oracular (3) Math::PlanePath::QuintetReplicate.3pm.gz
NAME
Math::PlanePath::QuintetReplicate -- self-similar "+" tiling
SYNOPSIS
use Math::PlanePath::QuintetReplicate;
my $path = Math::PlanePath::QuintetReplicate->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is a self-similar tiling of the plane with "+" shapes. It's the same kind of tiling as the
"QuintetCurve" (and "QuintetCentres"), but with the middle square of the "+" shape centred on the origin.
12 3
13 10 11 7 2
14 2 8 5 6 1
17 3 0 1 9 <- Y=0
18 15 16 4 22 -1
19 23 20 21 -2
24 -3
^
-4 -3 -2 -1 X=0 1 2 3 4
The base pattern is a "+" shape
+---+
| 2 |
+---+---+---+
| 3 | 0 | 1 |
+---+---+---+
| 4 |
+---+
which is then replicated
+--+
| |
+--+ +--+ +--+
| 10 | | |
+--+ +--+--+ +--+
| | | 5 |
+--+--+ +--+ +--+
| | 0 | |
+--+ +--+ +--+--+
| 15 | | |
+--+ +--+--+ +--+
| | | 20 |
+--+ +--+ +--+
| |
+--+
The effect is to tile the whole plane. Notice the centres 0,5,10,15,20 are the same "+" shape but
positioned around at angle atan(1/2)=26.565 degrees. The relative positioning in each of those parts is
the same, so at 5 the successive 6,7,8,9 are E,N,W,S like the base shape.
Complex Base
This tiling corresponds to expressing a complex integer X+i*Y as
base b=2+i
X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]
where each digit position factor a[i] corresponds to N digits
N digit a[i]
------- ------
0 0
1 1
2 i
3 -1
4 -i
The base b is at an angle arg(b) = atan(1/2) = 26.56 degrees as seen at N=5 above. Successive powers
b^2, b^3, b^4 etc at N=5^level rotate around by that much each time.
Npow = 5^level at b^level
angle(Npow) = level*26.56 degrees
radius(Npow) = sqrt(5) ^ level
The path can be reckoned bottom-up as a new low digit of N expanding each unit square to the base "+"
shape.
+---C
D-------C | 2 |
| | D---+---+---+
| | => | 3 | 0 | 1 |
| | +---+---+---B
A-------B | 4 |
A---+
Side A-B becomes a 3-segment S. Such an expansion is the same as the TerdragonCurve or GosperSide, but
here turns of 90 degrees. Like GosperSide there is no touching or overlap of the sides expansions, so
boundary length 4*3^level.
Rotate Numbering
Parameter "numbering_type => 'rotate'" applies a rotation to the numbering in each sub-part according to
its location around the preceding level.
The effect can be illustrated by writing N in base-5. Part 10-14 is the same as the middle 0-4. Part
20-24 has a rotation by +90 degrees. Part 30-34 has rotation by +180 degrees, and part 40-44 by +270
degrees.
21
/ |
22 20 24 12 numbering_type => 'rotate'
\ / / \ N shown in base-5
23 2 13 10--11
/ \ \
34 3 0-- 1 14
\ \
31--30 33 4 41
\ / / \
32 43 40 42
| /
41
Notice this means in each part the 11, 21, 31, etc, points are directed away from the middle in the same
way, relative to the sub-part locations.
Working through the expansions gives the following rule for when an N is on the boundary of level k,
write N in base-5 digits (empty string if k=0)
if length < k then non-boundary
ignore high digit and all 1 digits
if any pair 32, 33, 44 then non-boundary
A 0 digit is the middle of a block, so always non-boundary. After that the 4,1,2,3 parts variously
expand with rotations so that a 44 is enclosed on the clockwise side and 32 and 33 on the anti-clockwise
side.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::QuintetReplicate->new ()"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if "$n < 0" then
the return is an empty list.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 5**$level - 1)".
FORMULAS
Axis Rotations
The digits positions 1,2,3,4 go around +90deg each, so the N for rotation by +90 is each digit +1,
cycling around.
rot+90(N) = 0, 2, 3, 4, 1, 10, 12, 13, 14, 11, 15, ... decimal
= 0, 2, 3, 4, 1, 20, 22, 23, 24, 21, 30, ... base5
rot-90(N) = 0, 4, 1, 2, 3, 20, 24, 21, 22, 23, 5, ... decimal
= 0, 4, 1, 2, 3, 40, 44, 41, 42, 43, 10, ... base5
rot180(N) = 0, 3, 4, 1, 2, 15, 18, 19, 16, 17, 20, ... decimal
= 0, 3, 4, 1, 2, 30, 33, 34, 31, 32, 40, ... base5
X,Y Extents
The maximum X in a given level N=0 to 5^k-1 can be calculated from the replications. A given high digit
1 to 4 has sub-parts located at b^k*i^(d-1). Those sub-parts are all the same, so the one with maximum
real(b^k*i^(d-1)) contains the maximum X.
N_xmax_digit(j) = d=1,2,3,4 where real(i^(d-1) * b^j) is maximum
= 1,1,4,4,4,4,3,3,3,2,2,2,1,1, ...
k-1
N_xmax(k) = digits N_xmax_digit(j) low digit j=0
j=0
= 0, 1, 6, 106, 606, 3106, 15606, ... decimal
= 0, 1, 11, 411, 4411, 44411, 444411, ... base5
k-1
z_xmax(k) = sum i^d[j] * b^j
j=0 each d[j] with real(i^d[j] * b^j) maximum
= 0, 1, 3+i, 7-2*i, 18-4*i, 42+3*i, 83+41*i, ...
xmax(k) = real(z_xmax(k))
= 0, 1, 3, 7, 18, 42, 83, 200, 478, 1005, ...
For computer calculation these maximums can be calculated by the powers. The digit parts can also be
written in terms of the angle arg(b) = atan(1/2). For successive k, if adding atan(1/2) pushes the b^k
angle past +45deg then the preceding digit goes past -45deg and becomes the new maximum X. Write the
angle as a fraction of 90deg (pi/2),
F = atan(1/2) / (pi/2) = 0.295167 ...
This is irrational since b^k is never on the X or Y axes. That can be seen since imag(b^k) mod 5 == 1 if
k odd and == 4 if k even >= 2. Similarly real(b^k) mod 5 == 2,3 so not on the Y axis, or also anything
on the Y axis would have 3*k fall on the X axis.
Digits low to high successively step back in a cycle 4,3,2,1 so that (with mod giving 0 to 3),
N_xmax_digit(j) = (-floor(F*j+1/2) mod 4) + 1
The +1/2 is since initial direction b^0=1 is angle 0 which is half way between -45 and +45 deg.
Similarly the X,Y location, using -i for rotation back
z_xmax_exp(j) = floor(F*j+1/2)
= 0,0,1,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5, ...
z_xmax(k) = sum(j=0,k-1, (-i)^z_xmax_exp(j) * b^j)
By symmetry the maximum extent is the same for Y vertically and for X or Y negative, suitably rotated.
The N in those cases has the digits 1,2,3,4 cycled around as per "Axis Rotations" above.
If the +1/2 in the floor is omitted then the effect is to find the maximum point in direction +45deg, so
the point(s) with maximum sum S = X+Y.
N_smax_digit(j) = (-floor(F*j) mod 4) + 1
= 1,1,1,1,4,4,4,3,3,3,3,2,2,2,1, ...
k-1
N_smax(k) = digits N_smax_digit(j) low digit j=0
j=0
= 0, 1, 6, 31, 156, 2656, 15156, ... decimal
= 0, 1, 11, 111, 1111, 41111, 441111, ... base5
and also N_smax() + 1
z_smax_exp(j) = floor(F*j)
= 0,0,0,0,1,1,1,2,2,2,2,3,3,3,4,4,4,5,5,5, ...
z_smax(k) = sum(j=0,k-1, (-i)^z_smax_exp(j) * b^j)
= 0, 1, 3+i, 6+5*i, 8+16*i, 32+23*i, 73+61*i, ...
and also z_smax() + 1+i
smax(k) = real(z_smax(k)) + imag(z_smax(k))
= 0, 1, 4, 11, 24, 55, 134, 295, 602, 1465, ...
In the base figure points 1 and 2 are both on the same 45deg line and this remains so in subsequent
levels, so that for k>=1 N_smax(k) and N_smax(k)+1 are equal maximums.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A316657> (etc)
A316657 X coordinate
A316658 Y coordinate
A316707 norm X^2 + Y^2
SEE ALSO
Math::PlanePath, Math::PlanePath::QuintetCurve, Math::PlanePath::ComplexMinus,
Math::PlanePath::GosperReplicate
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde
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
<http://www.gnu.org/licenses/>.