ > 1. See Table 1. Again, any of these values could be unambiguously interpreted.

2.3 Chain species

2.3.1 General chain formulation.
2.3.1.1 All cations. In the case of a chain of polyhedra, each of the n Al³⁺ cations is again assumed to share ligands. Depending on its position in the chain, an Al³⁺ cation will share either two or four of its s ligands. At either end of the chain, the terminal Al³⁺ cations will each share only two of their ligands, with the next cation along. In a trimer or chain of greater length, each interior cation shares four. (In a dimer, only the two are shared; in a monomer, none.) This means that the total number of ligands p + q + r = s + (n − 1)(s − 2). In this case, r is not independently variable, but is determined by the chain constrained by the symmetry: r = (s − 2)n + 2 − (p + q).
2.3.1.2 Unipositive cations (z = 1). In this special case, 3n − 2p − q = 1. In this case, q is not independently variable, and is determined by the chain limitations: q = 3n − 2p − 1. However, it seems unnecessary to restrict our scope in this way.

2.3.2 Monomer. The monomer is the special case of a general chain where n = 1. This means that the number of ligands p + q + r = s. r = s − (p + q). Given that p = 0, r = s − q.

2.4 Keggin cage species

In the special case of the Keggin “cage”, it is the tetrahedral aluminate anion Al(OH)₄⁻ (again T_d) that is believed to be central, rather than an Al³⁺ cation (except in as much as Al³⁺ is central to the Al(OH)₄⁻ anion itself). It is surrounded by chains. A general Keggin formulation therefore may now be made as the positive species [Al_nO_p(OH)_q(OH₂)_r^c+]₄[AlO_y(OH)_4−y^(y+1)−]^z+, where z = 4c − (y + 1) and where in the cationic chain c_L = 3n_L − 2p_L − q_L. In general, each n_L value could be different (L = 1, 2…4). Typically, n_L = 3 is assumed with octahedral symmetry (s_L = 6) throughout. However, the Biliński case (n = 3 + δ, p = 0, q = 10ε) is also of interest (see below, Section 2.6.6 Mass spectra – Kepler–Biliński case.).

The leading Al³⁺ cation of each of the four (trimeric) chains will bond in some way with either an apex, edge or face of the aluminate tetrahedron. This bonding will involve either one, two or three of the cation's six ligands (assuming broadly octahedral symmetry, s_L = 6): K = 1, 2 or 3; and leave five, four or three of the six remaining. Of this remainder, two ligands will be shared with the next cation in the chain. As before, the terminal cation (now at the other end of the chain) will share two ligands with its neighbour. (Bonding through the tetrahedral apical oxygen seems intuitively the most likely possibility, retaining a high degree of symmetry such as T_d while distorting the tetrahedron the least severely.) This means that the total number of Keggin chain ligands p + q + r = s + (n − 2)(s − 2) + (s − 2 − K). r = n(s − 2) − (K − 2) − (p + q). Given the likely case (K = 1, n = 3, s = 6), r = 4n − (K − 2) − (p + q) = 13 − (p + q). In a mass spectrum containing the tridecameric species, m = 4m(n_L, p_L, q_L) + m(y).

Keggin-like combinations of tetrahedra and octahedra may be significant in crystal growth because, unlike tetrahedra alone, some such combinations have the capacity to fill space contiguously.¹³ An example is Biliński (see Fig. 2).

Fig. 2 3-d space-filling by Biliński rhombic dodecahedra. Biliński space-filling may be represented by the equation . Its relationship with the 2-d tiling equation cannot go unnoted (see Fig. 1). In m/z for the 3-d case, both {m, z} are linear in G(x); and, as with classical cubic space-filling, cubic in x. Sadly, however, it now seems unlikely that – save in mathematics – this beautiful crystal really exists.

Fig. 3 A rhombic dodecahedron. A dodecahedron of the Biliński kind comprises four golden rhombohedra, two prolate and two oblate: |G_ρD(x)〉 = 2|G_V1(x)〉 + 2|G_V2(x)〉. Were it to exist, an aluminal species structured with cations at the vertices (one internal) could be formulated as AlO_4α(OH)_8β[Al_3+δ(OH)_(10+4δ)ε(OH₂)_{s(3+δ)−(20+8δ)ε−2(α+β)}]₄ ^z+, where δ_i = {[0, 1], twice}_i=1⁴, s = s_v = 6; and where z_i = 3(4δ_i + 13) − 8[α + β + ε(2δ_i + 5)], or z = 45 − 8(α + β + 6ε). The Biliński rhombic dodecahedron is a space-filler (see Fig. 2). The formula makes explicit reference to a central tetrahedral aluminate structure. In such a structure the special case of δ_i = 0 (zero values only), x = 1 may be identified with the typical Keggin tridecamer [Al_nO_p(OH)_q(OH₂)_r^c+]₄[AlO_y(OH)_4−y^(y+1)−]^z+ (n = 3), in which y = 4α, α + 2β = 1 and z = 3(1 + 4n) − 4(1 + α + 2p + q), where q = 10ε and p = 0. The Biliński dodecahedron |G_ρD(x)〉 is central to the Fyodorov rhombic icosahedron, |G_ρI(x)〉, where it appears as a nuclear subtype within “Penrose in 3-d” (see Fig. 4): |G_ρI(x)〉 = |G_ρD(x)〉 + 3|G_V1(x)〉 + 3|G_V2(x)〉; so that the Fyodorov icosahedron has a total of five pairs of golden rhombohedra. It is subject to Penrose enlargement.

2.5 Prismatic species

2.5.1 General prism formulation. The prevalence in recent literature of references to implicit chain structures and the Keggin “cage” does not preclude the existence of other aluminal species. Besides the ring and the chain, the 2-dimensional sheet (where in the cation hn may be formulated, instead of simply n) and doubleSheet (2hn, as in a pseudoBoehmitic structure) are entirely possible. Moreover 3-dimensional systems may also be modelled (jhn), such as those with extended hexagonal rings, sheets and prisms.
2.5.1.1 Polygonal prism. Given an assembly of identical polygons each of size G(x), the extent of 3-dimensional growth G(x, y), is:

G(x) = jG(x), where j is a finite number of polygonal layers.

In this model of the prism, v O²⁻ ligands or OH⁻ ligands may be shared between each of the j − 1 pairs of adjacent layers in the prism. Where the layer itself contains no O²⁻ ligands, the formula for the cation is [Al_n(x)(OH)_q(x,h)−αv(OH₂)_r(x,h)^c+]_j[O_(j−1)v^2v(j−1)−]^z+, where z = cj − 2v(j − 1) and where in the cationic layer c = 3n − (q − αv). More generally, if prior to sharing the layer itself were already to contain p O²⁻ ligands, then after sharing, the formula would be [Al_n(x)O_p(x,h)(OH)_{q(x, h)−αv}(OH₂)_r(x,h)^c+]_j[O_(j−1)v^2v(j−1)−]^z+, where now c = 3n − 2p − (q − αv); v ≤ q; α ≤ 1. Throughout the length of the prism, the overall number of O²⁻ ligands shared between the j layers is αvj, which equals (j − 1)v; therefore α = 1 − (1/j), so that the formula becomes Al_n(x)O_p(x,h)(OH)_{q(x,h)−v[1−(1/j)]}(OH₂)_r(x,h)^c+]_j[O_(j−1)v^2v(j−1)−]^z+, where c = 3n − 2p − q + v[1 − (1/j)]; j ≥ 2 (j even). In a mass spectrum containing the prismatic species, m = jm(n, p, q, r) + m(j, v).

2.5.1.2 Hexagonal prism. For the hexagonal layer (above), the number of O²⁻ ligands within any one layer was taken to be zero: p = 0. n, q and r were computed above as functions of the row-length x. For the purpose of this computation h was defined as an explicit function of the row-length, h(x).z = 3nj − qj − αvj − 2vj + 2v = (3n − q)j −3v(j − 1).

Given hexagonal close packing of spheres and a unit-cell volume of 8πR³, the unit cell has length 4Rsqrt(2/3) and the Kepler maximum packing-density η = π/3sqrt(2). Cubic close packing gives the same maximum (though this may be surpassed by ellipsoidal packing). The hexagonal prism length is 4R(j − 2)sqrt(2/3). Without spherical close packing, η < η_Kepler, e.g. η = π/3sqrt(3), and the volume and length are correspondingly greater.

2.5.2 Sheet of octahedra. Hexagons may be unnecessary. If so, the model sheet comprises instead 2h + 1 rows of either x or x − 1 octahedra with row-length alternating x, x − 1, x… (x ≥ 3), finite integer h ≥ 1. Two edge rows exist, each comprising x octahedra. An edge would form the chain, Al(OH)₂(OH₂)₄[Al(OH)₂(OH₂)₂]_x−2Al(OH₂)₄^(x+2)+, except that this now shares x − 1 hydroxyl groups with the adjacent row, giving the reduced formula Al(OH)(OH₂)₄[Al(OH)(OH₂)₂]_x−2Al(OH₂)₄^(2x+1)+. The shorter, adjacent row comprises x − 1 octahedra (in which the hydroxyl groups shared between two rows have now become O²⁻ ligands); there are h rows of this shorter kind:

AlO₂(OH)₂(OH₂)₂[AlO₂(OH)₂]_x−3AlO₂(OH₂)₂^(3x−5)−

Also adjacent to this short interior row is either, the opposite edge of the sheet, or, a longer interior row comprising x octahedra; there are h − 1 such longer rows:

Al(OH)₂(OH₂)₂[Al(OH)₂]_x−2Al(OH₂)₂^(x+2)−.

Overall then, in the sheet: n = 2x + h(x − 1) + (h − 1)x, or x(2h + 1) − h. p = 2h(x − 1). q = 2 + 2(x − 2) + 2h(x − 2) + 2(h − 1)(x − 1) = 2h(2x − 3).

r = 16 + 4(x − 2) + 4h + 4(h − 1) = 16 + 4x − 8 + 4h + 4h − 4 = 4 + 4x + 8h = 4(1 + 2h + x).

The model sheet formula becomes Al_x(2h+1)−hO_2h(x−1)(OH)_2h(2x−3)(OH₂)_4(1+2h+x)^{(3x−2hx+7h)+}.

2.5.3 Double sheet. The doubleSheet is orthorhombic: a special case of the polygonal prism where the polygon is a rectangle and j = 2 (and therefore α = 1/2). Given that in the model one-half of the q hydroxyl groups are shared between the two sheets, then q/2 = αv, q = v and the reduced prismatic formula is [Al_n(x)O_p(x,h)(OH)_q(x,h)/2(OH₂)_r(x,h)^{(3n−2p−(q/2))+}]₂[O_q/2^q−]^z+, q even. In the doubleSheet, q/2 = h(2x − 3), 2p = 4h(x − 1) and so the special-case formula becomes [Al_{2[x(2h+1)−h]}O_h(6x−7)(OH)_2h(2x−3)(OH₂)_8(1+2h+x)]^{2(3x+7h−2hx)+}. See Table 1.

2.5.4 Extended double sheet. Now if αvj hydroxyl groups are shared between adjacent doubleSheets in a prism of length j ≥ 2 (α = 1/2, j even), each group becoming an O²⁻ ligand, the orthorhombic formula becomes [Al_{2[x(2h+1)−h]}O_h(6x−7)(OH)_{2h(2x−3)−v[1−(1/j)]}(OH₂)_8(1+2h+x)^c+]_j[O_(j−1)v^2v(j−1)−]^z+, where c = 6x + 14h − 4hx + v[1 − (1/j)]. Within the shorter rows of the sheet, 2h(x − 3) hydroxyl groups were available for sharing; but in the doubleSheet, half of these have already been shared. If the remaining half now comprise all the O²⁻ ligands shared with the adjacent doubleSheet, then v = h(x − 3) and the extended DoubleSheet formula becomes [Al_{2[x(2h+1)−h]}O_h(6x−7)(OH)_{3hx−3h+(h/j)(x−3)}(OH₂)_8(1+2h+x)^c+]_j[O_{h(j−1)(x−3)}^{2h(j−1)(x−3)−}]^z+, where c = 6x + 11h − 3hx − (h/j)(x − 3). Again, see Table 1.

Unlike the hexagonal or tetragonal prism and certain rhombohedra (not exclusively golden) where h is a function of x, in a triclinic, monoclinic or orthorhombic system such as the Sheet, doubleSheet or extended DoubleSheet, h and x are independently variable. In neither m(x) nor z(x) are the coefficients of x identical, and in the doubleSheet (or extended DoubleSheet) they are both linear in h. Without close packing, again η < η_Kepler; e.g., if cubic, η = π/6.

An attempt to equate m(x)/z(x) for the disparate crystal systems – to test for potential spectral ambiguity – leads to a quartic equation in h and x whose roots are not necessarily integers. In principle, therefore, mass spectrometry may well be capable of distinguishing between (e.g.) sheet growth and hexagonal growth patterns as early as nucleation.

Precise experimental design will depend on the extent of growth G(x, y), and the growth rate, G(x, y, t). In solution, Akihiro Wakisaka's experiment⁶ will be the preferred approach initially; however, once precipitation has begun,¹⁴ then other experimental methods may come into use: SIMS, for example, may become viable.

2.6 Other polyhedral species

2.6.1 Polyhedral formulation. Given that polygonal species may exist, such as the hexagon, it is natural to enquire whether a number of them might come together as faces of a polyhedron. Given that an Al³⁺ ion could be situated at each vertex, with two or fewer OH⁻ ligands on each edge and none elsewhere, the polyhedral formula would become Al_V(OH)_2εE(OH₂)_{(s−(4εE/V))V}^(3V−2εE)+ (0 < ε ≤ 1), where for a cationic aluminal species 3V > 2εE (in which V and E are the numbers of vertices and edges, respectively). In this formulation, ε cannot be empirical: 2εE must be a positive integer.

2.6.2 Platonic solids. Were the primitive icosahedron Al₁₂(OH)₂₄(OH₂)₂₄¹²⁺ to exist (s = 6, V = 12, E = 30, ε = 2/5), it would carry a positive charge: m/z = 97, again (further ambiguity); whereas, by contrast, Al₈(OH)₂₄ (s = 6, V = 8, E = 12, ε = 1) would be charge-neutral whatever the crystal system (cubic, golden rhombohedral or otherwise).

2.6.3 Archimedean solids. By Euler, E = F + V − χ (where the polyhedron has F polygonal faces). So, for a convex cation with two uni-negative ligands per edge (χ = 2, ε = 1), V > 2(F − 2). Archimedean solid geometry is a natural candidate; e.g., were Goldberg alumina or “buckminstalumina” to exist, with 60 Al³⁺ ions (V = 60, F = 32) arranged as a “soccer ball” or truncated icosahedron Al₆₀(OH)₁₈₀, it would be charge-neutral … although one OH⁻ group fewer (one more H₂O) would be sufficient to produce a value of m/z = m = 4663 + 18. Yet there is no reason to suppose that such species exist. Similarly, V = 12, 24, 48, 60, 120 could be paired with F = 8, 14, 26, 32, 62, respectively. (Invariably 4E/V = 6.) This would account for seven of the thirteen convex polyhedral arrangements, all neutral. Naturally, the charge-neutrality of a species does not preclude its existence. Neglecting water molecules, each is a multiple of Al(OH)₃!

The remaining six “semi-regular” species would each be anionic. For example, a snub cubic aluminate ion, were this unlikely species to exist, would comprise only squares and equilateral triangles, Al₂₄(OH)₁₂₀⁴⁸⁻ (V = 24, F = 38). (Now 4E/V ≤ 10.)

For the species to be cationic, the Euler characteristic χ could be relaxed such that χ > F − (V/2) and the species would be non-convex; although χ might be negative. If, for example, χ = −6, then for a cation, V would have to exceed 2F by at least 12. Alternatively perhaps, given that s ≥ 4, 3V/E > 2ε ≥ 4/5. Or the polyhedral cation might not exist.

2.6.4 Nucleation and growth in 3-d. Crystal growth in 3-d, if it is to produce space-filling convex polyhedra, could be taken to involve the sharing of a face between two polyhedra, P₀ and P₁ (and not merely an edge). Paradoxically then, although the crystal is growing, the overall number of external faces F₀ + F₁ decreases by 2. Since, following Penrose, one or two species of polyhedron P₁ or P₁ and P₂ may be adding to the crystal, a more general pair of growth equations in x may be written:

F_V(x+h)(x + h₁ + h₂) + h = F_V(x)(x) + c₁F_V1(x₁ + 1) + c₂F_V2(x₂ + 1) − 2h, and G_V(x+h)(x + h) = G_V(x)(x) + h,

where 1 + c₁ + c₂ is the number of polyhedra sharing faces (in which c₁ ≥ 0, c₂ ≥ 0; and either c₁ [and h₁] or c₂ [and h₂] is zero, though not simultaneously both c₁ and c₂), and h = h₁ + h₂ is the increase in the number of shared faces (h ≥ 2). F(0) = 0, G(0) = 0, V(0) = 0. The total number of vertices also diminishes: V(x + h) < V(x) + V₁ or V(x) + V₂. A more general expression in x_i could be written, but may be unnecessary. The paradox is explained if we accept that the system may attain a critical F_V(x) state where only 1/(1 + c₁ + c₂) polyhedra continue to grow, at the expense of both c₁/(1 + c₁ + c₂) and c₂/(1 + c₁ + c₂), each of which is diminishing. If the two additive species of polyhedron are identical (F_V1 = F_V2 and V₁ = V₂), then F_V(x+h)(x + h) + h = F_V(x)(x) + c₁F_V1(x₁ + 1) − 2h. An instance could be the nucleation of a 3-d tiling at x = 0, c₁ = h and its growth to x = 4: F₁₄(4) + 4 = 4F₈(1) − 8.

Suppose initially, at x₁ = 0, there exist 4 rhombohedra in 2 pairs (V = 8, F₈ = 6). If all four come together, each one sharing a face, the product is a rhombic dodecahedron F₁₄(4) with V = 14, F₁₄ = 12. This specific polyhedron could be significant as it can close off vertices and is potentially space-filling. Of the original 6 × 4 = 24 external faces, 12 survive – open to further growth – and 8 have vanished, while the remaining h = 4 have become internalised within the greater polyhedron: G₁₄(4) = 4.

Implicit in pure polyhedral growth – unlike specifically prismatic growth – is the characteristic that, beyond a requirement for mutual face congruence, there appears to be no geometrical constraint to preserve symmetry, provided that the grown structure remains convex. Thermodynamically there will be a tendency for a Wulff construction¹⁵ to form, lowering the Gibbs free energy; although the system seems unlikely to attain equilibrium short-term. (See below, Penrose in 3-d.) This may be so whether or not a 3-d tiling is a Penrose analogue, such as might grow from a Biliński F₁₄(2 + 2) species¹³ (where the 4 rhombohedra comprise 2 non-identical pairs of identical golden twins, each pair having the half-angle θ = cot⁻¹ τ). Continued growth, for example from x = 4 to x = 10, could produce a rhombic icosahedron: F₂₂(5 + 5) + 6 = F₁₄(2 + 2) + 6F₈(1) − 12; and might go further, to x = 20, say, and the F₃₂ triacontahedron attributed to Kepler: F₃₂(10 + 10) + 10 = F₂₂(5 + 5) + 10F₈(1) − 20.

Alternatively the rhombic dodecahedron, whether Kepler or Biliński, could potentially fill a 3-d volume (in a sense comparable with the way in which the hexagon fills a 2-d area).

2.6.5 3-d tiling?. It would be interesting to know, (a), whether, and under what conditions, potential tiling species can be identified at the nucleation stage; (b), if so, whether one, two or more distinct tiles can be distinguished from each other in the mass spectrum; and (c), whether 3-d tiling occurs; and, if it does, then whether this is Penrose or non-Penrose.

Unfortunately, however, the distinctive “first” rhombic dodecahedron, or classical Kepler F₁₄(4) species¹³ (whose four constituent rhombohedra are all identical, though these, having θ = cot⁻¹sqrt(2), are not golden), cannot be expected to show a mass spectrum distinguishable from its Biliński counterpart.

For 3-d tiling, and given the possibility of “Penrose in 3-d” with 2 unlike tiles, the capacity of any one species to fill space may matter less than contiguity of growth F_V(x + h) over a long range of x values irrespective of particular species in the growth pattern. F_V(x) is, of course, a positive integer; and we might assume that for contiguity, at least one of these external faces must be congruent with at least one face of either a F_V1(x₁ + 1) or a F_V2(x₂ + 1) species remaining. (At large x values, neither x₁ nor x₂ is necessarily still equal to zero.) The number of vertices on the two approaching faces must match.

If, for any given x, we knew the specific set of all polygons remaining, then to count the number of any such vertices could be a trivial task (e.g., for any rhombus, it would be 4); however, for large x, computation of the potential sets may not be straightforward; moreover, to establish facial congruence even in the case of smaller, regular polyhedra, the calculation of growth-angle θ and the edge-length is not always trivial. Again, thermodynamic considerations may help exclude some of the less likely possibilities.

2.6.6 Mass spectra.
2.6.6.1 General. Given that an Al³⁺ ion could be situated at each vertex and each closed vertex, with edges internalised there is also the possibility of two or fewer O²⁻ ligands on each internal edge; and, with none elsewhere, the polyhedral formula would become Al_C+VO_αD(OH)_2βD+2εE(OH₂)_{sV−4εE−2(α+β)D}^{[3C+3V−2(α+β)D−2εE]+} (with α + β ≤ 1, 0 < α, β, ε ≤ 1), where C(x) + V(x) is the total number of vertices, in which C(x) are closed points and V(x) completely open points or corners; and where the number of internal edges D(x) depends on the number of internal faces G(x). Again, neither α nor β nor ε may be empirical: any one of them must be either zero or a positive integer, or rational fractions such that αD, 2βD and 2εE are each either zero or a positive integer.
2.6.6.2 Kepler–Biliński case. For either of the two rhombic dodecahedra F₁₄(4) = 12, Kepler or Biliński, the specific formula becomes AlO_4α(OH)_8βAl₁₄(OH)_48ε(OH₂)_{14s−96ε−8(α+β)}^{[45−8(α+β+6ε)]+}. See Table 3.

The tetrahedral aluminate structure central to Biliński prompts the observation that the Keggin tridecamer (see above) is a special case with δ_i = 0 of the Biliński formula AlO_4α(OH)_8β[Al_3+δ(OH)_(10+4δ)ε(OH₂)_{s(3+δ)−(20+8δ)ε−2(α+β)}]₄^z+, where δ_i = {[0, 1], twice}_i=1⁴, s = s_v = 6. See Fig. 3.

2.6.6.3 Fyodorov case. For the rhombic icosahedron F₂₂(5 + 5) = 20 attributed to Fyodorov,¹¹ were it to exist, the specific formula would in all probability become AlO_4α(OH)_8βAl₁₀O_11α(OH)_22β+20ε[[Al(OH)_4ε]₃]₅(OH₂)_{22s−160ε−30(α+β)}^z+ or Al₂₆O_15α(OH)_30β+80ε(OH₂)_{22s−160ε−30(α+β)}^{[78−10(3α+3β+8ε)]+}. Again, see Table 3. Importantly, the Fyodorov rhombic icosahedron F_ρI(5 + 5) is also subject to Penrose enlargement; not as an independent species, but as the ρI part of a structural pattern characteristic of aperiodic 3-d tiling (see below, Section 2.6.8 Penrose in 3-d).

2.6.7 Polyhedral growth.
2.6.7.1 Stellar nucleus. For a stellar polyhedral crystal to be nucleated (with any χ) it appears sufficient for the additive species to add to faces of the nuclear species, e.g., F_V(28)(28) + 12 = F₁₄(4) + 6F₁₄(4) − 24. Growth might continue in this mode unless perhaps or until it were ever to become disallowed volumetrically (reminiscent of the term, “steric hindrance”!). A general formulation would comprise an indeterminate number of indefinite series, possibly complicated by branching. However, the special case of Ogawa's flower, which maintains a definite shape while still growing, is of particular interest (see below, Section 2.6.8 The growing flower).
2.6.7.2 Convex nucleus. To nucleate a convex polyhedron, however (χ = 2), the original edges must also be closed off, besides the faces. For example, if in addition to the faces all edges of the F_V(4) species were to be enclosed, then nucleation could initiate a Haüy growth pattern¹³ of the form x^d + ρF_V∑i² (where 0 ≤ ρ ≤1, and ρF_V is either zero or a positive integer), such as F_V(H(x))(H(x)) + 2h(x) = F_V(4) + h(x)F₁₄(4) − 4h(x), in which . In this example, F_V=14(4) = 12, ρ = 1/6, ρF_V = 2, d = 2 and H(x) = 4(h(x) + 1).

Such tiling by rhombic dodecahedra is in broad terms a 3-d counterpart of the 2-d tiling by hexagons, G(x). (See above, Section 2.2.4.) The first occurrence (x = 1) would be F_V(76)(76) + 36 = F₁₄(4) + [(3² − 1) + 2(1² + 2²)]F₁₄(4) − 72; the second (x = 2), F_V(340)(340) + 168 = F₁₄(4) + [(5² − 1) + 2(1² + 2² + 3² + 4²)]F₁₄(4) − 336; and so on.

A forerunner of this example is the classical cubic construction¹³ with the form , where d = 3. Ultimately, if a crystal were to be nucleated with this geometry, its growth could be expected to be asymptotic in the limit_x→∞ h(x) + 1.

2.6.7.3 Classical cubic model. If a species F₈(1) were to nucleate a rhombohedral species F_V(x³)=8(x³) = 6 with no facial growth (ρ = 0) and all six faces congruent – such as a cube, or golden rhombohedron – this would have (x + 1)³ cations, one at each of the 2³ vertices and the (x − 1)³ closed vertices. Of the remaining 6(x² − 1) cations, 12(x − 1) would be situated on the edges of the “cube” and 6(x − 1)² on its faces. Each corner unit of the V(x³) corners would present 3 open edges; and every other edge unit, 1: a total of 3V + (F + V − χ)(x − 2) or 12x open edges. On each face, the x² facial units would each present one open face: 2F_Vx(x − 1) additional “facial edges” overall. In total, then, the F_V(x³)(x³) species would present 24 + 12(x − 2) + 12x(x − 1) or 12x² open edges. All other edges would have been internalised.

In the F_V(x³)(x³) species the x³ units would bear a total of 3x³ + 6x² + 3x edges. Subtracting 12x², the number of internalised edges is 3x(x − 1)². The golden rhombohedral formula then becomes Al_(x+1)³O_{6αx(x−1)²}(OH)_{6βx(x−1)2+24εx²}(OH₂)_{4[s(3x−1)−12εx²]}^z+, where z = 3(x + 1)³ − 6(2α + β)x(x − 1)² − 24εx². Or expanding, , where:

m₀ = 9 − 24s, z₀ = 1

m₁ = 27 + 72s + 32α + 34β − 288ε, z₁ = [3 − 2(2α + β)]

m₂ = 27 − 64α − 68β + 136ε, z₂ = [3 + 4(2α + β) − 8ε]

m₃ = 9 + 32α + 34β, z₃ = [1 − 2(2α + β)].

See Table 3.

Table 3 Calculation of Al³⁺ m/z values for 3-d space-fillers

Special case	n	p	q	r	s	m	z	m/z	Text reference
Polyhedron (3-d space-filler, non-prismatic)									2.6.4–7
including
Non-Penrose: classical cubic (or rhombohedral)									2.6.7
	(x + 1)^3	6αx(x − 1)^2	6βx(x − 1)^2 + 24εx^2	4[s(3x − 1) − 12εx^2]	4 or 6	27n + 16p + 17q + 18r	3(x + 1)^3 − 6(2α + β)x(x − 1)^2 − 24εx^2	Both m and z are cubic in x:
of which the single rhombohedron (including the cube) is the special case, x = 1									2.6.7
	8	0	24ε	8(s − 6ε)	4 or 6		24(1 − ε): neutral if ε = 1	m/z = (9 + 6s − 19ε)/(1 − ε) e.g. s = 6, ε = 2/3
Rhombic dodecahedron – Biliński, ρD									2.6.6
	C + V = CρD(2 + 2) + VρD(2 + 2) = 1 + 14	αD = αDρD(2 + 2) = 4α	2βD + 2εE = 2βDρD(2 + 2) + 2εEρD(2 + 2) = 8(β + 6ε)	sV − 4εE − 2(α + β)D = 14s − 8(α + β) − 96ε	4 or 6	27n + 16p + 17q + 18r	45 − 8(α + β + 6ε)	If s = 6, then m/z = (1917 − 80α − 8β − 912ε)/[45 − 8(α + β + 6ε)]
Non-Penrose: non-classical – Biliński rhombic dodecahedra									2.6.7
	C + V = 4H(x − 1) + 8(4x^2 + 3x + 2), where	D = 4(H(x − 1) + 2(8x^2 + 1)); 2αD = 8α(H(x − 1) + 2(8x^2 + 1))	E = 8H(x − 1) + 8(6x^2 + 9x + 2); 2βD + 2εE = 8(β + 2ε)(H(x − 1)) + 16(8β + 6ε)x^2 + 144εx + 16(β + 2ε)	s(3H(x − 1) + 2(8x^2 + 12x + 7)) − 4ε(8H(x − 1) + 8(6x^2 + 9x + 2))	4 or 6	27n + 16p + 17q + 18r	Again both m and z are cubic in x; z = 4[3 − 2(2α + β + 2ε)]H(x − 1) + 24(4x^2 + 3x + 2) − 16(2α + β)(8x^2 + 1) − 16ε(6x^2 + 9x + 2)	Again, cubic: Biliński coefficients {(m, z)}i=0^3 differ from the classical cubic. If (e.g.) α = 0, β = 1, ε = 2/3, then m/z = 1301/5 or 260.2
Penrose in 3-d – the assembly									Table 4
	(C + V){[φ+, φ−] + ρ[φ+, φ−] + 3ρI}	αD {[φ+, φ−] + 3ρI}	2βD{[φ+, φ−] + 3ρI} + 2εE{[φ+, φ−] + ρ[φ+, φ−] + 3ρII}	sV − 4(βD + εE)	4 or 6	27n + 16p + 17q + 18r	Potentially comprises an aluminal ϕ+ species paired with a corresponding ϕ− species	[m1(2) + τ^−1m2(2)]/[z1(2) + τ^−1z2(2)], in the limit as n (in Φn) tends towards infinity
including
Ogawa's flower, {ϕ+, ϕ−}									Table 5
	(C + V){ϕ+, ϕ−}	αD{ϕ+, ϕ−}	2βD{ϕ+, ϕ−} + 2εE{ϕ+, ϕ−}
Fyodorov: the ρI component (cf. Biliński, ρD above)									2.6.6
	C + V = CρI(5 + 5) + VρI(5 + 5) = 4 + 22	αD = αDρI(5 + 5) = 15α	2βD + 2εE = 2βDρI(5 + 5) + 2εEρI(5 + 5) = 30β + 80ε	sV − 4εE − 2(α + β)D = 22s − 30(α + β) − 160ε	4 or 6	27n + 16p + 17q + 18r	78 − 10(3α + 3β + 8ε)	If s = 6, then m/z = (3078 − 300α − 30β − 1520ε)/[78 − 10(3α + 3β + 8ε)]
Gap-filling rhombohedra ρ{ϕ+, ϕ−}, with contributions to {m, z} reduced by vertex-, edge- or face-sharing									2.6.8
	Reduced from 8	0	Reduced from 24ε	Reduced from 8(s − 6ε)

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2.6.7.4 The golden rhombohedron. The two distinct golden rhombohedra F₈(1), prolate and oblate (sometimes called “acute” and “obtuse”, as if they were 2-dimensional), form one important special case (x = 1): Al₈(OH)_24ε(OH₂)_8(s−6ε)^24(1−ε)+. Charge-neutrality would arise if ε = 1: [Al(OH)₃]₈. Again, see Table 3.
2.6.7.5 Biliński tiling model. If in this model of nucleation, the rhombic dodecahedron F₁₄(4) is now substituted for the primitive cube or rhombohedron F₈(1), the model may support 3-d tiling of a Penrose or non-Penrose kind. The F₁₄(4) species is not itself primitive; it has G₁₄(4) or 4 internal edges already: D_internal = 4. If the Biliński species F₁₄(2 + 2) were to nucleate a crystal of the kind F_V(H(x))(H(x)), then the latter would again have cations from corner, edge, face and interior units. Additionally, because x = 4 initially, each unit would have already come with its own single internalised cation: C_internal = 1. (See above, Fig. 2.)

The number of Biliński components , or 2[1² + 2² + 3² +… + (2x)²] + (2x + 1)². Of these, H(x − 1) are interior units. Subtracting, the number of facial units, including corner, edge and facial dodecahedra, is not 8x(x + 1), but 2(8x² + 1) [because 2x is even]. H(0) = 1. We may refer to the subset (2x + 1)² as the one “principalSquare”, and to the other squares as “minor”. Of the 2(8x² + 1) units, 2 are polar, 4 are principalSquare corners, 4(2x − 1) are principalSquare edge units and each of the 2(2x − 1) minorSquares has 4 corners, leaving 8(2x² − 3x + 1) minorSquare edge units. Now, from the geometry:

V_interior = 3; E_interior = 8; V_polar = 5; E_polar = 12; V_{principalSquareCorner} = 5; E_{principalSquareCorner} = 10; V_{principalSquareEdge} = 2; E_{principalSquareEdge} = 6; V_{minorSquareCorner} = 2; E_{minorSquareCorner} = 6; V_{minorSquareEdge} = 1; E_{minorSquareEdge} = 3.

Multiplying out, and adding the products:

C = C_internalH(x) = H(x) = H(x − 1) + 2(8x² + 1); D = D_internalH(x) = 4H(x); V = V_interiorH(x − 1) + 2V_polar + 4V_{principalSquareCorner} + 4(2x − 1)V_{principalSquareEdge} + 8(2x − 1)V_{minorSquareCorner} + 8(2x² − 3x + 1)V_{minorSquareEdge};

V = 3H(x − 1) + 2(8x² + 12x + 7); C + V = 4H(x − 1) + 8(4x² + 3x + 2); similarly, E = 8H(x − 1) + 8(6x² + 9x + 2). Summing squares,

Unsurprisingly then, the Biliński equation for either m or z is also a cubic in x. See Table 3. The specific formula is

Al_{4H(x−1)+8(4x²+3x+2)}O_{8α(H(x−1)+2(8x²+1))}(OH)_{8(β+2ε)(H(x−1))+16(8β+6ε)x²+144εx+16(β+2ε)}(OH₂)_{s(3H(x−1)+2(8x²+12x+7))−4ε(8H(x−1)+8(6x²+9x+2))}^z+,

where z = 4[3 − 2(2α + β + 2ε)]H(x − 1) + 24(4x ² + 3x + 2) − 16(2α + β)(8x² + 1) − 16ε(6x² + 9x + 2). Or expanding again, in this instance:

m₀ = 4(81 + 67s + 32α + 34β − 76ε), z₀ = 4[9 − 2(2α + β + 2ε)]

m₁ = (4/3)[864 + 513s + 448α + 476β + 3116ε], z₁ = (16/3)[24 − 14α − 7β − 41ε]

m₂ = −16(9s − 64α − 68β − 38ε), z₂ = −32(4α + 2β − ε)

m₃ = (32/3)[57 + 27s + 64α + 68β − 152ε], z₃ = (64/3)[3 − 2(2α + β + 2ε)].

In this Biliński formulation, each & every coefficient differs from its classical cubic counterpart. Just as classical construction continues by adding a cube, here growth may continue through addition of a Biliński rhombic dodecahedron. Their space-filling capacity may not differ hugely, but the two non-Penrose tilings are distinct from each other. Compare 2-d area tiling by hexagons (see above, Section 2.2.3), in 3-d Biliński space-filling may be represented by the equation .

2.6.7.6 Non-Penrose nucleation. Despite the possible existence of a Biliński dodecahedral species comprising two unlike pairs of rhombohedra, a 3-d tiling of Haüy construction – such as the above examples – might always be regarded as non-Penrose, by dint of the degree of repetition under translation (albeit limited repetition). Either the cubic or Biliński nucleus would sit well on a classical lattice.

2.6.8 Penrose nucleation.
2.6.8.1 Penrose in 3-d. If evidence were to be found either of a rhombohedron (of either kind) or of a rhombic icosahedron, but none for a rhombic dodecahedron (after Haüy, Biliński above), then we might reasonably seek a distinct Penrose figure. Ogawa defines a 3-d Penrose transformation as a matrix operation¹⁶ on a set of space-filling polyhedra such that the enlarged Penrose figure comprises only polyhedra similar to those in the unenlarged set. The enlargement operation relies on a “self-similarity” factor. In formulating this definition Ogawa discovers a non-convex structure, again pentagon-based, with icosahedral symmetry: his “5-petalous floral dodecahedron”. (Temporarily I continue with what seems, in the absence of a sine curve, a misnomer; the structure is probably neither quinquepetalous nor dodecahedral but rather a stellated rhombic triacontahedron, a Great Stella stellation; or in our simple notation, F₆₂(20 + 0) = 5 × 12, or 60. Mathematically the triacontahedron itself, F₃₂ above, was discovered by Kepler in 1611. Its hexecontahedral stellation F₆₂ = 60 seems to have lain undiscovered until after 1940.)
2.6.8.2 Fibonacci growth. The space-filling polyhedra are taken to be either or both of the two golden rhombohedra F₈(1). In the F₆₂(20 + 0) example, all 20 are of the prolate kind; none is oblate. In the Penrose enlargement, both kinds of rhombohedron contribute: the prolate and the oblate. Ogawa discovered an enlarged oblate having two F₆₂(20 + 0) “flowers”, one at each of the two vertices of its principal diagonal, the two flowers sharing an enlarged prolate; each of the two larger rhombohedra comprising multiple smaller ones. The Penrose-nucleated crystal would grow in accordance with a pair of simultaneous equations

|G_V1(x + 1)〉 = Φ_n+1|G_V1(x)〉 + Φ_n|G_V2(x)〉

and

|G_V2(x + 1)〉 = Φ_n|G_V1(x)〉 + Φ_n−1|G_V2(x)〉,

where {Φ_k}_k=1^∞ is the Fibonacci number sequence, Φ_k+2 = Φ_k + Φ_k+1 (Φ₁ = Φ₂ = 1; here denoted Φ, as distinct from F in Euler's equation). Under the transform, n is invariant. The matrix is a 2 × 2 matrix. The two eigenstates |G_V1(1)〉 and |G_V2(1)〉 are both F₈(1): the prolate and oblate case, respectively (x = 1). In this instance, Ogawa invokes the special case Φ_n−1 = 21 and Φ_n = 34 (n = 9). The self-similarity is the golden rhombohedral edge length. See Fig. 4(a). More generally, whilst in Φ_n the value of n itself appears to depend on Fibonacci numbers Φ_r, in these Φ_r the value of r is not always determined by Φ_r and recently the series of atom cluster sizes {Φ(Φ_r)}_r=3⁶ has been recognised in Coelfen's silver nanoparticle precursors.¹⁷ (The requirement for the solution to comprise integer numbers obviates alternative formulations such as those based on exp((r − 2)/2) or its ceiling.)

Fig. 4 (a) “Penrose in 3-d”, showing self-similar enlargement. Ogawa's flower, at either end of the principal diagonal, is perhaps the most striking feature of Penrose in 3-d. The overall Penrose structure appears to be a special case n = Φ_r + 2 − i, r = Φ_r − 2 (i = 1, 2) of the Fibonacci enlargement |G_Vi(x + 1)〉 = Φ_n+1|G_V1(x)〉 + Φ_n|G_V2(x)〉. Each polyhedron is self-similar and enlarged by the linear factor, τ³ = 2 + sqrt(5); or volumetrically, τ⁹. In this diagram |G_V1(x)〉, |G_V1(x − 1)〉 and even |G_V1(x − 2)〉 may be discerned visually; though not the smaller moiety |G_V1(x − 3)〉. There is no semblance of a “unit cell” in the classical Bravais sense. The self-similar enlargement has a fractal dimension d = 3κ ln τ/ln 2. In fractal geometry this suggests asymmetric Cantor dust in 3-d, of capacity dimension κ_capacity = 3 (d is approximately 6.25). Because Φ_n+1Φ_n−1 − Φ_n² = (−1)ⁿ, it follows that |G_Vj(x + 3)〉 = (Φ_n+1 + Φ_n−1)|G_Vj(x + 2)〉 + |G_Vj(x + 1)〉 (n odd). Golden rhombohedra cement two floral Ogawa species ϕ+, ϕ− and three rhombic icosahedra ρI into a regular pentagonal shape. In the primitive state x = 1, this would potentially comprise an aluminal ϕ+ species Al₁₈₇O_12+45α(OH)_{30(4+3β)+482ε}(OH₂)_{30+100s−9[(α+β)+8ε]}^z+ where z = 417 − [90(α + β) + 482ε], paired with the ϕ− species Al₁₄₉O_12+45α(OH)_{30(4+3β)+268ε}(OH₂)_{30+70s−18[5(α+β)+28ε]}^z+ in which z = 303 − [90(α + β) + 268ε]. (b) Penrose in 3-d: aperiodicity. Ogawa's flower sits at either end of the principal diagonal of a major golden rhombohedron. The “floral” structure is here represented by a non-convex, stellar polyhedron, comprising twenty minor golden rhombohedra. (Fig. 5 appends a more precise description.) Along this diagonal, the two stellar polyhedra share one minor rhombohedron. Others – r₀, unshared – are positioned between the two. The stellations |G_62,ϕ+(x)〉 centre on the two major principal vertices: the diagonal joining these two centres is now taken to be the chord of a regular pentagon. On each of the pentagon's three remaining vertices sits a Fyodorov rhombic icosahedron, with the centre of its constituent Biliński dodecahedron positioned at the local vertex. This structure allows for five nuclei, each one equidistant from its two nearest neighbours. (Both these nuclear subtypes derive from Kepler's rhombic triacontahedron.) As described in Fig. 3, each Fyodorov icosahedron |G_ρI(x)〉 comprises the four Biliński and six extra rhombohedra of its own, all golden. Each Fyodorov is now orientated so as to allow a further r₁ + r₂ rhombohedra to be added (including three beyond the local vertex); with the remainder cementing the five gaps between nearest neighbour nuclei. Consistency with Ogawa requires |G_V1(x + 1)〉 equal to |G_62,ϕ+(x)〉 + |G_V1(x)〉 + r₀|G_V2(x)〉 + 3[|G_ρI(x)〉 + r₁|G_V1(x)〉 + r₂|G_V2(x)〉] + 5[((19 − 3r₁)/5)|G_V1(x)〉 + ((19 − r₀ − 3r₂)/5)|G_V2(x)〉]; with, for example, r₀ = 5 and r₁ = r₂ = 3: a special case of the equation |G_Vi(x + 1)〉 = Φ_n+1|G_V1(x)〉 + Φ_n|G_V2(x)〉.

Fig. 5 A stellation of Kepler's rhombic triacontahedron. A rhombic hexecontahedral stellation of the Great Stella class, this non-convex polyhedron has 62 vertices and an internal vertex, with icosahedral symmetry. Comprising 20 golden rhombohedra (all prolate), it gives a precise 3-d representation of Ogawa's flower: |G_62,ϕ+(x)〉 = 20|G_V1(x)〉; |G_62,ϕ−(x)〉 = 19|G_V1(x)〉. The stellation is shown enlarged, with self-similarity, by Penrose in 3-d: |G_V1(x + 1)〉 = |G_62,ϕ+(x)〉 + 20|G_V1(x)〉 + 19|G_V2(x)〉 + 3|G_ρI(x)〉. Given that with Shechtman icosahedral symmetry is now regarded as crystalline and that earlier an Al alloy¹⁹ was ascribed the stellar structure, it is reasonable to seek near-icosahedral aluminal species such as AlO₂[Al(OH)₅]₂[[Al₂OH₂(OH)₄]₅]₆⁵⁵⁺, with m/z = 81.5; or, as is perhaps more likely, AlO₂[Al₂O₂(OH)₂OH₂]₅[Al(OH)₅]₂[[Al₂(OH)₄OH₂]₅]₅⁴⁵⁺, where m/z = 99.4. In contrast to classical crystals generally and to Guyot's alloy, presumed dense, the fractal nature of the 3-d Penrose enlargement and its peculiar characterisation as Cantor dust suggests the possibility of a quasi-crystal which may be “nowhere dense” (my emphasis). My conjecture is that neither polyhedron of a pair {|G_V1(x)〉, |G_V2(x)〉} in which both are nowhere dense may be space-filled contiguously to every edge by similar polyhedra |G_V1(x − 1)〉 or |G_V2(x − 1)〉 in any combination (x ≥ 2). Can we then, using mass spectrometry, test for the physical existence of Cantor dust?

Ogawa constructed an aperiodic 3-d tiling with linear self-similarity factor τ³ (or volumetrically, τ⁹); τ = (1 + sqrt(5))/2. τ³ = 2 + sqrt(5). τ⁹ = 38 + 17sqrt(5).

The above equations suggest that Penrose enlargement may recur endlessly, and may be written in the form

or as the polynomial
where each coefficient C_ijn = c_ijΦ_n−1ⁱ (C = {A, B}). For example, in Ogawa's special case, Φ_n = 34 and

2.6.8.3 Mass spectrum. The formula of each of the smallest rhombohedra would be M₈P₀Q_24εR_8(s−6ε)^z+, z = 8(c − 3bε), where for instance M^c+ = Al³⁺, P^a− = O²⁻, Q^b− = OH⁻, R = OH₂: Al₈(OH)_24ε(OH₂)_8(s−6ε)^24(1−ε)+. Note that α = 0, initially; however, see below. In 3-d Penrose relations as written above there would be nothing to imply face- or edge-sharing were it not for the supposed quinquepetalousness (see above, Penrose in 3-d; and again below, The computation). Given its essential position in the structure, Ogawa's special-case equations with Φ_n = 34 are recast as

|G_V1(x + 1)〉 = |G_62,ϕ+(x)〉 + 20|G_V1(x)〉 + 19|G_V2(x)〉 + 3|G_ρI(x)〉

and

|G_V2(x + 1)〉 = |G_62,ϕ−(x)〉 + 6|G_V2(x)〉 + 3|G_ρI(x)〉,

where the rhombic icosahedron F_ρI(5 + 5) above is explicitly accommodated in the ρI enlargement (see below).

Now allowing for internal face-sharing when x = 1 (α > 0, but in the floral case we let α_ϕ = 2β_ϕ = 1), the generic formula becomes the pair of the sums of the floral (ϕ+, ϕ−), rhombohedral ρ and ρI contributors, MP₂[M₂P₂Q₂R]₅[MQ₅]₂[[M₂Q₄R]₅]₅M₈Q_28εR_4(s−6ε)[MQ_18εR_s−8ε]₄[MP_4αQ_8βM₁₀P_11αQ_22β+20ε[[MQ_4ε]₃]₅M₂Q_10εM₈Q_28εR_8s−56εR_{22s−160ε−30(α+β)}]₃M₄Q_28εR_2(s−8ε)^z+ or potentially the aluminal ϕ+ species Al₁₈₇O_12+45α(OH)_{30(4+3β)+482ε}(OH₂)_{10[3+10s−9(α+β+8ε)]}^z+, together with the ϕ− species Al₁₄₉O_12+45α(OH)_{30(4+3β)+268ε}(OH₂)_{30+70s−18[5(α+β)+28ε)]}^z+. See Table 4.

Table 4 Calculation of m/z for “Penrose in 3-d”

Special case	n	p	q	r	s	m	z	m/z	Text reference
Penrose in 3-d – the assembly									2.6.8
	(C + V){[φ+, φ−] + ρ[φ+, φ−] + 3ρI}	2αD{[φ+, φ−] + 3ρI}	2βD{[φ+, φ−] + 3ρI} + 2εE{[φ+, φ−] + ρ[φ+, φ−] + 3ρI}	sV − 4(βD + εE)
pairing
A species, denoted φ+ [deriving from Ogawa's flower, φ+]									2.6.8
	C + V = A*(x)[C(ϕ+,2) + V(ϕ+,2)] + B1*(x)[C(ϕ−,2) + V(ϕ−,2)] = νn+A*(x) + νn−B1*(x), where: A*(x) = A(A + B)^x−2 and B1*(x) = B(A + B)^x−2, in which A = Φn+1^2 + Φn^2 and B = Φn(Φn+1 + Φn−1); and where: νn+ = 187; νn− = 149. Note C = Φn−1^2 + Φn^2, also	2αD = 2α[A*(x)D(ϕ+,2) + B1*(x) D(ϕ−,2)] = νp+[A*(x)+B1*(x)], where: νp+ = 3[4 + 15α]	2βD + 2εE = 2A*(x)[βD(ϕ+,2) + εE(ϕ+,2)] + B1*(x)[βD(ϕ−,2) + εE(ϕ−,2)] = νq+A*(x) + νq−B1*(x), where: νq+ = 30(4 + 3β) + 482ε; νq− = νq+ − 214ε	sV − 4(βD + εE) = νr+A*(x) + νr−B1*(x), where: νr+ = 10[3 + 10s − 9(α + β + 8ε)]; νr− = 30 + 70s − 18[5(α + β) + 28ε]	4 or 6	27n + 16p + 17q + 18r	νz+A*(x) + νz−B1*(x), where: νz+ = 417 − [90(α + β) + 482ε]; νz− = 303 − [90(α + β) + 268ε]	m/z = [m(x)/z(x)]1 = [m1(2) + τ^−1m2(2)]/[z1(2) + τ^−1z2(2)], in the infinite limit (independent of x > 2); prior to correction for subtrahend. If α = β = ε and ε = 1/2, then m/z = 168.9.
	with
A corresponding species, denoted φ− (an approximation derived from vertex-sharing only)									2.6.8
	in which approximation (without subtrahend *)
						*An exact reduction may be obtained by formulating a precise subtrahend based on the preferred |G(2x)〉 interior (see text)
	C + V = B2*(x)[C(ϕ+,2) + V(ϕ+,2)] + C*(x)[C(ϕ−,2) + V(ϕ−,2)] = νn+B2*(x) + νn−C*(x), where: B2*(x) = B(B + C)^x−2 and C*(x) = C(B + C)^x−2	2αD = 2α[B2*(x)D(ϕ+,2) + C*(x)D(ϕ−,2)] = νp+[B2*(x) + C*(x)]	2βD + 2εE = 2B2*(x)[βD(ϕ+,2) + εE(ϕ+,2)] + 2C*(x)[βD(ϕ−,2) + εE(ϕ−,2)] = νq+B2*(x) + νq−C*(x)	sV − 4(βD + εE) = νr+B2*(x) + νr−C*(x)	4 or 6	27n + 16p + 17q + 18r	νz+B2*(x) + νz−C*(x)	m/z = [m(x)/z(x)]2, which in the limit again equals [m1(2) + τ^−1m2(2)]/[z1(2) + τ^−1z2(2)]. By the time Ogawa has been reached (n = 9), we may expect to discern a single line only. Mass spectrum independent of G(2x), the extent of 3-d tiling

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In this formulation the first M₈Q_28ε component derives from five rhombohedra (r₀) positioned between the two Ogawa flowers, while the M₂Q_10ε and M₈Q_28ε together derive from six (r₁ + r₂) added to each ρI; with M₄Q_28ε from three in the gap between the two ρI neighbours (see text below Fig. 4(b)). The [MQ_18ε]₄ component allows for three rhombohedra to cement each of the other four gaps between floral and ρI nuclei, where face-sharing necessitates only one additional vertex but nine additional edges. See Fig. 4(b).

For the Ogawa case Φ_n = 34 then, the result is a pair of expected spectral lines {(m/z)_j}_j=1², where the Al³⁺ values in this first instance (x = 1) are (m/z)₁ = [7821 + 1800s − (900α + 90β + 4766ε)]/[417 − [90(α + β) + 482ε]], and (m/z)₂ = [6795 + 1260s − (900α + 90β + 4992ε)]/[303 − [90(α + β) + 268ε]]; again, see Table 4. This pair derives from the primitive state: |G_V1(2)〉 and |G_V2(2)〉 (x = 1).

Alternatively, and without invoking Ogawa's flower, (m/z)₁ and (m/z)₂ could each be computed as the rational quotient m_j(Φ_n)/z_j(Φ_n) of paired polynomials {m_j(Φ_n), z_j(Φ_n)} of degree x in Φ_n; though to do so might imply the property α = 0, suggesting (dubiously for alumina) an absence of O²⁻ ligands. For this reason, it is physically significant to retain the floral and ρI characteristics explicitly.

2.6.8.4 The growing flower. The basic formula of Ogawa's flower F_62,ϕ+(20 + 0)(x = 1) is MP₂[M₂P_2αQ_4βR]₅[MQ₅]₂[[M₂Q₄R]₅]₅^z+. If this were to exist as an aluminal species, for example, its basic formula would be AlO₂[Al₂O_2α(OH)_4βOH₂]₅[Al(OH)₅]₂[[Al₂(OH)₄OH₂]₅]₅^z+ or Al₆₃O_2+10α(OH)_10(11+2β)(OH₂)₃₀^{[75−20(α+β)]+}. See Table 5. Correspondingly, F_62,ϕ−(19 + 0) might be formulated by subtracting from F_62,ϕ+(20 + 0) a single (prolate) rhombohedron, F₈(1), or M₈Q_24εR_8(s−6ε) or Al₈(OH)_24ε(OH₂)_8(s−6ε). However, if as expected there is vertex-, edge- or face-sharing by the rhombohedron, the subtrahend will be correspondingly smaller. For example, if three faces that in F_62,ϕ+(20 + 0) were shared are no longer shared in F_62,ϕ−(19 + 0), then Al₇(OH)_18ε(OH₂)_7(s−6ε) vanishes from the subtrahend, leaving the latter representing a single Al³⁺ cation with its 6ε hydroxyl (or for arithmetic purposes “hydroxide”) and s − 6ε water components. It will be assumed that, in the flower at least, this face-sharing does occur. Similarly, E_62,ϕ− and V_62,ϕ− may be computed.

Table 5 Calculation of m/z for a Great Stella stellation

Special case	n	p	q	r	s	m	z	m/z	Text reference
Non-convex, stellar polyhedron (Penrose in 3-d, stellation of Kepler's rhombic triacontahedron)									2.6.8
	(C + V){φ+, φ−}	αD{φ+, φ−}	2βD{φ+, φ−} + 2εE{φ+, φ−}
comprising
Ogawa's flower, φ+									2.6.8
	n62,ϕ+ = C + V = C62,ϕ+(20 + 0) + V62,ϕ+(20 + 0) = 1 + 62 = 63	D = D62,ϕ+(20 + 0) = 50; however, p62,ϕ+ = p62,ϕ = 2(1 + 5αϕ) – see basic formula	E = E62,ϕ+(20 + 0) = 120; however, q62,ϕ+ = 5(2 + 4(5 + βϕ)), or 10(11 + 2βϕ)	r62,ϕ+ = 5(1 + 5) = 30	6	27n62,ϕ+ + 16p62,ϕ+ + 17q62,ϕ+ + 18r62,ϕ+ = 4143 + 160αϕ − 340βϕ	75 − 20(αϕ + βϕ)	m/z = (4143 + 160αϕ − 340βϕ)/[75 − 20(αϕ + βϕ)] or if αϕ = 2βϕ then (1381 + 220βϕ)/5(5 − 4βϕ); e.g., if βϕ = 1/2, then m/z = 2982/30 or 99.4. Or if αϕ were 0 and βϕ were 1, then m/z would be 4483/55 or 81.5
paired with
a corresponding species, φ− (with subtrahend derived from assumed triple face-sharing)									2.6.8
	F = F62,ϕ−(19 + 0) = F62,ϕ+(20 + 0) − 3 = 60 − 3 = 57
from which subtrahend may be derived
	n62,ϕ− = C + V = C62,ϕ−(19 + 0) + V62,ϕ−(19 + 0) = C62,ϕ+(20 + 0) + V62,ϕ+(20 + 0) − 1 = n62,ϕ+ − 1 = 63 − 1 = 62	p62,ϕ− = p62,ϕ (which remains unchanged)	q62,ϕ− = q62,ϕ+ − 6εϕ = 10(11 + 2βϕ) − 6εϕ	r62,ϕ− = r62,ϕ+ − (s − 6εϕ) = 30 − (s − 6εϕ) = 6(4 + εϕ)	6	27n62,ϕ− + 16p62,ϕ− + 17q62,ϕ− + 18r62,ϕ−

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Table 6 A reinterpretation of m/z values observed^7,8 by ESI mass spectrometry in aqueous Al³⁺ solutions

m/z (observed)	Interpretation (most likely)							Possible ambiguity?
	Symmetry	n	p	q	r	z	Species
79	Td	2		4	2	2	Al2(OH)4(OH2)2^2+
97	Td	1		2	2	1	Al(OH)2(OH2)2^+	Y	Aln(OH)2n(OH2)2n^n+
									including
									n
									2	Dimer
										Al2(OH)4(OH2)4^2+
									4	Tetramer
										Al4(OH)8(OH2)8^4+
									6	Hexameric ring
										Al6(OH)12(OH2)12^6+
									8	Rhombohedron
										Al8(OH)16(OH2)16^8+
									(or similar formulation)
									OR
									AlO4α(OH)8β[Al3(OH)2(5+2δ)ε(OH2)6(3+δ)−4(5+2δ)ε−(α+2β)]4^z+, where α = 2β = 1/2, δ = 0, ε = 1/2
									which equals
									n
									13	Tridecamer
										AlO2(OH)2[Al3(OH)5(OH2)7]4^13+
115	Oh	2		4	6	2	Al2(OH)4(OH2)6^2+
133	Oh	1		2	4	1	Al(OH)2(OH2)4^+	Y	[An unlikely hexamer]
157	Td	2		5	1	1	Al2(OH)5(OH2)^+
217	Td	3		8		1	Al3(OH)8^+
277	Td	4	4	3	3	1	Al4O4(OH)3(OH2)3^+
337	Td	5	4	6	2	1	Al5O4(OH)6(OH2)2^+

---

A similar flower growing under the Penrose transformation, however, would now comprise not simply 20 prolate rhombohedra, or 19; but 20 enlarged prolate; or, pairwise, 19 enlarged prolate rhombohedra:

|G_62,ϕ+(x)〉 = 20|G_V1(x)〉: |G_62,ϕ−(x)〉 = 19|G_V1(x)〉.

See Fig. 5.

2.6.8.5 Fractal geometry. If Ogawa's flower |G_62,ϕ+(x)〉 is really the enlarged rhombicHexecontahedron ρH, such that |G_ρH(x)〉 = 20|G_V1(x)〉, then for edge length a₁ its most primitive volume will be V₁ = 4a₁³sqrt[2(5 + sqrt(5))]. Its enlargement by the linear self-similarity factor τ³ gives rise to a volume τ⁹ times greater: V₂ = 4τ⁹a₁³sqrt[2(5 + sqrt(5))]. The fractal dimension d = [ln(V₂) − ln(V₁)]/[ln(2) − ln(1)] = 9 ln τ/ln 2. Again, see Fig. 4(a).

This fractal characteristic of Penrose in 3-d closely resembles asymmetric Cantor dust¹⁸ in 3-d. As this is nowhere dense, it may suggest space filling that is less than contiguous. This is my conjecture.

2.6.8.6 The ρI enlargement. Again, under Penrose the enlarged rhombicIcosahedron ρI would now comprise not 10 rhombohedra (5 + 5), but 5 enlarged prolate + 5 enlarged oblate rhombohedra: |G_ρI(x)〉 = 5|G_V1(x)〉 + 5|G_V2(x)〉.
2.6.8.7 The computation. Bringing together the above pair of equations |G_Vj(x + i)〉, where i is a small number (j = 1, 2), we have for Penrose in 3-d

Φ_n−1|G_V1(x + 2)〉 − Φ_n+1Φ_n−1|G_V1(x + 1)〉 = Φ_n|G_V2(x + 2)〉 − Φ_n²|G_V1(x + 1)〉,

or

(Φ_n² − Φ_n+1Φ_n−1)|G_V1(x + 1)〉 = Φ_n|G_V2(x + 2)〉 − Φ_n−1|G_V1(x + 2)〉.

Now Φ_n+1Φ_n−1 − Φ_n² = (−1)ⁿ; from this, it follows that |G_Vj(x + 3)〉 = (Φ_n+1 + Φ_n−1)|G_Vj(x + 2)〉 + |G_Vj(x + 1)〉 (n odd). A special case of this relation is |G_Vj(4)〉 = (Φ_n+1 + Φ_n−1)|G_Vj(3)〉 + |G_Vj(2)〉.

Expanding |G_V1(3)〉, |G_V1(4)〉 = [(Φ_n+1² + (Φ_n+1Φ_n−1 + 1)]|G_V1(2)〉 + Φ_n(Φ_n+1 + Φ_n−1)|G_V2(2)〉. More generally for x ≥ 2, |G_V1(2x)〉 is the binomial expansion

so that |G_V1(2x)〉 = (A + B)^x−2[A|G_V1(2)〉 + B|G_V2(2)〉], where A = Φ_n+1² + Φ_n² and B = Φ_n(Φ_n+1 + Φ_n−1). Similarly |G_V1(2x + 1)〉 = (A + B)^x−2 [A|G_V1(3)〉 + B|G_V2(3)〉]. |G_Vj(2x)〉 and |G_Vj(2x − 1)〉 are linear combinations of |G_V1(2)〉 and |G_V2(2)〉. For the |G_Vj(2x)〉 pair therefore (x > 1), if the major |G_Vj(2)〉 rhombohedra share a M^c+ cation at a vertex only, a straight multiplication may be written (with no subtraction for edge- or face-sharing). See Table 4.

In the aluminal case for example, where again the Al³⁺ cation is shared only at a vertex, the ϕ+ species would be

paired with the ϕ− species

Since (A + B)^x−2 = A*(x)/A, B₁*(x) = (B/A)A*(x) and therefore [m(x)/z(x)]₁ = [m₁(2) + (Φ_2n/Φ_2n+1)m₂(2)]/[z₁(2) + (Φ_2n/Φ_2n+1)z₂(2)] = 168.9, independent of x. While Φ_2n/Φ_2n−1 is not equal to Φ_2n+1/Φ_2n below infinite n (the values oscillate), in the limit the ratio tends towards τ.

Either value [m(x)/z(x)]_i = [m₁(2) + τ⁻¹m₂(2)]/[z₁(2) + τ⁻¹z₂(2)], approximately (independent of x > 2). Again, [m(x)/z(x)]₂ = 168.9. By the time n = 9 (Ogawa) has been reached, we may expect to discern a single line only.

Should Penrose in 3-d exist as a pair of physical species, its mass spectrum would be independent of G(2x), the extent of the 3-d tiling development. This asymmetric form of Cantor dust, the Ogawa solution, is unlike any other crystal nucleus so far examined.

2.6.8.8 A subtrahend. The above approximation for [m(x)/z(x)]_i is predicated on the absence of all face- and edge-sharing between one primitive major |G_V1(2)〉 or |G_V2(2)〉 and another. Any sharing beyond that of a single vertex necessitates subtraction of numbers from m₁(2) and m₂(2), and the re-computation of z₁(2) and z₂(2). Again however either [m(x)/z(x)]_i value would be x-independent, and would still approximate to [m^*₁(2) + τ⁻¹m^*₂(2)]/[z^*₁(2) + τ⁻¹z^*₂(2)], in which m^*₁(2) now equals m₁(2) − σ₁(2); also m^*₂(2), correspondingly.

If, for each Fyodorov icosahedron |G_ρI(1)〉 in the |G_V1(2)〉 primitive, three golden rhombohedra – those beyond the local vertex – share faces with those in an adjoining |G_V1(2)〉 major, then the subtrahend σ₁(2) will derive from three M₄Q_28ε moieties, or three units of Al₄(OH)_28ε(OH₂)_2(s−8ε)^4(3−7ε)+. If σ₂(2) derives from one rhombohedron for each ρI, then this will represent MQ_6ε or Al(OH)_6ε(OH₂)_s−6ε^3(1−2ε)+.

If ε = 1/2 then [m^*₁(2) + τ⁻¹m^*₂(2)]/[z^*₁(2) + τ⁻¹z^*₂(2)] would equal 151, approximately (in contrast to the value from vertex-sharing alone). The question of face-, edge- and vertex-sharing between major rhombohedra assumes a critical importance.

2.6.8.9 A reduced subtrahend. It may be asserted that an exterior major rhombohedron |G_Vj(2x + 1)〉 is unlike an interior major, in that there is a limit to the extent of possible edge- and face-sharing (cf. Biliński tiling, above). The exterior major has no adjoining major |G_Vj(2x + 1)〉 in which its constituent rhombohedra |G_Vj(2x)〉 could share. The potential subtrahends σ₁(2x) and σ₂(2x) are diminished.

Because |G_V1(2x + 1)〉 comprises (Φ_n+1 + Φ_n−1) |G_V1(2x)〉 + |G_V1(2x − 1)〉, of which the interior comprises a single |G_V1(2x)〉, the interior fraction of rhombohedra is approximately (Φ_n+1 + Φ_n−1)⁻¹; consequently we may now expect m₁(2) to be decreased by the reduced subtrahend σ^*₁(2) = (Φ_n+1 + Φ_n−1)⁻¹σ₁(2), a reduction to the sharing fraction only. Similarly, σ^*₂(2) = (Φ_n+1 + Φ_n−1)⁻¹σ₂(2). Now [m^*₁(2) + τ⁻¹m^*₂(2)]/[z^*₁(2) + τ⁻¹z^*₂(2)] = 168.6, approximately.

Introducing the complication of non-sharing exterior rhombohedra appears to have restored the spectral line pair position to one more closely resembling the “vertex-only” result.

2.6.8.10 Exact reduction. Since |G_V1(2x + 1)〉 = (A + B)^x−2[A|G_V1(3)〉 + B|G_V2(3)〉], a precise subtrahend could be obtained by regarding the interior as a single |G_V1(2x − 1)〉, for then the interior fraction would be (A + B)^x−3/(A + B)^x−2, or 1/(A + B). Now σ^*₁(2) = σ₁(2)/(A + B). But A + B = 6765, and B + C = 4181: huge reduction factors compared with Φ_n+1 + Φ_n−1, which equals 76. (Φ_n+1 + Φ_n−1)⁻¹ = 0.0131579. A |G_V1(2x)〉 interior must be preferred. This comprises (A + B)^x−2[A|G_V1(2)〉 + B|G_V2(2)〉]. The fraction is x-independent, and equals [A|G_V1(2)〉 + B|G_V2(2)〉]/[A|G_V1(3)〉 + B|G_V2(3)〉]. Comparing like |G_Vj(2)〉 components, the two interior fractions are (Φ_n+1² + Φ_n²)/[Φ_n+1(Φ_n+1² + Φ_n²) + Φ_n²(Φ_n+1 + Φ_n−1)] and Φ_n(Φ_n+1 + Φ_n−1)/[Φ_n(Φ_n+1² + Φ_n²) + Φ_n−1 Φ_n(Φ_n+1 + Φ_n−1)]. Each fraction is equal to 0.0131556, which does differ slightly from (Φ_n+1 + Φ_n−1)⁻¹, though only by <200 ppm and not in the first 5 significant figures.
2.6.8.11 The emperor's new crystal?. Should crystalline evidence be found for Penrose nucleation and growth, any such structure would most certainly have been regarded as “amorphous” in the pre-Penrose era (we could say “X-ray amorphous”). No crystal lattice is implied. Perhaps the more disordered aperiodic tiling may be favoured – over Biliński, say – by the entropic part of its Gibbs free energy. As crystal growth tends to the infinite, however, then in the limit_n→∞(Φ_n/Φ_n−1) = τ. (Given the icosahedral symmetry,¹⁶ this limiting case appears fully consistent with non-repetition under translation, without any semblance of “unit cell”.)

Also, because at infinity the alternating series of inverse products 1/Φ_iΦ_i+1 converges to a value of τ − 1, an interesting question may arise: whether some Ogawa flowers grow less than others, and can wither as well as grow; whether there exists a Penrose transform under which a series or even an infinity of successive Fibonacci pairs {Φ_i−1, Φ_i}_i=1^∞ may be found [besides Ogawa's two successive pairs with n fixed in the crystal nucleus]; and, if so, whether τ – the infinite series – might represent a seemingly more jumbled-up or possibly truly chaotic precipitate;¹⁴ or now perhaps, Wakisaka's solution.⁶

2.6.9 Stochastic nucleation. If rhombohedral species Al₈(OH)_24ε(OH₂)_8(s−6ε)^24(1−ε)+ were to exist, but without immediately nucleating a well-ordered 3-d growth pattern such as a space-filler or Penrose moiety, two or more species could perhaps connect to one another through a shared vertex, edge or face. This could produce species seemingly at random such as Al[Al₇(OH)_24ε(OH₂)_(s−6ε)7]₂^{3(15−16ε)+}, Al₂O_2α(OH)_2β(OH₂)_{(s−(2β+8ε))2}[Al₆(OH)_22ε(OH₂)_(s−6ε)6]₂^{2[21−(2α+β+22ε)]+}, or Al₄O_8α(OH)_8β(OH₂)_{(s−(4β+4ε))4}[Al₄(OH)_16ε(OH₂)_(s−6ε)4]₂^{4[9−2(2α+β+4ε)]+}, and so on: including, perhaps, Biliński (from Keggin) and the stellation of Kepler's triacontahedron; any or all of which could develop into well-ordered crystal nuclei. Ultimately, however, we should not be surprised to discover some degree of structuring, even in an X-ray amorphous precipitate.

Adopting the general formula M_VQ_2εER_8(s−6ε)^(3V−2εE)+, the rhombohedron (having E = 12, V = 8) may be seen as merely one special case of a more general polyhedral formulation M_cVP_aαDQ_2bβD+2eεER_sV−4εE^{[3cV−2(aα+bβ)D−2eεE]+}, with a = 0, b = 0, c = 1, e = 1. A Poisson distribution may be expected!

2.6.10 N-Dimensional growth?. The n-dimensional hypercube, or n-cube, has 2ⁿ vertices and n2ⁿ⁻¹ edges. A general formulation is M_C+VP_αDQ_2β(D+E)R_{sV−4β(12+V)}^z+. If an aluminal species were to exist with such a structure, then approximating, z = 2ⁿ3 − [α(n − 1) + 2βn]2ⁿ⁻¹. See Table 2.

Charge-neutrality would arise when β = 3/n (if α = 0); or, more generally, when n=(6 + α)/(α + 2β), e.g. in the hexeractic [Al(OH)₃]₆₄(OH₂)₈ (α = 0, β = 1/2, s = 6). The icosahedritic icosahedron (see above, The regular icosahedron) has been represented as the 3-d projection of a 6-cubic structure.

The 3-cube may be regarded a special case of the rhombohedron with half-angle θ = π/4, or cot⁻¹(1). If an aluminal species were to exist with such an unlikely structure then again, if the sole difference were to be the direction of its crystal growth, this species too would remain indistinguishable by mass spectrometry. The 6-cube, however, might readily be distinguished as a special case of the n-cube described above.

2.7 Other species

Many kinds of 3-dimensional growth could potentially be modelled, with or without hexagonal rings; but including prismatic systems such as the monoclinic (e.g., incipient bayerite or gibbsite crystals) or orthorhombic. Any or all of these special cases could be the subject of further symmetry-based computations. Close packing may not be assumed. However, where it does exist, spherical expansion may give rise to space-filling polyhedra such as the hexagonal prism or rhombic dodecahedron. Spectral evidence could perhaps be sought for tessellated space or hyperspace. (In alumina, the 3-cube itself seems intuitively unlikely. However, given the icosandritic 6-cube projection it could be interesting to ask which symmetries besides the icosahedral are 3-d projections of an N-dimensional polytopic crystal, and what if any others might be expected.)

Formulation of the number of ligands p + q + r would again be determined by the specific polyhedron or polytope (again, s would equal 6 or 4). In the solid or 3-d geometry, vertex, edge, facial and internal polyhedra would differ from one another, as would their ligand numbers p, q, r.

To reduce gaps in the overall range of computed m/z values, and so improve interpretation of the mass spectra, there would be merit in performing a fuller range of such computations at an early stage. For the present, however, only the specific 3-dimensional arrays above have been selected (and so there may still be gaps). Despite the gaps, then, accepting the Group's interpretation⁷ of the interval of 18 between certain (m/z) values, few of the Group's observations^7,8 remain unexplained. As is sometimes the case, however, there are minor spectral lines, less intense, still requiring explanation.

3. Results and discussion

Table 6 shows a fresh interpretation of the observations made by Urabe and his group.^7,8 In their experiments, the intensity of the strong mass spectral line at m/z = 97 appears to provide good evidence that in the chloride solutions, AlCl₃ (described in the group's 2007 paper⁷), the monomer exists as a cation in a tetrahedral symmetry: Al(OH)₂(OH₂)₂⁺ (m = 97). (However see below, m/z = 133.) It is possible that, with no loss of symmetry, this species could readily give rise to an aluminate anion (observed by the same Group in 2011 (ref. 8)):

Al(OH)₂(OH₂)₂⁺ + 2OH⁻ ⇔ Al(OH)₄⁻ + 2H₂O

Dimeric cations appear to exist both in tetrahedral symmetry as Al₂(OH)₄(OH₂)₂²⁺ and Al₂(OH)₅(OH₂)⁺ (with lines at m/z = 79 and 157, respectively) and in octahedral symmetry as Al₂(OH)₄(OH₂)₆²⁺, with the mass spectral line of the latter at m/z = 115 by far the most intense of the three.

Trimeric cations appear to be tetrahedral only, e.g. Al₃(OH)₈⁺ (m/z = 217). It may be that the octahedral trimer has been converted into a more stable tetrahedral tetramer or pentamer (m/z = 277 and 337, respectively):

Al₃(OH)₈(OH₂)₆⁺ + Al(OH)₂(OH₂)₂⁺ + OH⁻ ⇔ Al₄O₄(OH)₃(OH₂)₃⁺ + 9H₂O

Al₄O₄(OH)₃(OH₂)₃⁺ + Al(OH)₂(OH₂)₂⁺ + OH⁻ ⇔ Al₅O₄(OH)₆(OH₂)₂⁺ + 3H₂O

Or dimer could react with dimer to form a tetramer directly (so bypassing the trimeric stage):

2Al₂(OH)₄(OH₂)₆²⁺ + 3OH⁻ ⇔ Al₄O₄(OH)₃(OH₂)₃⁺ + 9H₂O

Structural rearrangements between tetrahedral and octahedral symmetry are also not impossible, especially under the moderately acidic conditions pertaining in (e.g.) an Al₂(SO₄)₃ solution. In the pentamer, both cases (s = 4, m/z = 337 and s = 6, m/z = 173) are just visible, although the octahedral line is not at all intense:

Al₅O₄(OH)₆(OH₂)₂⁺ + 8H₂O + 2H₃O⁺ ⇔ Al₅(OH)₁₂(OH₂)₁₀³⁺.

Yet there remains the further possibility, especially in the nitrate solutions, Al(NO₃)₃ (described in the Group's 2011 paper⁸), that the trimer could have vanished under nucleophilic (S_Nx) attack by successive hydroxyl groups of the Al(OH)₄⁻ anion:

Al₃(OH)₈(OH₂)₆⁺ + Al(OH)₄⁻ ⇔ [Al₃(OH)₈(OH₂)₅][Al(OH)₄] + H₂O

[Al₃(OH)₈(OH₂)₅][Al(OH)₄] + Al₃(OH)₈(OH₂)₆⁺ ⇔ [Al₃(OH)₈(OH₂)₅]₂[Al(OH)₄]⁺ + H₂O…

[Al₃(OH)₈(OH₂)₅]₃[Al(OH)₄]²⁺ + Al₃(OH)₈(OH₂)₆⁺ ⇔ [Al₃(OH)₈(OH₂)₅]₄[Al(OH)₄]³⁺ + H₂O.

The tetrahedral structure of the tridecamer could stabilise the octahedral trimer chain particularly well, offering plenty of space for chloride or nitrate ions to surround it. This would explain its long-term persistence reported in quasi-amorphous alumina.⁸ Longer chains than the trimer could be similarly held (and n could be variable); but if formed from the trimer by accretion of the monomer, the tetramer (and therefore pentamer) might be relatively slow to appear. A hydroxyl group in the cationic monomer would be expected to carry a lesser negative charge density than the corresponding group in the aluminate anion. Assuming p = 0, q ≥ 2(n − 1) for simplicity, no evidence exists for a simple chain of length n = 13; nor indeed for any linear instance of n > 6 (save three unlikely exceptions with n > 10).

Computation of m/z for the hexameric ring Al₆(OH)₁₂(OH₂)₁₂⁶⁺ postulated on the basis of octahedral symmetry produces the value m/z = 97 (m = 582, z = 6). This value is indistinguishable from the value ascribed to the unipositive tetrahedral monomer, Al(OH)₂(OH₂)₂⁺. However, the spectral lines at m/z = 154 and 199, ascribable to linear hexamers, are exceedingly weak. The ring could be more stable than the chain! But because of the ambiguous interpretation of m/z = 97, its existence is neither supported nor disproven. Indeed, the observed m/z values of 154 and 199 distinguish a hexameric chain from the corresponding ring.

To disprove the existence of the ring, it would become important to know whether only unipositive cations may be found experimentally (as the group reported in 2007 (ref. 7)); or whether other cations may also be found, besides (as reported in 2011 (ref. 8)).

Amongst the observed spectral values there is, besides m/z = 97, one other instance of ambiguous symmetry: m/z = 133. This could be ascribed to a hexamer comprising six tetrahedra; however, it seems far more likely that m/z = 133 should be ascribed to the octahedral monomer, Al(OH)₂(OH₂)₄⁺, albeit dwarfed by the tetrahedral monomer (see above, m/z = 97).

To bring evidence for the special case (m = 582, z = 6), experimental variation might be required. For example, if an experiment could be devised where both m/z and m/z^u were measurable in the same system and u was determined or pre-determined (u > 0, say), then it would be possible to evaluate m = (m/z)^u/(u−1)(m/z^u)^−1/(u−1). For example, if u could be set equal to 2, then m would become calculable as (m/z)²/(m/z²).

Once such an experiment had become feasible it would then be worthwhile to study, besides AlCl₃ solutions, Al₂(SO₄)₃ solutions, where it seems that the tridecameric species do not form. Given that the SO₄²⁻ ion may block the path to the tridecamer it is possible that, especially in the sulphate case, formation of a hexamer may occur preferentially (a line at m/z = 307 is just visible):

Al₅(OH)₁₂(OH₂)₁₀³⁺ + 4H₃O⁺ ⇔ Al₅(OH)₈(OH₂)₁₄⁷⁺ + 4H₂O, followed by

Al₅(OH)₈(OH₂)₁₄⁷⁺ + Al(OH)₂(OH₂)₂⁺ + 6OH⁻ ⇔ Al₆(OH)₁₆(OH₂)₁₀²⁺ + 6H₂O

or possibly (to m/z = 97, m = 582):

Al₅(OH)₈(OH₂)₁₄⁷⁺ + Al(OH)₂(OH₂)₂⁺ + 2 OH⁻ ⇔ Al₆(OH)₁₂(OH₂)₁₂⁶⁺ + 4H₂O.

If the latter process were to occur, relative stability of the hexamer might explain the non-appearance of larger polymers (with n > 6) in solution.

If existence of the cyclic hexamer were to be confirmed, a follow-up question would be whether its concentration could be determined; and if so, whether the ratio of this to that of the hexameric chain would be any greater than would be expected statistically.

Clearly the hydroxide ion plays an important role in the crystal nucleation process. Moreover the results presented above are not exhaustive in terms of symmetrically similar permutations of the ligands, O²⁻, OH⁻ and OH₂. In some instances, multiple combinations are possible. At m/z = 217, for example, the trimer could exist variously as any one or more of the tetrahedral species Al₃(OH)₈⁺, Al₃O₂(OH)₄(OH₂)₂⁺ or Al₃O₄(OH₂)₄⁺. Usually the results presentation has been such as to show the species bearing the lowest charge z+ (where there is a choice) or the smallest variety of ligands, e.g. Al₃(OH)₈⁺ in this instance (having all hydroxyl groups); or to give a simple example of the possible reaction mechanisms.

Few of the group's observations^7,8 remain to be explained. The value of m/z = 415 reported in their 2011 paper⁸ equates to m/z = 397 + 18, where m/z = 397 could be ascribed to a partially developed chain of tetrahedra, such as Al₆O₈(OH)(OH₂)₅⁺, that is still reactive.

Curiously, the regular interval of 60 in the series m/z = 97 + 60(n − 1), where n = 1, 2… 6 and z is constant throughout (z = 1), equates numerically to the historical stoichiometry Al(OH)₃ minus a single water molecule! 27 + (17 × 3) − 18 = 60. Coincidentally, this matches the formula for the pseudoBoehmitic moiety, AlO(OH). (This reaffirms the merit that lies in computing the values of m/z that might arise from octahedra in a doubleSheet.)

By removing the cube as a classical “building block” and substituting instead either or both of two golden rhombohedra, any of a variety of possible structures might conceivably be built, including:

* a triclinic system,

* the Biliński rhombic dodecahedron, which is a 3-d space-filler (and of which the Keggin tridecamer is a special case),

* 3-d tiling by Biliński, which would be crystalline (and may never have been seen physicochemically),

* a non-convex stellar quasi-crystal, or

* “Penrose in 3-d”.

So far, however, it is believed that, beyond the tetrahedral and octahedral species, no simple polyhedral aluminal species has been observed in solution; neither classical (no “football”), nor Penrose tile nor Ogawa flower.

The next question is whether, given a suitable choice of salt solution, it would also be feasible to study for a given concentration the effect of a chemical reagent, such as an alkali, and possibly as a function of time. If in Akihiro Wakisaka's experiment⁶ we were to choose AlCl₃ instead of KCl, an issue might be whether the mass spectrum would be sensitive enough to identify, and distinguish between, clusters such as the ring Al₆(OH)₁₂(OH₂)₁₂⁶⁺ or chains of the kind Al_n(OH)_2(n−1)(OH₂)_2(n+2)⁽ⁿ⁺²⁾⁺. The effect of introducing the smallest aliquots of KOH into the solution might then be studied (and the K⁺(KCl)_x contribution perhaps subtracted out in some way). Over time, it might be anticipated that small n values would be superseded by large ones, for example; or possibly even results uniquely ascribable to unambiguous 2hn values.

4. Conclusions

The appearance in mass spectra of the line having a value of m/z = 97 is not sufficient to confirm the existence of the hexameric ring species with octahedral symmetry, Al₆(OH)₁₂(OH₂)₁₂⁶⁺ (m = 582). It seems far more likely that this spectral line originates primarily from the tetrahedral monomer, Al(OH)₂(OH₂)₂⁺ (m = 97). Yet the existence of the ring is not disproven. Further experimentation is needed; and, quite possibly, novel experimental design beforehand.

Crystal nucleation begins even in the acidic solution, prior to introduction of any alkali, through accretion of the monomeric species (though not beyond the hexamer). In the dimer, ESI quadrupole mass spectrometry distinguishes tetrahedral from octahedral symmetry. But the aluminate anion Al(OH)₄⁻ can, where present, arrest the aggregation process by removing chains of octahedra as small as the trimer and stabilising them with the tetrahedral form of the tridecamer, [Al₃(OH)₈(OH₂)₅⁺]₄[Al(OH)₄⁻]³⁺.

Of its nature, mass spectrometry cannot of course elucidate the geometry of the trimer or tetramer (irrespective of whether these comprise tetrahedra or octahedra); m/z is independent of any directional angle, θ, π/6, cot⁻¹sqrt(2) or cot⁻¹ τ. Yet the onset of this geometry may remain crucial in determining the ultimate crystal shape. Again, an additional experiment is required, possibly involving ²⁷Al NMR in solution; or, given the prospect of isotropy in the closed ring (the non-linear hexamer) but anisotropy in its trimeric (or tetrameric) precursor, perhaps some form of Raman spectroscopy.

Note added after first publication

This article replaces the version published on 16^th September 2016, which contained an error in the binomial expansion in Section 2.6.8.7.

Acknowledgements

This review stems from a 40-year fascination with the question of how the 3^rd cation attaches itself to a dimer in solution [whether in a straight line or at a 60° angle, and why], and whether the answer might explain the nucleation of hexagonal or hexagonal prismatic crystals. This particular enquiry has been stimulated and revitalised by Akihiro Wakisaka's 2007 presentation⁶ to the Faraday Division of the Royal Society of Chemistry. The paper is written in belated celebration of John C. Vickerman's 70^th birthday in 2013, and in recognition of his life's work in the field of mass spectrometry. It is dedicated to the memory of C. Stewart Hare.

Notes and references

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Footnotes

† Electronic supplementary information (ESI) available: Computational results may be found at www.alanhare.me.uk/RSC/Aluminal_Speciation-Mass_Spectral_Interpretation(symmetrical,non-polyhedral).ods or .xlsx (copy-paste full url into browser, if necessary). See DOI: 10.1039/c6ra11209a

‡ MRSC, without affiliation (retired)

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