This section makes uses of the notion of aggregation to reduce the restrictions that are imposed by the continual composition, or simple construction of discrete discs. It can be shown that it is possible in certain cases, which shall be exposited both here and in further sections where it is employed to show the approximation of the astroid metric.
As all points may be labelled by their distance over some arbitrary modulo value an extension to the current work is proposed whereby two differingr−neighbourhood sequences,AandA0 are used to label theZ2 lattice from the same pointp. At any point, p0, there are now two labels such that p0 = (i, j) where by k ≡ i mod m for the sequence A and k0 ≡ j mod m for the sequence A0. Here two differing functions for the aggregation of values of i and j which define new shapes on the lattice and in turn new metrics. The new shapes defined by the lattice are not necessarily convex.
As all of the discrete disks which make up the labellings of the lattice are composed of discrete lines it is possible to analyse the effects of the combinations of disks by considering only the intersections of the lines that make up the disks. Consider
a series of parallel lines expressed in the form y0 =m0x0+c0 where ci ∈ Z is an
arbitrary constant or offset,mi ∈Qis any gradient permitted by the line segments
of a discrete circle andx, y ∈Zare the usual Euclidean coordinates. These lines are such that each successive line is of a distance wi from the last which shall imitate
the width between iterations of the discrete circles, for the first line segment this is equivalent to b√r2c, and the discrete lines that they generate. All coordinates such that m0x0 +c0+kw0 ≤ y0 < m0x0 +c0+ (k+ 1)w0 are labelled as k0. A second set of parallel lines differing from the first are defined as y1 = m1x1 +c1 where all coordinates such that m1x1+c1+k1w1 ≤ y1 < m1x1+c1+ (k1+ 1)w1 are labelled k1. The intersection of these two areas is thus (k, k0). Increasing the offset ofk0 and k1 tok0+ 1 andk1+ 1 naturally results in the labelling of the area of their intersection as (k0+ 1, k1+ 1). The ordering of the tuple is not relevant to the functions that aggregate them.
The first of the functions here have previously been studied in [54] with regards to geometric computations on the lattice. It is of particular interest in the Broadcast- ing Automata model where the notion of waves as observed in nature are used to control a large number of distributed automata. Being able to predict the resultant shapes that are formed by the transmission of waves is, naturally, largely advan- tageous in that it affords the ability to predict and manipulate the formations on the plane. Such formations can then be used for a variety of computational duties such as partitioning and geometric computation.
Waves are represented as strings of the form (1,0,−1,0) and pairs of such values result in the nodal pattern, given in Definition. 3.1, that is resultant on the grid after distribution by broadcasting sequences. Here, summation is used over these values, much like the natural aggregation of the superposition of waves, to derive four sequences of the following form;
(2,0,−2,0) =a
,
,
(0,0,0,0) =c
and
(1,−1,−1,1) =d
which are all pairs of cyclic permutation.
Two functions for aggregation will now be introduced with the relevant equations for resultant patterning of the lattice.
Definition 6.14. The moir´e aggregation function is given in the table below.
moir´e(x,y)= ⊕ 0 1 2 3 0 a b c b 1 b a b c 2 c b a b 3 b c b a
Definition 6.15. TheAnti-moir´e aggregation function can be expressed simply as addition over modulo 4 and is given in the table below.
Anti-moir´e(x,y)= ⊕ 0 1 2 3 0 a b c d 1 b c d a 2 c d a b 3 d a b c
Proposition 6.16. The gradients of the new lines formed by the moir´e equation can be predicted using the equation
wo·m1−w1·m0 w0−w1
.
Proof. The form of the moir´e function is such that if the intersection of one area is labelled as (k, k0) then the next area that is labelled similarly will be of the form (k+ 1, k0+ 1). Encoding this as the intersections of linesy=m0x+c0+k·w0 with y=m1x+c1+k0·w1andy=m0x+c0+(k+1)·w0withy=m1x+c1+(k0+1)·w1. All that is left is to find the gradient of the two intersections. Solving the first case, x0 = c1−c0−k·w0−k 0·w 1 m0−m1 and y0 =m0( c1−c0−k·w0−k0·w1 m0−m1 ) +c0+k·w0 . The second, x1 = c1 −c0−(k+1)·w0−(k0+1)·w1 m0−m1 andy1 =m0( c1−c0−(k+1)·w0−(k0+1)·w1 m0−m1 ) +c0+ (k+ 1)·w0. The gradient of the two points can be found, y1−y0
x1−x0, which results in the gradient of the labelling applied to the lattice.
Proposition 6.17. The gradients of the new lines formed by the Anti-moir´e equa- tion can be predicted using the equation
wo·m1+w1·m0 w0 +w1
.
Variables refer to, line gradient, mi, and the width of the line, wi.
Proof. The form of the moir´e function is such that if the intersection of one area is labelled as (k, k0) then the next area that is labelled similarly will be of the form (k+ 1, k0+ 1). Encoding this as the intersections of linesy=m0x+c0+k·w0 with y=m1x+c1+(k0+1)·w1andy =m0x+c0+(k+1)·w0 withy=m1x+c1+k0·w1. All that is left is to find the gradient of the two intersections. Solving the first case, x0 = c1−c0−k·w0−(k0+1)·w1 m0−m1 and y0 =m0( c1−c0−k·w0−(k0+1)·w1 m0−m1 ) +c0+k·w0 . The second,x1 = c1 −c0−(k+1)·w0−k0·w1 m0−m1 andy1 =m0( c1−c0−(k+1)·w0−k0·w1 m0−m1 ) +c0+ (k+ 1)·w0. The gradient of the two points can be found, y1−y0
x1−x0, which results in the gradient of the labelling applied to the lattice.
Composition of this form is given as example in Figure.6.3. Here the two lines are shown on the lattice correspond to the gradients of the two differing aggregation functions.
(0,0) 0 1 2 3 3 2 1 0 0 1 2 3 2 1 (0,1) (0,2) (0,3) (0,3) (0,2) (0,1) (1,0) (1,1) (1,2) (3,3) (2,2) (3,3) (2,3) (3,0) (2,2) (1,1) (3,0) (2,3) (1,2) (3,1) (2,0) (2,1) (3,2) (3,2) (2,1) (1,0) (1,3) (2,0) (3,1) (3,1) (2,0) (1,2) (2,3) (3,0) (1,3) (3,0) (2,3) (1,2) (1,3) (2,0) (3,1) (1,0) (2,1) (3,2) (3,2) (1,3) (0,1) (3,0) (0,0) Pattern A Pattern B
Figure 6.3: The two forms of aggregation via the above functions. Here
pattern A gives the line formed by the anti-moire function and the pattern b gives the line as formed by the moire function.
Proposition 6.18. For any two discrete lines on the Z2 lattice it is possible to vary the observed gradients for resultant from aggregate functions by varying the width of the lines.
Proof. The proof is an observation of the equations for the gradients of the lines resultant from aggregation. Observing the results of two aggregating functions,
wo·m1+w1·m0
w0+w1 and
wo·m1−w1·m0
w0−w1 , any alteration to the width of the line which repre- sents results in a direct change in the gradient.
It is now possible to formulate the following statement which shows the increase in expressivity that comes from using the composition of two broadcast sequences.
The number of lines formed by aggregation are not limited by the restrictions that are demonstrated in Proposition. 6.11 where it is now possible to construct new lines through the varying of widths and indeed new polygons may now be formed with this technique such that they are non-convex.