Math::PlanePath::TriangleSpiral -- integer points drawn around an equilateral triangle
use Math::PlanePath::TriangleSpiral; my $path = Math::PlanePath::TriangleSpiral->new; my ($x, $y) = $path->n_to_xy (123);
This path makes a spiral shaped as an equilateral triangle (each side the same length).
16 4 / \ 17 15 3 / \ 18 4 14 ... 2 / / \ \ \ 19 5 3 13 32 1 / / \ \ \ 20 6 1-----2 12 31 <- Y=0 / / \ \ 21 7-----8-----9----10----11 30 -1 / \ 22----23----24----25----26----27----28----29 -2 ^ -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8
Cells are spread horizontally to fit on a square grid as per "Triangular Lattice" in Math::PlanePath. The horizontal gaps are 2, so for instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The diagonals are 1 across and 1 up or down, so n=3 is at x=1,y=1. Each alternate row is offset from the one above or below.
This grid is the same as the
HexSpiral and the path is like that spiral except instead of a flat top and SE,SW sides it extends to triangular peaks. The result is a longer loop and each successive loop is step=9 longer than the previous (whereas the
HexSpiral is step=6 more).
The triangular numbers 1, 3, 6, 10, 15, 21, 28, 36 etc, k*(k+1)/2, fall one before the successive corners of the triangle, so when plotted make three lines going vertically and angled down left and right.
The 11-gonal "hendecagonal" numbers 11, 30, 58, etc, k*(9k-7)/2 fall on a straight line horizontally to the right. (As per the general rule that a step "s" lines up the (s+2)-gonal numbers.)
The default is to number points starting N=1 as shown above. An optional
n_start can give a different start with the same shape etc. For example to start at 0,
n_start => 0 15 / \ 16 14 / \ 17 3 13 / / \ \ 18 4 2 12 ... / / \ \ \ 19 5 0-----1 11 30 / / \ \ 20 6-----7-----8-----9----10 29 / \ 21----22----23----24----25----26----27----28
With this adjustment the X axis N=0,1,11,30,etc is the hendecagonal numbers (9k-7)*k/2. And N=0,8,25,etc diagonally South-East is the hendecagonals of the second kind which is (9k-7)*k/2 for k negative.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::TriangleSpiral->new ()
$path = Math::PlanePath::TriangleSpiral->new (n_start => $n)
Create and return a new triangle spiral object.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number
$n on the path.
$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
$y are each rounded to the nearest integer, which has the effect of treating each
$n in the path as a square of side 1.
Only every second square in the plane has an N. If
$x,$y is a position without an N then the return is
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
n_start=1 (default) A010054 turn 1=left,0=straight, extra initial 1 A117625 N on X axis A081272 N on Y axis A006137 N on X negative axis A064226 N on X=Y leading diagonal, but without initial value=1 A064225 N on X=Y negative South-West diagonal A081267 N on X=-Y negative South-East diagonal A081589 N on ENE slope dX=3,dY=1 A038764 N on WSW slope dX=-3,dY=-1 A060544 N on ESE slope dX=3,dY=-1 diagonal A063177 total sum previous row or diagonal n_start=0 A051682 N on X axis (11-gonal numbers) A062741 N on Y axis A062708 N on X=Y leading diagonal A081268 N on X=Y+2 diagonal (right of leading diagonal) A062728 N on South-East diagonal (11-gonal second kind) A062725 N on South-West diagonal A081275 N on ENE slope from X=2,Y=0 then dX=+3,dY=+1 A081266 N on WSW slope dX=-3,dY=-1 A081271 N on X=2 vertical n_start=-1 A023531 N position of turns (to the left) 1 at N=k*(k+3)/2
n_start=-1 to match its "offset=0" for the first turn, being the second point of the path. A010054 which is 1 at triangular numbers k*(k+1)/2 is the same except for an extra initial 1.
Copyright 2010, 2011, 2012, 2013 Kevin Ryde
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