X Axis

The starting Nring=1,7,19,37 etc on the X axis for the default step=6 is 6*d*(d-1)/2 + 1, counting the innermost ring as d=1. In general Nring is a multiple of the triangular numbers d*(d-1)/2, plus 1,

    Nring = step*d*(d-1)/2 + 1

This is the centred polygonal numbers, being the cumulative count of points making concentric polygons or rings in the style of this path.

Straight line radials further around arise from adding multiples of d, so for example in step=6 shown above the line N=3,11,25,etc is Nring + 2*d. Multiples k*d with k>=step give lines which are in between the base ones from the innermost ring.

Step 1

For step=1 the first ring is 1 point and each subsequent ring has 1 further point.

                24
                               23
             18       12    17
    25              8
          13     5
    19     9     3  1  2  4  7 11 16 22     <- Y=0
          14     6
    26             10
             20       15    21
                               28
                27
                    ^
     -5 -4 -3 -2-1 X=0 1  2  3  4  5  6

The rings are

    polygon        radius     N values
    ------------   ------     --------
    single point     0         1
    two points       1         2, 3
    triangle         2         4, 5, 6
    square           3         7, 8, 9,10
    pentagon         4        11,12,13,14,15
    hexagon          5        16,17,18,19,20,21
     etc

The X axis as described above is the triangular numbers plus 1, ie. k*(k+1)/2 + 1.

Step 2

For step=2 the arrangement is roughly

                   34
          35                33
                24 15 23
    36 25                      22 32
             16  9  4  8 14
    37 26 17 10  5  2  1  3  7 13 21 31
             18 11  6 12 20
    38 27                      30 42
                28 19 29
          39                41
                   40

The pattern is similar to the "SacksSpiral" (see Math::PlanePath::SacksSpiral). In "SacksSpiral" each spiral loop is 2 more points than the previous the same as here, but the positioning differs. Here the X axis is the pronic numbers and the squares are to the left, whereas in "SacksSpiral" rotated around to squares on X axis and pronics to the left.

Ring Shape

Option "ring_shape => 'polygon'" puts the points on concentric polygons of "step" many sides, so each concentric polygon has 1 more point on each of its sides than the previous polygon. For example step=4 gives 4-sided polygons, ie. diamonds,

                  16
                /    \                ring_shape=>'polygon', step=>4
             17    7   15
           /    /     \   \
        18    8    2    6   14
      /     /   /    \    \    \
    19   9    3         1    5   13
      \     \   \    /    /    /
        20   10    4   12   24
           \    \    /    /
             21   11   23
                \    /
                  22

The polygons are scaled to keep points 1 unit apart. For step>=6 this means 1 unit apart sideways. step=6 is in fact a honeycomb grid where each points is 1 away from all six of its neighbours.

For step=3, 4 and 5 the polygon sides are 1 apart radially, as measured in the centre of each side. This makes points a little more than 1 apart along the sides. Squeezing them up to make the closest points exactly 1 apart is possible, but may require iterating a square root for each ring. step=3 squeezed down would in fact become a variable spacing with successively four close then one wider.

For step=2 and step=1 in the current code the default circle shape is used. Should that change? Is there a polygon style with 2 sides or 1 side?

The polygon layout is only a little different from a circle, but it lines up points on the sides and that might help show a structure for some sets of points plotted on the path.

Step 3 Pentagonals

For step=3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2, are a radial going up to the left, and the second pentagonal numbers 2,7,15,26, S(k) = (3k+1)*k/2 are a radial going down to the left, respectively 1/3 and 2/3 the way around the circles.

As described in "Step 3 Pentagonals" in Math::PlanePath::PyramidRows, those P(k) and preceding P(k)-1, P(k)-2, and S(k) and preceding S(k)-1, S(k)-2 are all composites, so plotting the primes on a step=3 "MultipleRings" has two radial gaps where there's no primes.

N to X,Y - Circle

As per above, each ring begins at

    Nring = step*d*(d-1)/2 + 1

This can be inverted to get the ring number d for a given N, and then subtract Nring for a remainder into the ring. (N-1)/step in the formula effectively converts into triangular number style.

    d = floor((sqrt(8*(N-1)/step + 1) + 1) / 2)
    Nrem = N - Nring

Rings are sized so that points are spaced 1 unit apart. There are three cases,

    circle,  step<=6     unit radially on X axis
    polygon, step<=6     unit radially on sides centre
             step>=7     unit chord between points

For the circle shape the integer points are on a circle and fractional N is on a straight line between those integer points. This means it's a polygon too, but one with ever more sides whereas ring_shape=polygon is a fixed "step" many sides.

    circle       numsides = d*step
    polygon      numsides = step

The radial distance to a polygon corner is calculated as

                           base               varying with d
    ----------------     ---------------------------------------
    circle,  step<=6     0.5/sin(pi/step) + d-1
    polygon, step<=6     0.5/sin(pi/step) + (d-1)/cos(pi/step)
    circle,  step>=7     0                + 0.5/sin(pi/(d*step))
    polygon, step>=7     0                + d * 0.5/sin(pi/step)

The step<=6 cases are an initial polygon of "step" many unit sides, then unit spacing d-1 for circle, or for polygon (d-1)/cos(pi/step) which is bigger and ensures the middle of the sides have unit spacing radially.

The 0.5/sin(pi/step) for radius of a unit sided polygon arises from

          r      ___---*
           ___---      | 1/2 = half the polygon side
     ___--- alpha      |
    o------------------+
    alpha = (2pi/numsides) / 2 = pi/numsides
    sin(alpha) = (1/2) / base_r
    r = 0.5 / sin(pi/numsides)

The angle theta to a polygon vertex is simply a full circle divided by numsides.

    side = circle   Nrem
           polygon  floor(Nrem / step)
    theta = side * (2pi / numsides)
    vertex X = r * cos(theta)
           Y = r * sin(theta)
    next_theta = (side+1) * (2pi / numsides)
    next_vertex X = r * cos(next_theta)
                Y = r * sin(next_theta)
    frac into side
    f = circle   frac(Nrem)    = Nrem modulo 1
        polygon  Nrem - side*d = Nrem modulo d
    X = vertex_X + f * (next_vertex_X - vertex_X)
    Y = vertex_Y + f * (next_vertex_Y - vertex_Y)

If Nrem is an integer for circle, or multiple of d for polygon, then the vertex X,Y is the final X,Y, otherwise a fractional distance between the vertex X,Y and next vertex X,Y.

For a few cases X or Y are exact integers. Special case code for these cases can ensure floating point rounding of pi doesn't give small offsets from integers.

For step=6 the base r is r=1 exactly since the innermost ring is a little hexagon. This means for the circle step=6 case the points on the X axis (positive and negative) are all integers X=1,2,3,etc.

       P-----P
      /   1 / \ 1  <-- innermost points 1 apart
     /     /   \
    P     o-----P   <--  base_r = 1
     \      1  /
      \       /
       P-----P

If theta=pi, which is when 2*Nrem==d*step, then the point is on the negative X axis. Returning Y=0 exactly for that avoids sin(pi) giving some small non-zero due to rounding.

If theta=pi/2 or theta=3pi/2, which is 4*Nrem==d*step or 4*Nrem==3*d*step, then N is on the positive or negative Y axis (respectively). Returning X=0 exactly avoids cos(pi/2) or cos(3pi/2) giving some small non-zero.

Points on the negative X axis points occur when the step is even. Points on the Y axis points occur when the step is a multiple of 4.

If theta=pi/4, 3*pi/4, 5*pi/4 or 7*pi/4, which is 8*Nrem==d*step, 3*d*step, 5*d*step or 7*d*step then the points are on the 45-degree lines X=Y or X=-Y. The current code doesn't try to ensure X==Y in these cases. The values are not integers and floating point rounding might mean sin(pi/4)!=cos(pi/4) resulting in X!=Y.

N to RSquared - Step 1

For step=1 the rings are point, line, triangle, square, pentagon, etc, with vertices at radius=numsides-1. For fractional N the triangle, square and hexagon cases are quadratics in the fraction part, allowing exact values from "n_to_rsquared()".

           Ring                    R^2
    ---------------------     --------------
    triangle   4 <= N < 7      4 - 12*f*(1-f)
    square     7 <= N < 11     9 - 18*f*(1-f)
    hexagon   16 <= N < 22    25 - 25*f*(1-f)
    f = N - int(N)  fractional part of N

For example for the square at N=7.5 have f=0.5 and R^2=4.5 exactly. These quadratics arise because sine of 2pi/3, 2pi/4 and 2pi/6 are square roots, which on squaring up in R^2=X^2+Y^2 become integer factors for the fraction f along the polygon side.