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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.)

N Start

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 
      /                                      \ 

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.

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 $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 undef.


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

      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

      A023531     turn 1=left,0=straight, being 1 at N=k*(k+3)/2
      A023532     turn 1=straight,0=left

A023531 is 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.


Math::PlanePath, Math::PlanePath::TriangleSpiralSkewed, Math::PlanePath::HexSpiral



Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <>.