Math::PlanePath::SierpinskiTriangle -- self-similar triangular path traversal
use Math::PlanePath::SierpinskiTriangle; my $path = Math::PlanePath::SierpinskiTriangle->new; my ($x, $y) = $path->n_to_xy (123);
This path is an integer version of Sierpinski's triangle from
Waclaw Sierpinski, "Sur une Courbe Dont Tout Point est un Point de Ramification", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, volume 160, January-June 1915, pages 302-305. http://gallica.bnf.fr/ark:/12148/bpt6k31131/f302.image.langEN
Unit triangles are numbered numbered horizontally across each row.
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 15 57 58 59 60 61 62 63 64 14 49 50 51 52 53 54 55 56 13 45 46 47 48 12 37 38 39 40 41 42 43 44 11 33 34 35 36 10 29 30 31 32 9 27 28 8 19 20 21 22 23 24 25 26 7 15 16 17 18 6 11 12 13 14 5 9 10 4 5 6 7 8 3 3 4 2 1 2 1 0 <- Y=0 X= ... -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 ...
The base figure is the first two rows shape N=0 to N=2. Notice the middle "." position X=0,Y=1 is skipped
1 . 2 0
This is replicated twice in the next row pair as N=3 to N=8. Then the resulting four-row shape is replicated twice again in the next four-row group as N=9 to N=26, etc.
See the SierpinskiArrowheadCentres
path to traverse by a connected sequence rather than rows jumping across gaps.
The number of points in each row is always a power of 2. The power is the count of 1-bits in Y. (This count is sometimes called Gould's sequence.)
rowpoints(Y) = 2^count_1_bits(Y)
For example Y=13 is binary 1101 which has three 1-bits so in row Y=13 there are 2^3=8 points.
Because the first point is N=0, the N at the left of each row is the cumulative count of preceding points,
Ndepth(Y) = rowpoints(0) + ... + rowpoints(Y-1)
Since the powers of 2 are always even except for 2^0=1 in row Y=0, this Ndepth(Y) total is always odd. The self-similar nature of the triangle means the same is true of the sub-triangles, for example odd N=31,35,41,47,etc on the left of the triangle at X=8,Y=8. This means in particular the primes (being odd) fall predominately on the left side of the triangles and sub-triangles.
Counting the single point N=0 as level=0, then N=0,1,2 as level 1, each replication level goes from
Nstart = 0 Nlevel = 3^level - 1 inclusive
For example level 2 is from N=0 to N=3^2-1=8. Each level doubles in size,
0 <= Y <= 2^level - 1 - 2^level + 1 <= X <= 2^level - 1
Optional align=>"right"
puts points to the right of the Y axis, packed next to each other and so using an eighth of the plane.
align => "right" 7 | 19 20 21 22 23 24 25 26 6 | 15 16 17 18 5 | 11 12 13 14 4 | 9 10 3 | 5 6 7 8 2 | 3 4 1 | 1 2 Y=0 | 0 +------------------------- X=0 1 2 3 4 5 6 7
Optional align=>"left"
puts points to the left of the Y axis, ie. into negative X. The rows are still numbered starting from the left, so it's a shift across, not a negate of X.
align => "left" 19 20 21 22 23 24 25 26 | 7 15 16 17 18 | 6 11 12 13 14 | 5 9 10 | 4 5 6 7 8 | 3 3 4 | 2 1 2 | 1 0 | <- Y=0 -------------------------+ -7 -6 -5 -4 -3 -2 -1 X=0
Optional align=>"diagonal"
puts rows on diagonals down from the Y axis to the X axis. This uses the whole of the first quadrant, with gaps according to the pattern.
align => "diagonal" 15 | 65 ... 14 | 57 66 13 | 49 67 12 | 45 50 58 68 11 | 37 69 10 | 33 38 59 70 9 | 29 39 51 71 8 | 27 30 34 40 46 52 60 72 7 | 19 73 6 | 15 20 61 74 5 | 11 21 53 75 4 | 9 12 16 22 47 54 62 76 3 | 5 23 41 77 ... 2 | 3 6 17 24 35 42 63 78 1 | 1 7 13 25 31 43 55 79 Y=0 | 0 2 4 8 10 14 18 26 28 32 36 44 48 56 64 80 +------------------------------------------------- X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
This form visits all points X,Y where X and Y written in binary have no 1-bits in the same bit positions, ie. where X bitand Y == 0. For example X=13,Y=3 is not visited because 13="1011" and 6="0110" both have bit "0010" set.
This bit-and rule is an easy way to test for visited or not visited cells of the pattern. The visited cells can be calculated by this diagonal X,Y bitand, but then plotted X,X+Y for the "right" align or X-Y,X+Y for "triangular".
The triangle arises in Stephen Wolfram's 1-D cellular automatons (per Math::PlanePath::CellularRule and Cellular::Automata::Wolfram).
align rule ----- ---- "triangular" 18,26,82,90,146,154,210,218 "right" 60 "left" 102
http://mathworld.wolfram.com/Rule90.html
In each row the rule 18 etc pattern turns a cell "on" in the next row if one but not both its diagonal predecessors are "on". This is a mod 2 sum giving Pascal's triangle mod 2.
Some other cellular rules are variations on the triangle,
The default is to number points starting N=0 as shown above. An optional n_start
parameter can give a different start, with the same shape. For example starting at 1, which is the numbering of CellularRule
rule=60,
n_start => 1 20 21 22 23 24 25 26 27 16 17 18 19 12 13 14 15 10 11 6 7 8 9 4 5 2 3 1
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::SierpinskiTriangle->new ()
$path = Math::PlanePath::SierpinskiTriangle->new (align => $str, n_start => $n)
Create and return a new path object. align
is a string, one of the following as described above.
"triangular" (the default) "right" "left" "diagonal"
$n = $path->n_start()
Return the first N in the path. This is 0 by default, or the given n_start
parameter.
@n_children = $path->tree_n_children($n)
Return the children of $n
, or an empty list if $n < n_start
(ie. before the start of the path).
The children are the points diagonally up left and right on the next row (Y+1). There can be 0, 1 or 2 such points. At even depth there's 2, on depth=1mod4 there's 1. On depth=3mod4 there's some 0s and some 1s. See "N to Number of Children" below.
For example N=3 has two children N=5,N=6. Then in turn N=5 has just one child N=9 and N=6 has no children. The way points are numbered across a row means that when there's two children they're consecutive N values.
$n_parent = $path->tree_n_parent($n)
Return the parent node of $n
, or undef
if $n <= n_start
(the top of the triangle).
$depth = $path->tree_n_to_depth($n)
Return the depth of node $n
, or undef
if there's no point $n
. In the "triangular", "right" and "left" alignments this is the same as the Y coordinate from n_to_xy()
. In the "diagonal" alignment it's X+Y.
$n = $path->tree_depth_to_n($depth)
$n = $path->tree_depth_to_n_end($depth)
Return the first or last N at tree level $depth
. The start of the tree is depth=0 at the origin X=0,Y=0.
This is the N at the left end of each row. So in the default triangular alignment it's the same as xy_to_n(-$depth,$depth)
.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 3**$level - 1)
.
For calculation it's convenient to turn the X,Y coordinates into the "right" alignment style, so that Y is the depth and X is in the range 0<=X<=Y.
The starting position of each row of the triangle is given by turning 1-bits of the depth into powers-of-3.
Y = depth = 2^a + 2^b + 2^c + 2^d ... a>b>c>d... Ndepth = first N at this depth = 3^a + 2 * 3^b + 2^2 * 3^c + 2^3 * 3^d + ...
For example depth=6=2^2+2^1 starts at Ndepth=3^2+2*3^1=15. The powers-of-3 are the three parts of the triangle replication. The power-of-2 doubling is the doubling of the row Y when replicated.
Then the bits of X at the positions of the 1-bits of the depth become an N offset into the row.
a b c d depth = 10010010010 binary X = m00n00p00q0 Noffset = mnpq binary N = Ndepth + Noffset
For example in depth=6 binary "110" then at X=4="100" take the bits of X where depth has 1-bits, which is X="10_" so Noffset="10" binary and N=15+2=17, as per the "right" table above at X=4,Y=6.
If X has any 1-bits which are a 0-bits in the depth depth then that X,Y is not visited. For example if depth=6="110" then X=3="11" is not visited because the low bit X="__1" has depth="__0" at that position.
The row containing N can be found by working down the Ndepth formula shown above. The "a" term is the highest 3^a <= N, thus giving a bit 2^a for the depth. Then for the remaining Nrem = N - 3^a find the highest "b" where 2*3^b <= Nrem. And so on until reaching an Nrem which is too small to subtract any more terms.
It's convenient to go by bits high to low of the prospective depth, deciding at each bit whether Nrem is big enough to give the depth a 1-bit there, or whether it must be a 0-bit.
a = floor(log3(N)) round down to power-of-3 pow = 3^a Nrem = N - pow depth = high 1-bit at bit position "a" (counting from 0) factor = 2 loop bitpos a-1 down to 0 pow /= 3 if pow*factor <= Nrem then depth 0-bit, factor *= 2 else depth 1-bit factor is 2^count1bits(depth) Noffset = Nrem offset into row 0 <= Noffset < factor
N is turned into depth and Noffset as per above. X in "right" alignment style is formed by spreading the bits of Noffset out according to the 1-bits of the depth.
depth = 100110 binary Noffset = abc binary Xright = a00bc0
For example in depth=5 this spreads an Noffset=0to3 to make X=000, 001, 100, 101 in binary and in "right" alignment style.
From an X,Y in "right" alignment the other alignments are formed
alignment from "right" X,Y --------- ---------------- triangular 2*X-Y, Y so -Y <= X < Y right X, Y unchanged left X-Y, Y so -Y <= X <= 0 diagonal X, Y-X downwards sloping
The number of children follows a pattern based on the depth.
depth number of children ----- ------------------ 12 2 2 2 2 11 1 0 0 1 1 0 0 1 10 2 2 2 2 9 1 1 1 1 8 2 2 7 1 0 0 0 0 0 0 1 6 2 2 2 2 5 1 1 1 1 4 2 2 3 1 0 0 1 2 2 2 1 1 1 0 2
If depth is even then all points have 2 children. For example row depth=6 has 4 points and all have 2 children each.
At odd depth the number of children is either 1 or 0 according to the Noffset position in the row masked down by the trailing 1-bits of the depth.
depth = ...011111 in binary, its trailing 1s Noffset = ...00000 \ num children = 1 = ...11111 / = ...other num children = 0
For example depth=11 is binary "1011" which has low 1-bits "11". If those two low bits of Noffset are "00" or "11" then 1 child. Any other bit pattern in Noffset ("01" or "10" in this case) is 0 children. Hence the pattern 1,0,0,1,1,0,0,1 reading across the depth=11 row.
In general when the depth doubles the triangle is replicated twice and the number of children is carried with the replications, except the middle two points are 0 children. For example the triangle of depth=0to3 is repeated twice to make depth=4to7, but the depth=7 row is not children 10011001 of a plain doubling from the depth=3 row, but instead 10000001 which is the middle two points becoming 0.
The number of siblings of a given node is determined by its depth,
depth number of siblings ----- ------------------ 4 0 0 3 1 1 1 1 2 0 0 1 1 1 0 0 depth number of siblings ----- ------------------ odd 1 even 0
In an even row the points are all spread apart so there are no siblings. The points in such a row are cousins or second cousins, etc, but none share a parent.
In an odd row each parent node (an even row) has 2 children and so each of those points has 1 sibling.
The effect is to conflate the NumChildren=1 and NumChildren=0 cases in the picture above, those two becoming a single sibling.
num children of N num siblings of N ----------------- ----------------- 0 or 1 1 2 0
An easy range can be had just from the Y range by noting the diagonals X=Y and X=-Y are always visited, so just take the Ndepth of Ymin and Nend of Ymax,
# align="triangular" Nmin = N at X=-Ymin,Y=Ymin Nmax = N at X=Ymax,Y=Ymax
Or in "right" style the left end is at X=0 instead,
# align="right" Nmin = N at X=0,Ymin Nmax = N at Ymax,Ymax
For less work but a bigger over-estimate, invert the Nlevel formulas given in "Row Ranges" above to round up to the end of a depth=2^k replication,
level = floor(log2(Ymax)) + 1 Nmax = 3^level - 1
For example Y=11, level=floor(log2(11))+1=4, so Nmax=3^4-1=80, which is the end of the Y=15 row, ie. rounded up to the top of the replication block Y=8 to Y=15.
The Sierpinski triangle is in Sloane's Online Encyclopedia of Integer Sequences in various forms,
http://oeis.org/A001316 (etc)
A001316 number of cells in each row (Gould's sequence) A001317 rows encoded as numbers with bits 0,1 A006046 total cells to depth, being tree_depth_to_n(), A074330 Nend, right hand end of each row (starting Y=1)
A001316 is the "rowpoints" described above. A006046 is the cumulative total of that sequence which is the "Ndepth", and A074330 is 1 less for "Nend".
align="triangular" (the default) A047999 0,1 cells by rows A106344 0,1 cells by upwards sloping dX=3,dY=1 A130047 0,1 cells of half X<=0 by rows
A047999 etc is every second point in the default triangular lattice, or all points in align="right" or "left".
align="triangular" (the default) A002487 count points along dX=3,dY=1 slopes is the Stern diatomic sequence A106345 count points along dX=5,dY=1 slopes
dX=3,dY=1 sloping lines are equivalent to opposite-diagonals dX=-1,dY=1 in align="right".
align="right" A075438 0,1 cells by rows including 0 blanks at left of pyramid align="right", n_start=0 A006046 N on Y axis, being Ndepth A074330 N on Diagonal starting from Y=1, being Nend align="left", n_start=0 A006046 N on NW diagonal, being Ndepth A074330 N on Y axis starting from Y=1, being Nend A080263 Dyck encoding of the tree structure A080264 same in binary A080265 position in list of all balanced binary A080268 Dyck encoding breadth-first A080269 same in binary A080270 position in list of all balanced binary A080318 Dyck encoding breadth-first of branch-reduced (duplicate each bit) A080319 same in binary A080320 position in list of all balanced binary
For the Dyck encoding see for example "Binary Trees" in Math::NumSeq::BalancedBinary. The position in all balanced binary which is A080265 etc corresponds to value_to_i()
in that NumSeq
.
A branch-reduced tree has any single-child node collapsed out, so that all remaining nodes are either a leaf node or have 2 (or more) children. The effect of this on the Sierpinski triangle in breadth-first encoding is to duplicate each bit, so A080269 with each bit repeated gives the branch-reduced A080319.
Math::PlanePath, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::SierpinskiArrowheadCentres, Math::PlanePath::CellularRule, Math::PlanePath::ToothpickUpist
Math::NumSeq::SternDiatomic, Math::NumSeq::BalancedBinary
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015 Kevin Ryde
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