Tiling Spaces: Quasicrystals & Geometry Daniel Bragg

1)Bravais Lattices in N-dimensional space. 2)Crystallographic Restriction Theorem. 3)Voronoi Cell Construction. 4)Projection Lattices. 5)Penrose Tilings of the Plane Last time

1)Multigrids 2)Pentagrids 3)Projection Method for Constructing Tilings of the Plane. 4)Penrose Tilings as Projections 5)Other Areas of Research Overview of Presentation

These are methods of constructing tilings of the plane and were first introduced by the late N.G. de Bruijn. He noticed that any edge to edge tiling of the plane by rhombs contains “ribbons” made by rhombs with parallel edges. We can take these ribbons and replace them with straight lines orthogonal to the set of edges which defines it. We obtain a superposition of k infinite families of lines, where k is the number of directions of edges of the tiling. This is known as a multigrid. Multigrids

The multigrid construction is itself a tiling. de Bruijn coined the name meshes to describe the tiles of the multigrid.

The multigrid construction can be reversed. That is we can create the corresponding tiling of the plane from the multigrid. Steps: 1)Choose a star of k unit vectors. 2)Superimpose k grids onto the plane, where each one is orthogonal to one of the star vectors. It is important that no more than n grid lines meet at any point. 3)Begin to construct prototiles. Tile corresponding to a particular vertex is spanned by the star vectors that meet there. 4)With the prototile copies, using the multigrid as a blueprint, we can construct the tiling of the plane.  Definition A shift vector for the k-grid is the k-tuple of real numbers, for. If is such that no more than n hyperplanes meet at any point, it is said to be regular. Else it is known as singular. Multigrids

The multigrid construction allows us to assign coordinates to the vertices of tiles. Take each grid and index the parallel hyperplanes with +1/2. The equations of the hyperplanes are thus: Where, and is the star vector of grid j. Define a slab as the area which two successive hyperplanes bound in the plane. Every mesh is an intersection of k slabs. We can label each slab with the integers and thus label the vertices of the corresponding tiling with the same integers. The Mesh Condition: There are integers,, such that the k inequalities have simultaneous solutions. Multigrids

Pentagrids are a special case of multigrids when k=5. Their grid star vectors point to the vertices of a regular pentagon. They define T and t rhombs For a pentagrid to correspond to a Penrose tiling of the plane, it must satisfy the sum condition and the shift vector must be regular. Pentagrids

We can rewrite the mesh condition as Where. Summing all five inequalities, we obtain: In the special case when we can say. This is known as the index of the vertex that corresponds to a mesh. Sum Condition: Pentagrids

Projection Method for Constructing Tilings of the Plane We want to find a way of directly projecting tiling spaces from higher dimensional spaces onto lower dimensional planes. Main ingredients of the canonical projection method were an n- dimensional lattice (integral), a totally irrational subspace and a regular shift vector. We projected, onto ε, the points of the lattice whose Voronoi cells are cut by ε. The vertices of the tiles are simply the lattice points projected onto our subspace, but what are the tiles themselves images of? The answer is in the multigrid construction.

It can be shown that the n-grid is the intersection of the Voronoi tessellation of, induced by our integral lattice, with a d- dimensional subspace ε. Moreover the n-tuples of integers that satisfy the mesh condition are precisely the coordinates, in, of the lattice points whose Voronoi cells are cut by ε.  Definition Two tilings are orthogonally dual if, for k>0, the corresponding k and (n-k) faces are orthogonal.  Definition Two tilings of are said to be dual if there is a one-to-one inclusion-reversing map between the k- dimensional faces of one tiling and the (n-k) dimensional faces of the other, k=0,...,n Projection Method for Constructing Tilings of the Plane

The dual of the Voronoi tiling of is called the Delone tiling and is the tiling which has lattice points as vertices. When ε cuts the Voronoi tiling, we select a subset of faces of the Voronoi tiling but at the same time select a subset of faces of the Delone tiling. Therefore when we construct the tiling of ε, we are projecting the selected faces of the Delone tiling. Projection Method for Constructing Tilings of the Plane

The multigrid related to Penrose tilings is the pentagrid. The pentagrid star is an orthogonal projection, onto a two- dimensional plane, of vectors of an orthonormal basis of. can be interpreted as a translation in. The 5-tuple of integers which located a particular mesh in the pentagrid is now a point in. The grid equations represent parallel hyperplanes orthogonal to the corresponding basis vector of. is the location of a mesh in the pentagrid if and only if Penrose Tilings as Projections

We denote the unit hypercube in five-dimensional space by. It has ten four dimensional facets with facet vectors: It has thirty-two vertices which have the form:, When we project this hypercube onto the plane orthogonal to the body diagonal vector we get the following image. Penrose Tilings as Projections

We can show that is invariant under five-fold rotation, implying that the integer lattice is also invariant under this rotation. The rotation matrix is given by Eigen values are the fifth roots of unity. We can decompose into two two-dimensional planes, denoted by ε and ε’, and a fixed space Δ generated by.. Penrose Tilings as Projections

Projection principle: Project onto ε, the duals of the facets of the Voronoi tessellation that are cut by ε. ε meets the faces of in dimensions 3,4 and 5 so the corresponding faces of the Delone tiling are of dimension 2,1 and 0 respectively. It is these two dimensional faces which go on to form the rhomb Penrose tiling. The set of points we choose are decided by the canonical choice – the points whose orthogonal projection via is in the acceptance domain. Penrose Tilings as Projections

Consider the star of a vertex v of a Penrose tiling. By duality this corresponds to the intersection of ε with a cluster of Voronoi cells of. Let for some. Then. The facets of K will overlap when projected. Hence will lie in the intersection of several facets. The decomposition of K by the projected facets determines the vertex configuration in the projected tiling. Penrose Tilings as Projections

To create K, we are projecting onto which is three dimensional. This is a rhombic icosahedron. The facets of (hypercubes ) are projected as rhombic dodecahedra. Penrose Tilings as Projections

The 22 vertices of K satisfy. The set of lattice points to be projected onto ε are projected into four cross sections of K. These are the cross sections which are orthogonal to at levels. The index of a vertex is therefore the level on which the pre-image of the vertex lies in K. Penrose Tilings as Projections

Level 1: where S,K and Q correspond to the stars Penrose Tilings as Projections respectively

Level 2: where D, J, 3, 4, and 5 correspond to the stars Penrose Tilings as Projections respectively

Diffraction Geometry: N-point diffraction, Fourier transforms, Diffraction condition. Topological Bragg peaks One dimensional Crystals: Fibonacci sequences as crystals. Other Areas of Research

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