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# NAME

Math::PlanePath::AztecDiamondRings -- rings around an Aztec diamond shape

# SYNOPSIS

`````` use Math::PlanePath::AztecDiamondRings;
my \$path = Math::PlanePath::AztecDiamondRings->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path makes rings around an Aztec diamond shape,

``````                 46-45                       4
/     \
47 29-28 44                    3
/  /     \  \
48 30 16-15 27 43  ...            2
/  /  /     \  \  \  \
49 31 17  7--6 14 26 42 62           1
/  /  /  /     \  \  \  \  \
50 32 18  8  2--1  5 13 25 41 61    <- Y=0
|  |  |  |  |  |  |  |  |  |
51 33 19  9  3--4 12 24 40 60          -1
\  \  \  \     /  /  /  /
52 34 20 10-11 23 39 59             -2
\  \  \     /  /  /
53 35 21-22 38 58                -3
\  \     /  /
54 36-37 57                   -4
\     /
55-56                      -5

^
-5 -4 -3 -2 -1  X=0 1  2  3  4  5``````

This is similar to the `DiamondSpiral`, but has all four corners flattened to 2 vertical or horizontal, instead of just one in the `DiamondSpiral`. This is only a small change to the alignment of numbers in the sides, but is more symmetric.

Y axis N=1,6,15,28,45,66,etc are the hexagonal numbers k*(2k-1). The hexagonal numbers of the "second kind" 3,10,21,36,55,78, etc k*(2k+1), are the vertical at X=-1 going downwards. Combining those two is the triangular numbers 3,6,10,15,21,etc, k*(k+1)/2, alternately on one line and the other. Those are the positions of all the horizontal steps, ie. where dY=0.

X axis N=1,5,13,25,etc is the "centred square numbers". Those numbers are made by drawing concentric squares with an extra point on each side each time. The path here grows the same way, adding one extra point to each of the four sides.

``````    *---*---*---*
|           |
| *---*---* |     count total "*"s for
| |       | |     centred square numbers
* | *---* | *
| | |   | | |
| * | * | * |
| | |   | | |
| | *---* | |
* |       | *
| *---*---* |
|           |
*---*---*---*``````

## N Start

The default is to number points starting N=1 as shown above. An optional `n_start` can give a different start, in the same pattern. For example to start at 0,

``````    n_start => 0

45 44
46 28 27 43
47 29 15 14 26 42
48 30 16  6  5 13 25 41
49 31 17  7  1  0  4 12 24 40
50 32 18  8  2  3 11 23 39 59
51 33 19  9 10 22 38 58
52 34 20 21 37 57
53 35 36 56
54 55``````

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::AztecDiamondRings->new ()`
`\$path = Math::PlanePath::AztecDiamondRings->new (n_start => \$n)`

Create and return a new Aztec diamond spiral object.

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the X,Y coordinates of point number `\$n` on the path.

For `\$n < 1` the return is an empty list, it being considered the path starts at 1.

`\$n = \$path->xy_to_n (\$x,\$y)`

Return the point number for coordinates `\$x,\$y`. `\$x` and `\$y` are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1, so the entire plane is covered.

`(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)`

The returned range is exact, meaning `\$n_lo` and `\$n_hi` are the smallest and biggest in the rectangle.

# FORMULAS

## X,Y to N

The path makes lines in each quadrant. The quadrant is determined by the signs of X and Y, then the line in that quadrant is either d=X+Y or d=X-Y. A quadratic in d gives a starting N for the line and Y (or X if desired) is an offset from there,

``````    Y>=0 X>=0     d=X+Y  N=(2d+2)*d+1 + Y
Y>=0 X<0      d=Y-X  N=2d^2       - Y
Y<0  X>=0     d=X-Y  N=(2d+2)*d+1 + Y
Y<0  X<0      d=X+Y  N=(2d+4)*d+2 - Y``````

For example

``````    Y=2 X=3       d=2+3=5      N=(2*5+2)*5+1  + 2  = 63
Y=2 X=-1      d=2-(-1)=3   N=2*3*3        - 2  = 16
Y=-1 X=4      d=4-(-1)=5   N=(2*5+2)*5+1  + -1 = 60
Y=-2 X=-3     d=-3+(-2)=-5 N=(2*-5+4)*-5+2 - (-2) = 34``````

The two X>=0 cases are the same N formula and can be combined with an abs,

``    X>=0          d=X+abs(Y)   N=(2d+2)*d+1 + Y``

This works because at Y=0 the last line of one ring joins up to the start of the next. For example N=11 to N=15,

``````    15             2
\
14          1
\
13   <- Y=0

12         -1
/
11            -2

^
X=0 1  2``````

## Rectangle to N Range

Within each row N increases as X increases away from the Y axis, and within each column similarly N increases as Y increases away from the X axis. So in a rectangle the maximum N is at one of the four corners of the rectangle.

``````              |
x1,y2 M---|----M x2,y2
|   |    |
-------O---------
|   |    |
|   |    |
x1,y1 M---|----M x1,y1
|``````

For any two rows y1 and y2, the values in row y2 are all bigger than in y1 if y2>=-y1. This is so even when y1 and y2 are on the same side of the origin, ie. both positive or both negative.

For any two columns x1 and x2, the values in the part with Y>=0 are all bigger if x2>=-x1, or in the part of the columns with Y<0 it's x2>=-x1-1. So the biggest corner is at

``````    max_y = (y2 >= -y1              ? y2 ? y1)
max_x = (x2 >= -x1 - (max_y<0)  ? x2 : x1)``````

The difference in the X handling for Y positive or negative is due to the quadrant ordering. When Y>=0, at X and -X the bigger N is the X negative side, but when Y<0 it's the X positive side.

A similar approach gives the minimum N in a rectangle.

``````    min_y = / y2 if y2 < 0, and set xbase=-1
| y1 if y1 > 0, and set xbase=0
\ 0 otherwise,  and set xbase=0

min_x = / x2 if x2 < xbase
| x1 if x1 > xbase
\ xbase otherwise``````

The minimum row is Y=0, but if that's not in the rectangle then the y2 or y1 top or bottom edge is the minimum. Then within any row the minimum N is at xbase=0 if Y<0 or xbase=-1 if Y>=0. If that xbase is not in range then the x2 or x1 left or right edge is the minimum.

# OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

``````    n_start=1 (the default)
A001844    N on X axis, the centred squares 2k(k+1)+1

n_start=0
A046092    N on X axis, 4*triangular
A139277    N on diagonal X=Y
A023532    abs(dY), being 0 if N=k*(k+3)/2``````

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