.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is >0, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{\ . if \nF \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" ======================================================================== .\" .IX Title "Math::PlanePath::PyramidSpiral 3pm" .TH Math::PlanePath::PyramidSpiral 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation" .\" For nroff, turn off justification. Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::PyramidSpiral \-\- integer points drawn around a pyramid .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::PyramidSpiral; \& my $path = Math::PlanePath::PyramidSpiral\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a pyramid shaped spiral, .PP .Vb 10 \& 31 3 \& / \e \& 32 13 30 2 \& / / \e \e \& 33 14 3 12 29 1 \& / / / \e \e \e \& 34 15 4 1\-\-2 11 28 ... <\- Y=0 \& / / / \e \e \e \& 35 16 5\-\-6\-\-7\-\-8\-\-9\-10 27 52 \-1 \& / / \e \e \& 36 17\-18\-19\-20\-21\-22\-23\-24\-25\-26 51 \-2 \& / \e \& 37\-38\-39\-40\-41\-42\-43\-44\-45\-46\-47\-48\-49\-50 \-3 \& \& ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 .Ve .PP The perfect squares 1,4,9,16 fall one before the bottom left corner of each loop, and the pronic numbers 2,6,12,20,30,etc are the vertical upwards from X=1,Y=0. .IX Xref "Square numbers Pronic numbers" .SS "Square Spiral" .IX Subsection "Square Spiral" This spiral goes around at the same rate as the \f(CW\*(C`SquareSpiral\*(C'\fR. It's as if two corners are cut off (like the \f(CW\*(C`DiamondSpiral\*(C'\fR) and two others extended (like the \f(CW\*(C`OctagramSpiral\*(C'\fR). The net effect is the same looping rate but the points pushed around a bit. .PP Taking points up to a perfect square shows the similarity. The two triangular cut-off corners marked by \*(L".\*(R"s are matched by the two triangular extensions. .PP .Vb 9 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+ 7x7 square \& | . . . 31 . . .| \& | . . 32 13 30 . .| \& | . 33 14 3 12 29 .| \& |34 15 4 1 2 11 28| \& 35|16 5 6 7 8 9 10|27 \& 36 17|18 19 20 21 22 23 24|25 26 \& 37 38 39|40 41 42 43 44 45 46|47 48 49 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+ .Ve .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, with the same shape etc. For example to start at 0, .PP .Vb 9 \& 12 n_start => 0 \& / \e \& 13 2 11 \& / / \e \e \& 14 3 0\-\-1 10 \& / / \e \& 15 4\-\-5\-\-6\-\-7\-\-8\-\-9 \& / \& 16\-17\-18\-19\-20\-21\-22\-... .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::PyramidSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::PyramidSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::PyramidSpiral->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::PyramidSpiral\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::PyramidSpiral\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::PyramidSpiral->new (n_start => $n)" .PD Create and return a new pyramid spiral object. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the effect of treating each N in the path as centred in a square of side 1, so the entire plane is covered. .SH "OEIS" .IX Header "OEIS" This path is in Sloane's Online Encyclopedia of Integer Sequences as .Sp .RS 4 (etc) .RE .PP .Vb 4 \& n_start=1 (the default) \& A053615 abs(X), distance to next pronic, but starts n=0 \& A054552 N on X axis, 4n^2 \- 3n + 1 \& A033951 N on South\-East diagonal, 4n^2 + 3n + 1 \& \& A214250 sum N of eight surrounding cells \& \& A217013 permutation N of points in SquareSpiral order \& rotated +90 degrees \& A217294 inverse .Ve .PP In the two permutations the pyramid spiral is conceived as starting to the left and the square spiral starting upwards. The paths here start in the same direction (both to the right), hence rotate 90 to adjust the orientation. .PP .Vb 11 \& n_start=0 \& A329116 X coordinate \& A329972 Y coordinate \& A053615 abs(X) \& A339265 dX\-dY increments (runs +1,\-1) \& A001107 N on X axis, decagonal numbers \& A002939 N on Y axis \& A033991 N on X negative axis \& A002943 N on Y negative axis \& A007742 N on diagonal South\-West \& A033954 N on diagonal South\-East, decagonal second kind \& \& n_start=2 \& A185669 N on diagonal South\-East .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidRows, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangleSpiralSkewed .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE.\s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .