1Math::PlanePath::PyramiUdsSeirdeCso(n3t)ributed Perl DocMuamtehn:t:aPtliaonnePath::PyramidSides(3)
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6 Math::PlanePath::PyramidSides -- points along the sides of pyramid
7
9 use Math::PlanePath::PyramidSides;
10 my $path = Math::PlanePath::PyramidSides->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path puts points in layers along the sides of a pyramid growing
15 upwards.
16 21 4
18 20 13 22 3
19 19 12 7 14 23 2
20 18 11 6 3 8 15 24 1
21 17 10 5 2 1 4 9 16 25 <- Y=0
22 ------------------------------------
23 ^
24 ... -4 -3 -2 -1 X=0 1 2 3 4 ...
25 N=1,4,9,16,etc along the positive X axis is the perfect squares.
27 N=2,6,12,20,etc in the X=-1 vertical is the pronic numbers k*(k+1) half
28 way between those successive squares.
29 The pattern is the same as the "Corner" path but turned and spread so
31 the single quadrant in the "Corner" becomes a half-plane here.
32 The pattern is similar to "PyramidRows" (with its default step=2), just
34 with the columns dropped down vertically to start at the X axis. Any
35 pattern occurring within a column is unchanged, but what was a row
36 becomes a diagonal and vice versa.
37 Lucky Numbers of Euler
39 An interesting sequence for this path is Euler's k^2+k+41. The low
40 values are spread around a bit, but from N=1763 (k=41) they're the
41 vertical at X=40. There's quite a few primes in this quadratic and
42 when plotting primes that vertical stands out a little denser than its
43 surrounds (at least for up to the first 2500 or so values). The line
44 shows in other step==2 paths too, but not as clearly. In the
45 "PyramidRows" for instance the beginning is up at Y=40, and in the
46 "Corner" path it's a diagonal.
47 N Start
49 The default is to number points starting N=1 as shown above. An
50 optional "n_start" can give a different start, in the same pyramid
51 pattern. For example to start at 0,
52 n_start => 0
54 20 4
56 19 12 21 3
57 18 11 6 13 22 2
58 17 10 5 2 7 14 23 1
59 16 9 4 1 0 3 8 15 24 <- Y=0
60 --------------------------
61 -4 -3 -2 -1 X=0 1 2 3 4
62
64 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
65 classes.
66 "$path = Math::PlanePath::PyramidSides->new ()"
68 "$path = Math::PlanePath::PyramidSides->new (n_start => $n)"
69 Create and return a new path object.
70 "($x,$y) = $path->n_to_xy ($n)"
72 Return the X,Y coordinates of point number $n on the path.
73 For "$n < 0.5" the return is an empty list, it being considered
75 there are no negative points in the pyramid.
76 "$n = $path->xy_to_n ($x,$y)"
78 Return the point number for coordinates "$x,$y". $x and $y are
79 each rounded to the nearest integer which has the effect of
80 treating points in the pyramid as a squares of side 1, so the half-
81 plane y>=-0.5 is entirely covered.
82 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
84 The returned range is exact, meaning $n_lo and $n_hi are the
85 smallest and biggest in the rectangle.
86
88 Rectangle to N Range
89 For "rect_to_n_range()", in each column N increases so the biggest N is
90 in the topmost row and and smallest N in the bottom row.
91 In each row N increases along the sequence X=0,-1,1,-2,2,-3,3, etc. So
93 the biggest N is at the X of biggest absolute value and preferring the
94 positive X=k over the negative X=-k.
95 The smallest N conversely is at the X of smallest absolute value. If
97 the X range crosses 0, ie. $x1 and $x2 have different signs, then X=0
98 is the smallest.
99
101 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
102 this path include
103 <http://oeis.org/A196199> (etc)
105 n_start=1 (the default)
107 A049240 abs(dY), being 0=horizontal step at N=square
108 A002522 N on X negative axis, x^2+1
109 A033951 N on X=Y diagonal, 4d^2+3d+1
110 A004201 N for which X>=0, ie. right hand half
111 A020703 permutation N at -X,Y
112 n_start=0
114 A196199 X coordinate, runs -n to +n
115 A053615 abs(X), runs n to 0 to n
116 A000196 abs(X)+abs(Y), being floor(sqrt(N)),
117 k repeated 2k+1 times starting 0
118
120 Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner,
121 Math::PlanePath::DiamondSpiral, Math::PlanePath::SacksSpiral,
122 Math::PlanePath::MPeaks
123
125 <http://user42.tuxfamily.org/math-planepath/index.html>
126
128 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
129 2020 Kevin Ryde
130 This file is part of Math-PlanePath.
132 Math-PlanePath is free software; you can redistribute it and/or modify
134 it under the terms of the GNU General Public License as published by
135 the Free Software Foundation; either version 3, or (at your option) any
136 later version.
137 Math-PlanePath is distributed in the hope that it will be useful, but
139 WITHOUT ANY WARRANTY; without even the implied warranty of
140 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
141 General Public License for more details.
142 You should have received a copy of the GNU General Public License along
144 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
145perl v5.32.1 2021-01-27 Math::PlanePath::PyramidSides(3)