Eight-fold, with the golden ratio taken out

Everything this collection has built about quasicrystals has been built on five. Penrose’s two rhombs are five-fold. The Fibonacci chain has the golden ratio in it. Cut-and-project slices a five-dimensional lattice. What Shechtman measured was a ten-fold pattern.

That is a great deal of weight on one number, and a method that works only on φ is a method tuned to its answer. So this essay runs the same construction on eight, where the irrational is √2 and the lattice is four-dimensional, and changes nothing else.

The result is a tiling with a name of its own — Ammann–Beenker, after the two people who found it independently in the 1970s and 1980s, one of them an amateur working from a job as a postal sorter. It is the eight-fold analogue of Penrose’s tiling in every respect that matters here, and the essay’s point is that “analogue” is too weak a word: it is the same construction.

Why eight is available in the same sense five is #

The crystallographic restriction forbids both, and it says how much room each needs: a rotation of order n is an integer matrix in exactly φ(n) dimensions and no fewer, where φ is Euler’s function.

φ(5) = 4 and φ(8) = 4. Both orders need four dimensions and neither needs more, so a plane pattern with five-fold symmetry and one with eight-fold symmetry are in precisely the same position: forbidden a lattice in the plane, permitted one in four dimensions, and available as a projection.

What differs between them is the ring. Five-fold arithmetic happens in the integers of ℚ(√5) and eight-fold in the integers of ℚ(√2), and both rings have what the construction needs: a unit larger than one whose conjugate is smaller than one.

The rings are worth setting side by side, because the resemblance is exact and not an analogy. ℚ(√5) contains the golden ratio φ = (1 + √5)/2, whose conjugate is (1 − √5)/2 ≈ −0.618; ℚ(√2) contains 1 + √2, whose conjugate is 1 − √2 ≈ −0.414. Both are units — their product with their conjugate is ±1 — and both have a conjugate inside the unit circle. Those two properties are the whole of what the construction below needs, and every order n with φ(n) = 4 has a ring supplying them.

That is why five, eight, ten and twelve are the orders the literature is full of. It is not that they are the beautiful ones; it is that four is the smallest dimension in which a rotation of a forbidden order becomes an integer matrix, and those are the orders with φ(n) = 4.

The window is the hypercube’s shadow #

Four unit vectors at forty-five degrees to one another span the plane and no rational combination of them vanishes, so the map from ℤ⁴ to the plane sending each basis vector to one of them lands on a dense set. Its companion sends the same basis vectors to a different set of four directions — the odd ones — and together the two place ℤ⁴ in four-dimensional space as a lattice on which the eight-fold rotation acts by the integer matrix above.

Then the cut. Keep the lattice points whose companion image falls inside a window, and project the survivors into the plane.

The window being the hypercube’s shadow is not a convenience. It is what makes the result a tiling rather than a scattering of points: the projected points are the vertices of a face-to-face tiling exactly when the acceptance region is the projection of the unit cell. Choosing a circle instead — which looks tidier — gives a point set with holes and overlaps in it.

The acceptance test is one line and it is the whole algorithm. Walk a box of lattice points; for each, compute the companion image; keep it if the image is inside the octagon; project the survivors. There is no growth rule, no matching condition, no backtracking and no choice. Compare that with the way the same tiling is usually introduced — as a substitution on two tiles with a rule about how their edges may meet — and the cut is the shorter of the two by a long way.

It is also the construction the Fibonacci chain is built by, one dimension lower and drawn there rather than here: a strip of fixed width laid through a square lattice at an irrational slope, and the shadow of the points it catches. Everything above is that picture with two more integers in it. The strip becomes a four-dimensional slab, its width becomes the octagon, and the irrational slope becomes the pair of projections — but the test a point is put to is the same test, and it is worth noticing that the harder-sounding case needed no new idea to state.

The two descriptions are equivalent and the equivalence is not obvious. This collection has met the pair before: the Fibonacci chain has three completely different constructions which are required here to agree. The eight-fold case has the same pair, and only the cut is performed above.

What comes out #

Two tiles, a square and a rhomb with a forty-five degree angle, both with unit edges. Eight directions of edge. No period anywhere, and the eight-fold symmetry the plane forbids, present as the symmetry of the point set rather than of any lattice.

The tiles come in a ratio, and it is measured. In an infinite tiling there are √2 rhombs for every square, and a finite patch approaches that from below because a boundary is where the squares are missing.

That is exactly the role φ plays for Penrose: the ratio of thick tiles to thin ones. The number is different and its job is identical, which is the point of running the construction twice.

And the tiling has vertices where nothing crystallographic could. Look at the patch and there are points where eight tiles meet in an eight-fold rosette — a configuration the restriction forbids outright to any periodic pattern, present here as a local arrangement that never repeats at any fixed spacing. That is the whole content of the phrase “order without repetition”, drawn rather than argued: the eight-fold symmetry is exact at those points and there is no translation carrying one of them to another.

A finite patch is not the tiling and the measurement says so. Both numbers above — the tile ratio and the direction count — are measured on a patch cut from a box of lattice points, and only one of them is exact at any size. The direction count is eight in the smallest patch that has any tiles at all, because it is fixed by the projection rather than by the sample. The tile ratio is not, and watching it climb is watching a boundary effect shrink.

Distinguishing the two is worth doing every time this collection measures something on a patch. The patch census for Penrose makes the same separation, and the window measurement is entirely about it.

The inflation is an integer matrix #

Self-similarity in this family is a statement about a linear map: stretch the plane by some factor λ, shrink the companion plane by its conjugate λ′, and ask whether the combined map carries the lattice onto itself.

For eight-fold the factor is 1 + √2, the silver ratio, and its conjugate is 1 − √2, whose modulus is 0.414. The matrix has whole-number entries; applied to each basis vector it multiplies the parallel part by 2.4142 and the companion part by 0.4142, both exactly.

That conjugate being smaller than one is the whole of why the diffraction is sharp. Which inflation factors exist makes the argument in general: an inflation factor is an eigenvalue of an integer matrix, so it is an algebraic integer, and sharp Bragg peaks require its conjugates to be small — the Pisot condition. The golden ratio satisfies it with a conjugate of 0.618; the silver ratio satisfies it with 0.414. Same condition, different number, same verdict.

Both numbers are on the permitted side of that condition, and for the same reason rather than by two arguments. φ satisfies x² = x + 1 and 1 + √2 satisfies x² = 2x + 1; each is the larger root of a monic quadratic with integer coefficients, and in each case the other root has modulus below one. That is the entire qualification. It is not a property either number has because of what it measures — it is a property of the quadratic, and every order with φ(n) = 4 supplies one.

What the four dimensions are and are not #

It is worth being exact about the four-dimensional lattice, because the phrase invites a misunderstanding this collection has been careful about before.

The lattice is real and the extra dimensions are bookkeeping. ℤ⁴ here is not a claim that the material occupies four dimensions; it is the observation that a plane point set with eight-fold symmetry has a description in which four integers index a point and the eight-fold rotation is an integer matrix. The tiling is in the plane. What is in four dimensions is the arithmetic that makes it decidable.

That is the same distinction the extra dimension that makes it periodic makes for incommensurate structures, and six integers and the lattice that holds them makes for the icosahedral case. Three constructions, three dimensions of embedding — four, four and six — and one idea.

The companion plane is where the information about aperiodicity lives. Two points close together in the plane can be far apart in the companion plane, and it is that separation which prevents the pattern from repeating: a translation of the tiling would have to be a lattice vector whose companion image is zero, and the only such vector is zero itself. Aperiodicity is not asserted here; it is a consequence of the lattice having no vector with a vanishing companion part.

The matrix has another use worth naming. Because the inflation is an integer matrix on the lattice, its powers are too, and applying it repeatedly generates the hierarchy of supertiles a substitution description would build by hand. The two accounts of the tiling — a cut and a substitution — meet here: the substitution’s inflation factor is this matrix’s eigenvalue on the plane, and its shrinking of the companion plane is what makes the hierarchy consistent.

Eight directions, and why that matters #

Every other eight-fold diffraction picture in this collection — including the one that opens the quasicrystal field — is drawn from the star: integer combinations of eight unit vectors, laid down where a quasicrystal’s reflections are known to fall. That is a picture of the answer, and it cannot establish that this tiling diffracts eight-fold, because eight-fold is what it was told to draw. The figure above is the other thing, and the difference is the essay’s whole method in miniature.

The check it makes is sharper than it looks, and a square lattice is what sharpens it. Eight strong reflections forty-five degrees apart sounds like a quasicrystal and is not: an ordinary square lattice has exactly that, because its (1,0) family and its (1,1) family stand at forty-five degrees to one another. What separates the two is that the eight here lie on one ring — the measured radii agree to 0.12%, where the square lattice’s spread by 29% — and that is the assertion the file makes, checked against a square lattice fed to the same measurement.

The Ammann–Beenker tiling’s edges point in eight directions — four lines, each used both ways — and that finiteness is what allows its diffraction to have eight-fold symmetry rather than none.

Set that beside the pinwheel, whose tiles point in infinitely many directions and whose diffraction is therefore circularly symmetric. Two aperiodic tilings, two completely different answers to the same question about orientations, and the difference is whether the substitution’s turning angle is a rational part of a turn. Eight-fold is; the pinwheel’s is not.

So aperiodicity does not settle the orientation question, and this collection now has both answers in the same field.

And the window may be slid without changing any of it. Move the octagon a little in the companion plane and a different set of lattice points passes the test, so the tiling rearranges — some squares and rhombs trade places along the lines where the window’s edge swept past a point. What does not move is the density, the tile ratio or the diffraction, because none of the three depends on where the window sits, only on its shape and area. The rearrangement has a name in the physics of these materials, a phason flip, and the freedom a crystal has not is where this collection treats it. The point to carry here is that the construction produces not one tiling but a continuum of them, all with the same measurements, and that this is another respect in which eight behaves exactly as five does.

Which orders four dimensions reach, exactly #

The essay says five and eight arrive together at four dimensions. The full list is short and worth writing down, because it is exactly the list the experimental literature contains.

The dimension a rotation of order n needs is φ(n), Euler’s totient — the count of numbers below n coprime to it. The orders with φ(n) = 4 are 5, 8, 10 and 12, and there are no others. Everything with a smaller totient is crystallographic, and everything else needs six dimensions or more.

So the plane has exactly four non-crystallographic rotation orders reachable by a four-dimensional lattice, and every one of them has been found in a real alloy: ten-fold in the aluminium–manganese phase Shechtman measured, eight-fold in chromium–nickel–silicon and related systems, twelve-fold in nickel–chromium. The literature’s concentration on those four orders is not a fashion; it is the totient.

Each brings its own quadratic irrational, and the correspondence is tidy. Five and ten give the golden ratio and the ring generated by √5. Eight gives the silver ratio 1 + √2. Twelve gives 2 + √3. All three are Pisot numbers — algebraic integers whose conjugates are smaller than one in modulus — which is the condition an inflation factor has to satisfy for the diffraction to be sharp.

That is the strongest form of the essay’s argument. Running the construction on eight shows the method does not depend on five; the totient list shows there are exactly four numbers it could have been run on; and the Pisot condition shows why each of those four works. Nothing in the chain mentions the golden ratio at any point, and the golden ratio’s prominence is a fact about which alloy was measured first.

What running it twice establishes #

That the method is a method. Every step of the construction was carried over unchanged — the projection, the window as a hypercube shadow, the acceptance test, the search for an integer inflation matrix — and every step produced an answer of the same kind with a different number in it. A construction that needed adjusting to work on eight would have been a construction about five with machinery attached.

That the golden ratio was never doing the work. Read the Penrose essays on this site and φ appears in the inflation, in the tile ratio, in the chain’s spacings and in the eigenvalue of the substitution matrix, which makes it look like the subject. Run the construction on eight and every one of those appearances has a counterpart in 1 + √2, filling the same slot for the same reason. What was doing the work was the property — a unit with a small conjugate — and φ was one number with it.

That five is not special. The reason quasicrystallography is full of five-fold and ten-fold examples is that a five-fold pattern is what an electron microscope found in 1982, not that five is the only order available. Eight-fold and twelve-fold quasicrystals were found within a few years of the first, in alloys of chromium and nickel and manganese, and they need no new theory.

That the checks transfer. Every refusal this file makes is one the five-fold work would make in the same words with a different number: an order that has no integer trace in the plane and does in four dimensions, a window that is a shadow rather than a choice, an inflation that is integral and Pisot. A set of checks that holds up when the subject changes is a set of checks about the method rather than about the example, which is the strongest thing running a construction twice can establish.

And that the plane’s refusal is one refusal. The crystallographic restriction forbids five-fold and eight-fold by the same line of arithmetic — a trace that is not a whole number — and both escape it in the same way. The essays that treat five as the interesting case are treating an accident of history as a fact about the subject.

And nothing here is a claim about a material. Eight-fold quasicrystals exist and were found in alloys within a few years of the first ten-fold one; what this essay builds is a tiling, and a tiling is a geometric object that a structure may or may not be modelled by. The step from a point set to a set of atoms is the same step the modulation essays make and is not made here.

Nor is the tiling unique. Sliding the window through the companion plane gives a different tiling every time, all with the same tiles in the same proportions, the same directions and the same diffraction — which is the freedom a crystal has not appearing for the third time on this site. The patch drawn above is one member of an uncountable family and nothing distinguishes it.

What this does not do is derive the tiling’s diffraction. The peaks are indexed by four integers rather than two, and computing them is a sum this collection performs for the one-dimensional chain and not here. What is established above is the input to that computation — the point set, its inflation, its ring — and where the essay’s own arithmetic stops.

There is a caution to attach to that tidiness. The four orders are the ones a four-dimensional lattice reaches, and nothing says a plane quasicrystal must be built at the minimum dimension — a seven-fold pattern would need six, and such patterns exist as constructions. What is true is that no seven-fold alloy has been found, and the natural reading is that six dimensions of bookkeeping correspond to a structure with more constraints to satisfy than a real material readily meets. That is a plausible story rather than a theorem, and it is worth labelling as one.

The same reading applies to the three-dimensional case. Icosahedral symmetry needs six dimensions and occurs abundantly; the plane’s four orders need four and occur; and nothing between or beyond them has been reported in a material. Two data points do not make a rule, and the pattern is at least consistent with the obvious guess — that the number of extra dimensions is a measure of how much a structure is being asked to satisfy.