In space, a five-fold axis is a line

Five-fold centres stay few, and mirrors do not counted the patches of a Penrose tiling by the local symmetry they keep, and found three behaviours at once. Patches about a vertex with a five-fold centre stopped at four, however large they were drawn. Patches with one mirror grew in proportion to their radius. Patches with no symmetry grew as its square. The explanation was the same in all three cases, and it was the dimension of the set each operation leaves fixed in the tiling’s internal space. A five-fold rotation fixes a point there, a mirror a line, and the identity fixes everything. The number of patches keeping an operation grows as the radius raised to that dimension.

That essay ended with a prediction. The icosahedral quasicrystals are sliced from a six-dimensional lattice, not a four-dimensional one, and their internal space has three dimensions. There the icosahedral group’s operations fix points, lines and planes, and the same argument predicts four columns: bounded, linear, quadratic and cubic. It also predicts something the plane cannot show. A five-fold rotation, which fixes a single point of the Penrose tiling’s two-dimensional internal space, fixes a whole line of a three-dimensional one.

This essay makes the count. The prediction holds in every column it can reach. The patches about a vertex with a five-fold axis grow from 5 per axis at a radius of 1.34 edges to 24 at 4.11 and 46 at 8.01. Patches with a three-fold or two-fold axis grow the same way. Patches with a mirror grow from 111 to 3,763 per mirror plane by 4.99 edges, roughly as the square of the radius once the radius is large enough. Patches with a centre of inversion, the one operation besides the rotoinversions that fixes only a point, number one up to 1.9 edges and none at all from 2.15 on. And the count could not be made by looking at vertices, as the plane’s was. It had to be made in internal space, along the fixed sets themselves.

A quasicrystal cut from six dimensions #

The point set is the canonical one. Six integers and the lattice that holds them built the six-dimensional integer lattice the icosahedral group acts on: the group permutes the six five-fold axes of an icosahedron with signs, so in coordinates along those axes each of its operations is a signed permutation of six integers. Six-dimensional space splits into two three-dimensional pieces the group preserves. One is physical space, where the six axes appear as the icosahedron’s five-fold directions. The other is internal space, where the same six integers are paired with the axes of a second icosahedron, the one obtained by replacing the golden ratio $\tau$ by its conjugate $-1/\tau$.

As in cut and project, a lattice point is kept when its internal image falls inside a window. The window here is the shadow of the six-dimensional unit cube in internal space, a rhombic triacontahedron with thirty rhombic faces, shifted by a small generic vector so that no lattice point falls on its surface. The points kept are the vertices of the Ammann–Kramer–Neri tiling of space by two golden rhombohedra, the three-dimensional relative of the Penrose rhombs, with every edge of unit length along one of the six five-fold directions.

A vertex’s patch of radius $r$ is the set of vertices within $r$ of it, recorded as the integer differences between their six coordinates and the vertex’s own. That is exact: two patches are the same exactly when the two sets of integer vectors are equal. And the full icosahedral group acts on the vectors by signed permutations, so whether a patch is carried onto itself by one of the 120 operations is also a comparison of integers.

The table is the prediction made explicit. The full icosahedral group has 120 operations, the rotation group that before the lattice has a say lists among the finite groups of space, doubled by the inversion: the identity, 24 five-fold rotations, 20 three-fold rotations, 15 half-turns, the inversion, 24 five-fold and 20 three-fold rotoinversions, and 15 mirrors. Each acts on internal space through the conjugate icosahedron, and each fixes a subspace of a definite dimension there, computed from its action rather than read off a table. A rotation fixes its axis. A mirror fixes its plane. The inversion fixes only the centre, and so does every rotoinversion, since a rotoinversion is a rotation followed by the inversion and its only fixed point is the one the rotation’s axis shares with the inversion. Four dimensions, and four predicted rates of growth.

Why the vertices cannot be counted directly #

The plane’s census sampled tens of thousands of vertices, took the patch of each and counted the distinct ones. It converged, because a Penrose tiling’s number of distinct patches of radius six edges is a few thousand, as how many patches of each size had measured, and a sample of fifty thousand vertices sees them all. The same census in space does not converge, and the table shows how badly.

A sample of 26,291 vertices, every vertex within 16 edges of the origin, has 12,221 vertices far enough from its edge to have complete patches of radius 2.15 edges, and they show 9,325 distinct patches. The smaller sample, within 12 edges, has 3,747 such vertices and 3,669 distinct patches: 98 per cent of its vertices have a patch no other vertex in it shares. The larger sample finds two and a half times as many. The number being measured is the size of the sample, not a property of the quasicrystal. The total number of distinct patches of a three-dimensional quasicrystal grows as the cube of the radius, and the constant in front is large.

The symmetric patches are rarer still. At a radius of 1.34 edges the samples find 13 patches with a five-fold axis, the same thirteen in both. That looks converged and is not. The count in internal space below finds 25, and each of the thirteen the samples find is among them. The twelve that were missed occupy regions of internal space so thin that no vertex of a sample this size falls in them. Sampling vertices underestimates the symmetric columns most of all, which are the columns the question is about.

Counting in internal space #

The way round is the one the plane essay used to explain its counts, turned into the method of counting. A vertex’s patch is decided by where the vertex sits in the window. A vertex with internal position $x$ has a neighbour at difference $\Delta$ exactly when $x$ plus the internal image of $\Delta$ is inside the window as well. So the patch of radius $r$ changes only when $x$ crosses the surface of the window shifted by one of the finitely many differences $\Delta$ with physical length at most $r$. These shifted windows cut the window into cells, and each cell is one distinct patch.

A local symmetry is a condition on $x$. The window, measured from its centre, is carried onto itself by all 120 operations. So when $x$ lies on the fixed set of an operation’s internal action, the operation carries the vertex’s patch onto itself. The patches keeping an operation are the cells meeting its fixed set.

That turns each column into a count that can be made exactly or nearly so. Along a line the cells are intervals, and their ends are the points where the line crosses the surface of a shifted window, found from the fifteen pairs of the window’s parallel faces with no approximation beyond floating point. Across a plane the cells are polygons, counted by evaluating the patch on a fine grid of the plane, with the grid made finer until the count stops moving. At a point there is nothing to count but the cells that touch it.

A line’s worth of five-fold symmetry #

The figure shows the five-fold axis of internal space crossing the window, drawn three times, with a tick at every point where the patch changes. Between ticks the patch is constant, and each interval is a distinct patch with a five-fold axis. At 1.34 edges the axis carries 5 of them. At 2.55 it carries 12, and at 4.11, 24.

That is the plane’s result reversed. A Penrose tiling’s five-fold patches stop at four because the five-fold rotation of its internal plane fixes a point, and a point lies in a bounded number of cells however finely the window is cut. Here the five-fold rotation of internal space fixes a line, and a line crosses more and more cells as the radius grows and the cells shrink. Five-fold symmetry about a vertex, a point’s worth in the plane, is a line’s worth in space. The five-fold rotation of space, which no lattice permits in three dimensions, is permitted locally around more and more vertices as the patches grow.

The count continues far beyond the radii the plane essay could reach, since a line is cheap. Per five-fold axis the patches number 26 at 4.99 edges, 34 at 6.00, 38 at 7.00 and 46 at 8.01. Over the last few edges the count grows by about six patches for every edge of radius, a fixed rate, which is linear growth. The three-fold axes behave the same way, 24 at 4.11 and 46 at 8.01, and the two-fold axes run higher at 36 and 72. Each of the six five-fold axes, ten three-fold axes and fifteen two-fold axes carries its own patches, so with orientations counted the totals are those numbers multiplied by six, ten and fifteen.

A plane’s worth of mirror symmetry #

A mirror fixes a plane of internal space, and the picture shows what that plane looks like inside the window. The window’s section is a decagon, and every shifted window that crosses the plane cuts a polygon out of it. Their edges divide the decagon into cells, and each cell is a patch with that mirror. At 1.34 edges there are 111 or so, at 1.9 about 350, and the cells visibly shrink as their number grows.

The full series, per mirror plane, is 111, 223, 353, 550, 838, 1,134, 1,522 and 1,931 at radii from 1.34 to 3.5 edges, then 2,598 at 4.11 and 3,763 at 4.99. Over the first part of that range the count grows faster than the square of the radius, with a local exponent near 2.6. Over the last two steps it falls to 1.85 and then 1.91. The lines show the same drift on a smaller scale: their local exponent is between 1.3 and 1.7 at the start of the range and about 1.05 between six and eight edges.

The picture at the head of this essay puts the columns side by side. On logarithmic axes a power law is a straight line, and its slope is the exponent: the three kinds of axis run parallel at a slope near one by the largest radii, the mirror plane at a slope near two, and the point column has fallen to nothing.

The drift has a clear source. Every cell boundary is a piece of a face of some shifted window, and the faces come in fifteen orientations. Along a line, the number of distinct crossings contributed by each orientation grows in proportion to the radius only once the radius is several edges, because at small radii new differences keep adding new offsets faster than the eventual rate. Across a plane the cells are formed by fifteen families of such segments crossing one another, so the plane’s count goes as roughly the square of the line’s. At 1.34 edges the mirror plane’s 111 cells are 4.4 times the square of the five-fold axis’s 5. At 3.5 edges its 1,931 are 4.8 times the square of 20. The exponents’ ratio is two even where the exponents themselves have not settled, and the plane’s own exponent is approaching two from above where the radius is large enough to tell.

A point’s worth: the inversion #

The inversion and the rotoinversions fix only the centre of internal space, so the patches keeping them are the cells that touch that single point. Up to a radius of 1.9 edges there is exactly one such cell. It contains the centre, and its patch, a vertex surrounded by the twelve vertices of an icosahedron at unit distance along the six five-fold axes and their opposites, keeps all 120 operations. Both vertex samples find it.

At 2.15 edges the centre stops being inside a cell. Shifted windows’ faces pass through it, and 120 cells meet there, carried into one another by the 120 operations, none carried onto itself. So none of them is centrosymmetric. From 2.15 edges to the largest radius computed there is no patch about a vertex with a centre of inversion, and none with a rotoinversion. The point column is bounded, as predicted, and the bound is zero.

That is the same thing the plane essay found for the Penrose tiling’s half-turn, whose only patch was a bare rhomb at one edge and which vanished beyond it. In both cases an operation fixing a single point of internal space is the most fragile local symmetry the quasicrystal has, as the reflection of the Fibonacci chain’s circle was in a chain with no mirror has mirrors everywhere, where two fixed points gave one or two palindromes of each length, and in space the inversion takes the role that belonged to the half-turn and, in a sense, to the five-fold rotation in the plane.

The fourth column #

The fourth column is the patches with no symmetry, whose cells fill the window’s whole volume. Their number should grow as the cube of the radius. The census here does not measure it. The vertex samples show only that the number is enormous by two edges, beyond what any sample of tens of thousands of vertices reaches. Counting the cells of a three-dimensional arrangement of shifted windows exactly would take the arrangement itself, polyhedron by polyhedron, which this census does not build. The argument that gave the other three columns predicts the cube, and the general results on how fast the number of patches of a cut-and-project tiling grows agree with it for a canonical tiling of this kind, whose window is the shadow of the lattice’s own cube. On that reading the asymmetric patches are almost all of them. That is quoted here, not counted here.

What the picture cannot show #

Every figure in this essay is a picture of internal space or a table of counts, and none shows a patch of the quasicrystal. The patches themselves are clusters of a few dozen to a few hundred vertices in three dimensions. A drawing of one is a projection that would hide exactly the symmetry the essay is counting.

The counts are also counts per axis and per plane, with every orientation counted separately. Two five-fold patches on different axes are different patches, related by a rotation. Counting up to rotation divides each column by the number of axes or planes and changes no exponent.

And the convention the counts depend on is the window. A generic shift keeps every vertex off the window’s surface. The window centred exactly on the origin instead puts lattice points on its surface, so that whether those points are vertices is undecided. The resulting point set is not a generic member of the family, and its census would include patches that a generic one does not have. The census refuses that window rather than counting it.

Where the counting comes from #

Patch counting for quasicrystals goes back to the 1980s, when the number of distinct neighbourhoods of a Penrose tiling was first related to the geometry of its window. How the count grows for cut-and-project tilings in general, as a power set by the dimensions and the rank of the lattice, was worked out by Antoine Julien in 2010. The Ammann–Kramer–Neri tiling itself was found in 1984 by Peter Kramer and Roberto Neri, and independently in Robert Ammann’s unpublished work on three-dimensional rhombohedral tilings, the same year Shechtman’s icosahedral diffraction pattern was published.

The step taken here is to split the count by symmetry, and to use the symmetry’s fixed set as the place to count. The split shows that the exponent of each column is the dimension of a fixed set. The place shows why the vertices could not be counted. Their symmetric patches occupy sets of zero volume in internal space, and a random vertex lands in a neighbourhood of one only with a probability that falls as the radius grows.

What the census must refuse #

The checks are chosen so that each can fail. The dimension of the fixed set of each of the 120 operations is computed from its action on internal space, and it must match its kind: three, two, one or zero. The internal-space count must contain everything the vertex samples find: every one of the thirteen five-fold patches in the sample at 1.34 edges is among the twenty-five found along the axes with their orientations. The centred window must be refused, and it is, with a lattice point exactly on its surface. And the claim the essay exists to refuse, that the five-fold axis keeps a bounded number of patches in space as it does in the plane, is refused by its own column: 5, 7, 9, 10, 12, 14 over the first six radii.

Still open: the fourth column, counted #

The asymmetric column is the one this census leaves to a theorem. Counting it means building the arrangement of shifted triacontahedra in the window and counting its three-dimensional cells, which grow in number as the cube of the radius and shrink in size as its inverse. A count of that kind would also settle a question the plane essay could answer and this one cannot: how the share of vertices with any local symmetry falls as the radius grows. In the plane that share fell as the inverse of the radius, set by the mirrors. In space the mirrors’ cells are a plane’s worth inside a volume, so the share should fall again as the inverse of the radius, while the share keeping an axis falls as its square. Measuring it would take the three-dimensional cells’ volumes, not just their number, because a cell’s volume is the frequency of its patch, the quantity the average is the same wherever it is taken showed is well defined.