A Class of Pleasing Periodic Designs Travis Clohessy and Kenneth Gibson December 14, 2006
Background Federico Fern´ andez is a retired architect that has had a lifelong interest in finding methods for creating esthetically pleasing designs.
Azulejos Azulejos are ceramic tiles from Spain that were traditionally blue and white. The designs discussed in Federico’s article ”A Class of Pleasing Periodic Designs” are examples of periodic designs that could be suitable patterns for azulejos .
Purpose What we hope to do is to understand Federico’s ”semiautomatic” method for creating unique designs, and apply it by creating a few of our own. To do this, we first need to understand a few concepts.
The integer lattice Λ 0 There exists a lattice Λ 0 that contains all linear combinations of vectors with integer coefficients. Essentially forming a 1 by 1 grid �� 1 � � 0 �� of points. This would have a basis of β = where all 0 1 linear combinations of the elements would have integer coefficients.
The sublattice Λ � a � � c � If v = and w = are two linearly independent vectors with b d integer entries, the group Λ of all integer linear combinations m v + n w is called the sublattice with basis { v , w } of the integer � a � c lattice Λ 0 . We’ll express the basis { v , w } as a matrix A = . b d Now, if x = [ x , y ] ∈ Λ, the equation x = m v + n w can be expressed in the matrix form as � x � � a � � m � c = . y b d n That is, the members of the sublattice Λ are the vectors obtained by multiplying elements of the integer lattice Λ 0 by the basis matrix A . We can therefore compactly express the statement that A is a basis for the sublattice Λ of Λ 0 by writing Λ A = A Λ 0 .
Example What this means is all when the points of matrix Λ 0 as vector coordinates are multiplied by the basis matrix of a sublattice Λ A , the resultant vector coordinate will have integer coefficients and thus be member of Λ 0 . For example, � 1 1 � β Λ A = A = 2 1 � 2 � ( m , n ) = 1 Remember v and w are the column space of the basis of Λ A , and the elements of v and w are integers. So when put into the equation: � x � � a � � m � c = , y b d n we can expect the resulting [x,y] coordinates to have integer elements.
Let’s see When we get our resultant [x,y] values, � 1 � � 2 � � 3 � 1 = 2 1 1 5 � 3 � Yes, ∈ Λ 0 as 3 and 5 are both integers. 5
Visually (Think subspaces!) All the points contained in the sublattice Λ all have integer elements and are therefore contained within Λ 0 .
Unimodularity Let Z 2 × 2 the set of all 2 × 2 matrices with integer entries. A matrix U is said to be unimodular if ◮ U ∈ Z 2 × 2 ◮ det U = ± 1 From the formula for the inverse it follows that the inverse of a unimodular matrix is unimodular. Since UU − 1 = I , the standard � 1 � � 0 � basis vectors for Λ 0 ( , ) are integer linear combinations of 0 1 the columns of U . Thus, the sublattice generated by the columns of U is the entire integer lattice Λ 0 . That is, U Λ 0 = Λ 0
The Nets of class n ◮ Two matrices A and B in Z 2 × 2 are bases for the same sublattice Λ if and only if there exists a unimodular matrix U such that A = BU ; that is, the product B − 1 A is unimodular. ◮ This relationship regard lets us categorize different classes of sublattices by natural numbers. ◮ Therefore, for any natural number n , a sublattice of Λ 0 whose basis matrix has determinant n will be called a net of class n . ◮ This is useful later as we will be able to determine the number of different sublattices that can be generated by a nay given net n .
nets By using the word net to denote a sublattice it is easier to imagine what is happening in the making of these designs. If you can imagine points of a sublattice lying on a net, you can imagine laying different mutations of the sublattice on different nets and overlapping them.
result When four distinct sublattices are found and overlapped, it usually creates an esthetically pleasing design.
Theorem 1 For each natural number n there are exactly σ ( n ) distinct nets of class n . In fact, every net Λ of class n has one of the σ ( n ) matrices � d � k as a basis, for d a divisor of n and 0 ≤ k ≤ d − 1. We 0 n / d � d � k will call the canonical basis for Λ 0 n / d
Simpler Terms This is a long proof and it took Dave and Bruce to do it. We’ll spare you the details.
Basically... ◮ For any natural number n , there are d nets where d represents the sum of the divisors of n . ◮ For instance, a net of class 6 would have 12 nets because 1 , 2 , 3 , 6 | n and 1 + 2 + 3 + 6 = 12. ◮ We call this function σ ( n ).
Back to the Theorem � d � k So we can use matrices of the form as bases for our nets 0 n / d of class n . Let’s consider nets of class 5. The sum of the divisors of 5 is 1 + 5 = 6 so there should be 6 bases. All equations of the form � d � k are below. 0 n / d � 5 � � 5 � � 5 � � 5 � � 5 � � 1 � 0 1 2 3 4 0 { , , , , , } 0 1 0 1 0 1 0 1 0 1 0 5
Dihedral Group ◮ There are only eight possibilities for an orthogonal 2 × 2 matrix with integer entries: � 0 � 1 � � � 0 � � 1 � 0 1 1 0 I = , α = , β = , αβ = 0 1 − 1 0 1 0 0 − 1 and their negatives. ◮ These eight matrices form a nonabelian group under matrix multiplication, the dihedral group D 4 . The group is generated by α , an element of order 4 (clockwise rotation through a right angle), and by β , an element of order 2 (reflection in the line y = x in R 2 .
Semiautomatic Federico’s ”semiautomatic” method for producing an azulejo design is to choose a net Λ and superimpose the four nets Λ, α (Λ), β (Λ), αβ (Λ), provided these four are distinct. Cases when they reduce to just two nets or one may be considered degenerate, since a single net or the overlay of just two nets is not of esthetic interest. More intricate designs can be produced by overlaying the nets of two or more different types in some class.
Degenerate Types There exists some nets that only yield one or two distinct forms. These types of nets are the degenerate cases. The nets of class n always contain the two degenerate types R and D. Degenerate types originate from an ill chosen set of bases so that when dihedral operations are performed on them redundancies occur.
Degeneracy Elaborated If we have a diagonal lattice resembling this: And reflect across y = x or y = − x we will get:
And if we superimpose the images we get the beautiful: This is the basic concept of degeneracy, and we can find various versions through any of the dihedral sets with various bases.
n is prime number greater than 2 Let p be an odd prime, so σ ( p )= p +1. If p ≡ 3(mod 4), then except for the types R and D, which contain just two nets each, the other types all contain four nets. If p ≡ 1(mod 4), then besides the types R and D and one additional type that contains just two nets, all other types contain four nets. Superimposing the four nets of any such type can be expected to yield an esthetically pleasing design.
One Eighth of a Lattice Something that ought to be noted is when you focus your attention on an eighth of a lattice you can predict the rest of the lattice. The other seven-eighths of the lattice are just the operations of the dihedral group applied to the first eighth.
What we were able to make Ole Federico pulled through and we were able to make a few designs and even recreate one of Federico’s.
� 13 � � 13 � 5 10 Bases are and . 0 1 0 1
OK, this one is just a 1 × 1 grid rotated 30 degrees twice so it’s only 3 nets...and totally cheating...but come on, it’s pretty cool.
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