If we put the centre on a point where four regions meet - i.e. on one of our lines - it will simultaneously enclose two adjacent lattice points. If we put the centre of our hypercube on the border between two regions - i.e. Then that hypercube will, in turn, enclose the associated lattice point L. If we place the centre of a unit hypercube anywhere in the interior of one of those regions of P - that is, at a point on P that is enclosed by the hypercube centred on some lattice point L. (The hyperplanes x i = odd-half-integer all lie on the boundaries of these dual fundamental domains.) Unit hypercubes centred on all the points of the lattice. Wherever two of these lines intersect at a point Q, a unit hypercube centred at Q will contain a complete square from the lattice.Īnother way to think about this is to consider the lines to be dividing P into regions which are its intersections with the dual fundamental domains of the lattice: With these families of hyperplanes will be sets of parallel lines in P. If we define n families of hyperplanes in R n by fixing each of the coordinates x i at odd-half-integer values, then the intersections of P In other words, those coordinates will have odd-half-integer values, midway between the integer values of the lattice points. Then the hypercube’s centre must have the two coordinates corresponding to the square’s edge directions equal to those of the centre of the square itself. We are interested in identifying squares within the lattice that “approximate” the plane P, in the sense that all four of these squares’ vertices can fit simultaneously withinĪ unit-side-length hypercube whose centre lies somewhere on P and which is oriented parallel to the hypercubes of the lattice itself.Ī spacing of 1, and the hypercube whose centre lies on the plane has a side length of 1, if a whole square from the lattice ever lies inside the hypercube If it were, the projections of the 2 n vectors would overlap each other to form an n-pointed star. Note that for odd values of n, P is orthogonal to one of the hypercubic lattice diagonals, (1,1,1,1,…,1), but this is not the case for even n The projections of the n coordinate axes and their opposites onto P form a symmetrical 2 n-pointed star. These are not unit vectors if we want to normalise them we need to divide by √( n/2). Suppose we choose an affine plane P in R n parallel to the two-dimensional subspace spanned by the two vectors: x = (1, –cos(π/ n), cos(2π/ n), …, (–1) ( n–1)cos(( n–1)π/ n)) These points comprise the vertices of an infinite number of unit hypercubes packed together to fill R n, so we will refer to this as the “hypercubic lattice”. We define the points of the lattice Z n to be those ( x 1, …, x n) in R n where all the x i have integer Quasiperiodic tilings of the plane from the hypercubic lattice Construction Quasiperiodic tilings of the plane from the hypercubic lattice.It also discusses the generalisation of these methods to higher dimensional tilings, and to other lattices (including the A n lattice used by the Tübingen applet). This page explains in detail how the Penrose tiling, and other related quasiperiodic tilings drawn by the deBruijn applet, are constructed from a hypercubic lattice Back to home page | Site Map | Side-bar Site Map.SO(3) | Escher | Cantor | Laplace | Schwarz | Gummelt | QuantumWell | Flowers | LiquidMoon | Tesla | SoapBubbles | deBruijn | Kaleidoscope | Prisms | Lissajous | MirrorRind | Clouds | KaleidoHedron | Syntheme | Subluminal | Dirac | SO(4) | Spin | Platonic | Solid | Wythoff | Slice | Crystalline | Hypercube | Lattice | Tübingen | Girih | Girih Scroll | QuasiMusic | Antipodal.If you link to this page, please use this URL:.DeBruijn (Technical Notes) - Greg Egan deBruijn Mathematical Details
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