Kronecker product based fractals
You are encouraged to solve this task according to the task description, using any language you may know.
This task is based on Kronecker product of two matrices.
If your language has no a built-in function for such product then you need to implement it first.
The essence of fractals is self-replication (at least, self-similar replications).
So, using n times self-product of the matrix (filled with 0/1) we will have a fractal of the nth order.
Actually, "self-product" is a Kronecker power of the matrix.
In other words: for a matrix M and a power n create a function like matkronpow(M, n),
which returns MxMxMx... (n times product).
A formal recurrent algorithm of creating Kronecker power of a matrix is the following:
- Algorithm
- Let M is an initial matrix, and Rn is a resultant block matrix of the Kronecker power, where n is the power (a.k.a. order).
- Self-product of M, i.e., M x M producing R2 (resultant matrix with order/power 2).
- To receive the next order/power matrix use this recurrent formula: Rn = R(n-1) x M.
- Plot this Rn matrix to produce the nth order fractal.
Even just looking at the resultant matrix you can see what will be plotted.
There are virtually infinitely many fractals of this type. You are limited only by your creativity and
the power of your computer.
- Task
Using Kronecker product implement and show two popular and well-known fractals, i.e.:
The last one ( Sierpinski carpet) is already here on RC, but built using different approaches.
- Test cases
These 2 fractals (each order/power 4 at least) should be built using the following 2 simple matrices:
│ 0 1 0 │ and │ 1 1 1 │
│ 1 1 1 │ │ 1 0 1 │
│ 0 1 0 │ │ 1 1 1 │
- Note
- Output could be a graphical or ASCII-art representation, but if an order is set > 4 then printing is not suitable.
- The orientation and distortion of the fractal could be your language/tool specific.
- It would be nice to see one additional fractal of your choice, e.g., based on using a single (double) letter(s) of an alphabet, any sign(s) or already made a resultant matrix of the Kronecker product.
See implementations and results below in JavaScript, PARI/GP and R languages. They have additional samples of "H", "+" and checkerboard fractals.
11l
F kroneckerProduct(a, b)
V m = a.len
V n = a[0].len
V p = b.len
V q = b[0].len
V result = [[0] * (n * q)] * (m * p)
L(i) 0 .< m
L(j) 0 .< n
L(k) 0 .< p
L(l) 0 .< q
result[i * p + k][j * q + l] = a[i][j] * b[k][l]
R result
F kroneckerPower(m, n)
V result = m
L 2..n
result = kroneckerProduct(result, m)
R result
F to_str(m)
V result = ‘’
L(row) m
L(val) row
result ‘’= I val == 0 {‘ ’} E ‘ *’
result ‘’= "\n"
R result
V a1 = [[0, 1, 0], [1, 1, 1], [0, 1, 0]]
print(‘Vicsek fractal:’)
print(to_str(kroneckerPower(a1, 4)))
print()
V a2 = [[1, 1, 1], [1, 0, 1], [1, 1, 1]]
print(‘Sierpinski carpet fractal:’)
print(to_str(kroneckerPower(a2, 4)))- Output:
The same as in Nim solution.
Action!
The user must type in the monitor the following command after compilation and before running the program!
SET EndProg=*
CARD EndProg ;required for ALLOCATE.ACT
INCLUDE "D2:ALLOCATE.ACT" ;from the Action! Tool Kit. You must type 'SET EndProg=*' from the monitor after compiling, but before running this program!
DEFINE PTR="CARD"
DEFINE MATRIX_SIZE="4"
TYPE Matrix=[
BYTE width,height
PTR data]
PTR FUNC CreateEmpty(BYTE w,h)
Matrix POINTER m
m=Alloc(MATRIX_SIZE)
m.width=w
m.height=h
m.data=Alloc(w*h)
RETURN (m)
PTR FUNC Create(BYTE w,h BYTE ARRAY a)
Matrix POINTER m
m=CreateEmpty(w,h)
MoveBlock(m.data,a,w*h)
RETURN (m)
PROC Destroy(Matrix POINTER m)
Free(m.data,m.width*m.height)
Free(m,MATRIX_SIZE)
RETURN
PTR FUNC Product(Matrix POINTER m1,m2)
Matrix POINTER m
BYTE x1,x2,y1,y2
INT i1,i2,i
BYTE ARRAY a1,a2,a
m=CreateEmpty(m1.width*m2.width,m1.height*m2.height)
a1=m1.data
a2=m2.data
a=m.data
i=0
FOR y1=0 TO m1.height-1
DO
FOR y2=0 TO m2.height-1
DO
FOR x1=0 TO m1.width-1
DO
FOR x2=0 TO m2.width-1
DO
i1=y1*m1.width+x1
i2=y2*m2.width+x2
a(i)=a1(i1)*a2(i2)
i==+1
OD
OD
OD
OD
RETURN (m)
PROC DrawMatrix(Matrix POINTER m INT x,y)
INT i,j
BYTE ARRAY d
d=m.data
FOR j=0 TO m.height-1
DO
FOR i=0 TO m.width-1
DO
IF d(j*m.width+i) THEN
Plot(x+i,y+j)
FI
OD
OD
RETURN
PROC DrawFractal(BYTE ARRAY a BYTE w,h INT x,y BYTE n)
Matrix POINTER m1,m2,m3
BYTE i
m1=Create(w,h,a)
m2=Create(w,h,a)
FOR i=1 TO n
DO
m3=Product(m1,m2)
IF i<n THEN
Destroy(m1)
m1=m3 m3=0
FI
OD
DrawMatrix(m3,x,y)
Destroy(m1)
Destroy(m2)
Destroy(m3)
RETURN
PROC Main()
BYTE CH=$02FC,COLOR1=$02C5,COLOR2=$02C6
BYTE ARRAY a=[0 1 0 1 1 1 0 1 0],
b=[1 1 1 1 0 1 1 1 1],
c=[1 0 1 0 1 0 1 0 1]
Graphics(8+16)
AllocInit(0)
Color=1
COLOR1=$0C
COLOR2=$02
DrawFractal(a,3,3,12,55,3)
DrawFractal(b,3,3,120,55,3)
DrawFractal(c,3,3,226,55,3)
DO UNTIL CH#$FF OD
CH=$FF
RETURN- Output:
Screenshot from Atari 8-bit computer
Ada
Using multiplication function from Kronecker product.
with SDL.Video.Windows.Makers;
with SDL.Video.Renderers.Makers;
with SDL.Events.Events;
with SDL.Events.Mice;
procedure Kronecker_Fractals is
Width : constant := 800;
Height : constant := 800;
Order : constant := 6;
Window : SDL.Video.Windows.Window;
Renderer : SDL.Video.Renderers.Renderer;
type Matrix is array (Positive range <>, Positive range <>) of Integer;
function "*" (Left, Right : in Matrix) return Matrix is
Result : Matrix
(1 .. Left'Length (1) * Right'Length (1),
1 .. Left'Length (2) * Right'Length (2));
LI : Natural := 0;
LJ : Natural := 0;
begin
for I in 0 .. Result'Length (1) - 1 loop
for J in 0 .. Result'Length (2) - 1 loop
Result (I + 1, J + 1) :=
Left (Left'First (1) + (LI), Left'First (2) + (LJ))
* Right
(Right'First (1) + (I mod Right'Length (1)),
Right'First (2) + (J mod Right'Length (2)));
if (J + 1) mod Right'Length (2) = 0 then
LJ := LJ + 1;
end if;
end loop;
if (I + 1) mod Right'Length (1) = 0 then
LI := LI + 1;
end if;
LJ := 0;
end loop;
return Result;
end "*";
function "**" (Base : Matrix; Exp : Positive) return Matrix is
(case Exp is
when 1 => Base,
when 2 => Base * Base,
when others => Base * Base ** (Exp - 1));
procedure Draw_Matrix (LX, LY : Integer; M : Matrix) is
use SDL.C;
begin
for Y in M'Range (1) loop
for X in M'Range (2) loop
if M (Y, X) /= 0 then
Renderer.Draw (Point => (int (LX + X), int (LY + Y)));
end if;
end loop;
end loop;
end Draw_Matrix;
type Fractals is (Cross, H, X, Sierpinski, U);
Base : Fractals := Fractals'First;
M : constant array (Fractals) of Matrix (1 .. 3, 1 .. 3) :=
(Cross => ((0, 1, 0), (1, 1, 1), (0, 1, 0)),
H => ((1, 0, 1), (1, 1, 1), (1, 0, 1)),
X => ((1, 0, 1), (0, 1, 0), (1, 0, 1)),
Sierpinski => ((1, 1, 1), (1, 0, 1), (1, 1, 1)),
U => ((1, 0, 1), (1, 0, 1), (1, 1, 1)));
procedure Draw is
begin
Renderer.Set_Draw_Colour ((0, 0, 0, 255));
Renderer.Fill (Rectangle => (0, 0, Width, Height));
Renderer.Set_Draw_Colour (Colour => (0, 220, 0, 255));
Draw_Matrix (10, 10, M (Base) ** Order);
Window.Update_Surface;
Base := (if Base = Fractals'Last
then Fractals'First
else Fractals'Succ (Base));
end Draw;
procedure Event_Loop is
use type SDL.Events.Event_Types;
Event : SDL.Events.Events.Events;
begin
loop
SDL.Events.Events.Wait (Event);
case Event.Common.Event_Type is
when SDL.Events.Quit => return;
when SDL.Events.Mice.Button_Down => Draw;
when others => null;
end case;
end loop;
end Event_Loop;
begin
if not SDL.Initialise (Flags => SDL.Enable_Screen) then
return;
end if;
SDL.Video.Windows.Makers.Create (Win => Window,
Title => "Kronecker fractals (Click to cycle)",
Position => SDL.Natural_Coordinates'(X => 10, Y => 10),
Size => SDL.Positive_Sizes'(Width, Height),
Flags => 0);
SDL.Video.Renderers.Makers.Create (Renderer, Window.Get_Surface);
Draw;
Event_Loop;
Window.Finalize;
SDL.Finalise;
end Kronecker_Fractals;
ALGOL 68
BEGIN # Kronecker product based fractals - translated from the Kotlin sample #
MODE MATRIX = FLEX[ 1 : 0, 1 : 0 ]INT;
PROC kronecker product = ( MATRIX a in, b in )MATRIX:
BEGIN
MATRIX a = a in[ AT 0, AT 0 ], b = b in[ AT 0, AT 0 ];
INT m = 1 UPB a + 1, n = 2 UPB a + 1;
INT p = 1 UPB b + 1, q = 2 UPB b + 1;
INT rtn = m * p, ctn = n * q;
[ 0 : rtn - 1, 0 : ctn - 1 ]INT r;
FOR i FROM 0 TO rtn - 1 DO FOR j FROM 0 TO ctn - 1 DO r[ i, j ] := 0 OD OD;
FOR i FROM 0 TO m - 1 DO
FOR j FROM 0 TO n - 1 DO
FOR k FROM 0 TO p - 1 DO
FOR l FROM 0 TO q - 1 DO
r[ p * i + k, q * j + l ] := a[ i, j ] * b[ k, l ]
OD
OD
OD
OD;
r
END # kronecker product # ;
PROC kronecker power = ( MATRIX a, INT n )MATRIX:
BEGIN
MATRIX pow := a;
TO n - 1 DO pow := kronecker product( pow, a ) OD;
pow
END # kronecker power # ;
PROC show matrix = ( STRING text, MATRIX m )VOID:
BEGIN
print( ( text, " fractal :", newline ) );
FOR i FROM 1 LWB m TO 1 UPB m DO
FOR j FROM 2 LWB m TO 2 UPB m DO
print( ( IF m[ i, j ] = 1 THEN "*" ELSE " " FI ) )
OD;
print( ( newline ) )
OD;
print( ( newline ) )
END # show matrix # ;
BEGIN
MATRIX a := MATRIX( ( 0, 1, 0 )
, ( 1, 1, 1 )
, ( 0, 1, 0 )
);
show matrix( "Vicsek", kronecker power( a, 4 ) );
a := MATRIX( ( 1, 1, 1 )
, ( 1, 0, 1 )
, ( 1, 1, 1 )
);
show matrix( "Sierpinski carpet", kronecker power( a, 4 ) )
END
END- Output:
Same as the Kotlin sample.
C
Although this task is related to Kronecker product, this is computationally a more complex task as the matrix has to be raised to an arbitrary power. Assume matrix A, order i x j has to be raised to power n, the final result will have (i^n)x(j^n) elements. Doing this "conventionally" will require at least (i^n)x(j^n) operations with storage for the same number of elements. This means a storage requirement of 4 x (i^n) x (j^n) bytes for an integer matrix.
However, if half of the elements of the initial matrix A are zeroes, computations and storage for such elements are wasted as they will never be plotted. The only relevant elements are the 1s.
Thus this implementation treats the initial matrix as a Sparse matrix. Doing so cuts down drastically on the required storage and number of operations. The graphical part needs the WinBGIm library.
#include<graphics.h>
#include<stdlib.h>
#include<stdio.h>
typedef struct{
int row, col;
}cell;
int ROW,COL,SUM=0;
unsigned long raiseTo(int base,int power){
if(power==0)
return 1;
else
return base*raiseTo(base,power-1);
}
cell* kroneckerProduct(char* inputFile,int power){
FILE* fp = fopen(inputFile,"r");
int i,j,k,l;
unsigned long prod;
int** matrix;
cell *coreList,*tempList,*resultList;
fscanf(fp,"%d%d",&ROW,&COL);
matrix = (int**)malloc(ROW*sizeof(int*));
for(i=0;i<ROW;i++){
matrix[i] = (int*)malloc(COL*sizeof(int));
for(j=0;j<COL;j++){
fscanf(fp,"%d",&matrix[i][j]);
if(matrix[i][j]==1)
SUM++;
}
}
coreList = (cell*)malloc(SUM*sizeof(cell));
resultList = (cell*)malloc(SUM*sizeof(cell));
k = 0;
for(i=0;i<ROW;i++){
for(j=0;j<COL;j++){
if(matrix[i][j]==1){
coreList[k].row = i+1;
coreList[k].col = j+1;
resultList[k].row = i+1;
resultList[k].col = j+1;
k++;
}
}
}
prod = k;
for(i=2;i<=power;i++){
tempList = (cell*)malloc(prod*k*sizeof(cell));
l = 0;
for(j=0;j<prod;j++){
for(k=0;k<SUM;k++){
tempList[l].row = (resultList[j].row-1)*ROW + coreList[k].row;
tempList[l].col = (resultList[j].col-1)*COL + coreList[k].col;
l++;
}
}
free(resultList);
prod *= k;
resultList = (cell*)malloc(prod*sizeof(cell));
for(j=0;j<prod;j++){
resultList[j].row = tempList[j].row;
resultList[j].col = tempList[j].col;
}
free(tempList);
}
return resultList;
}
int main(){
char fileName[100];
int power,i,length;
cell* resultList;
printf("Enter input file name : ");
scanf("%s",fileName);
printf("Enter power : ");
scanf("%d",&power);
resultList = kroneckerProduct(fileName,power);
initwindow(raiseTo(ROW,power),raiseTo(COL,power),"Kronecker Product Fractal");
length = raiseTo(SUM,power);
for(i=0;i<length;i++){
putpixel(resultList[i].row,resultList[i].col,15);
}
getch();
closegraph();
return 0;
}
C++
This program produces image files in PNG format. The C++ code from Kronecker product is reused here.
#include <cassert>
#include <vector>
#include <QImage>
template <typename scalar_type> class matrix {
public:
matrix(size_t rows, size_t columns)
: rows_(rows), columns_(columns), elements_(rows * columns) {}
matrix(size_t rows, size_t columns,
const std::initializer_list<std::initializer_list<scalar_type>>& values)
: rows_(rows), columns_(columns), elements_(rows * columns) {
assert(values.size() <= rows_);
size_t i = 0;
for (const auto& row : values) {
assert(row.size() <= columns_);
std::copy(begin(row), end(row), &elements_[i]);
i += columns_;
}
}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
const scalar_type& operator()(size_t row, size_t column) const {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
scalar_type& operator()(size_t row, size_t column) {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
private:
size_t rows_;
size_t columns_;
std::vector<scalar_type> elements_;
};
// See https://en.wikipedia.org/wiki/Kronecker_product
template <typename scalar_type>
matrix<scalar_type> kronecker_product(const matrix<scalar_type>& a,
const matrix<scalar_type>& b) {
size_t arows = a.rows();
size_t acolumns = a.columns();
size_t brows = b.rows();
size_t bcolumns = b.columns();
matrix<scalar_type> c(arows * brows, acolumns * bcolumns);
for (size_t i = 0; i < arows; ++i)
for (size_t j = 0; j < acolumns; ++j)
for (size_t k = 0; k < brows; ++k)
for (size_t l = 0; l < bcolumns; ++l)
c(i*brows + k, j*bcolumns + l) = a(i, j) * b(k, l);
return c;
}
bool kronecker_fractal(const char* fileName, const matrix<unsigned char>& m, int order) {
matrix<unsigned char> result = m;
for (int i = 0; i < order; ++i)
result = kronecker_product(result, m);
size_t height = result.rows();
size_t width = result.columns();
size_t bytesPerLine = 4 * ((width + 3)/4);
std::vector<uchar> imageData(bytesPerLine * height);
for (size_t i = 0; i < height; ++i)
for (size_t j = 0; j < width; ++j)
imageData[i * bytesPerLine + j] = result(i, j);
QImage image(&imageData[0], width, height, bytesPerLine, QImage::Format_Indexed8);
QVector<QRgb> colours(2);
colours[0] = qRgb(0, 0, 0);
colours[1] = qRgb(255, 255, 255);
image.setColorTable(colours);
return image.save(fileName);
}
int main() {
matrix<unsigned char> matrix1(3, 3, {{0,1,0}, {1,1,1}, {0,1,0}});
matrix<unsigned char> matrix2(3, 3, {{1,1,1}, {1,0,1}, {1,1,1}});
matrix<unsigned char> matrix3(2, 2, {{1,1}, {0,1}});
kronecker_fractal("vicsek.png", matrix1, 5);
kronecker_fractal("sierpinski_carpet.png", matrix2, 5);
kronecker_fractal("sierpinski_triangle.png", matrix3, 8);
return 0;
}
- Output:
Media:Kronecker fractals sierpinski carpet.png
Media:Kronecker fractals sierpinski triangle.png
Media:Kronecker fractals vicsek.png
EasyLang
func[][] krpr a[][] b[][] .
for m = 1 to len a[][]
for p = 1 to len b[][]
r[][] &= [ ]
for n = 1 to len a[m][]
for q = 1 to len b[p][]
r[$][] &= a[m][n] * b[p][q]
.
.
.
.
return r[][]
.
func[][] krpow a[][] n .
r[][] = [ [ 1 ] ]
for i to n
r[][] = krpr a[][] r[][]
.
return r[][]
.
proc show p[][] . .
clear
n = len p[][]
sc = 100 / n
for r to n
for c to n
x = (c - 1) * sc
y = (r - 1) * sc
move x y
if p[r][c] = 1
rect sc sc
.
.
.
.
show krpow [ [ 1 1 1 ] [ 1 0 1 ] [ 1 1 1 ] ] 5
sleep 2
show krpow [ [ 0 1 0 ] [ 1 1 1 ] [ 0 1 0 ] ] 5
Factor
USING: io kernel math math.matrices.extras sequences ;
: mat-kron-pow ( m n -- m' )
1 - [ dup kronecker-product ] times ;
: print-fractal ( m -- )
[ [ 1 = "*" " " ? write ] each nl ] each ;
{ { 0 1 0 } { 1 1 1 } { 0 1 0 } }
{ { 1 1 1 } { 1 0 1 } { 1 1 1 } }
{ { 0 1 1 } { 0 1 0 } { 1 1 0 } }
[ 3 mat-kron-pow print-fractal ] tri@
Output shown at order 4 and 25% font size.
- Output:
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*** *** *** *** *** *** *** ***
* * * * * * * * * * * * * * * *
*** *** *** *** *** *** *** ***
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
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Fortran
A Fortran 90 implementation. Uses dense matrices and dynamic allocation for working arrays.
program Kron_frac
implicit none
interface
function matkronpow(M, n) result(Mpowern)
integer, dimension(:,:), intent(in) :: M
integer, intent(in) :: n
integer, dimension(size(M, 1)**n, size(M,2)**n) :: Mpowern
end function matkronpow
function kron(A, B) result(M)
integer, dimension(:,:), intent(in) :: A, B
integer, dimension(size(A,1)*size(B,1), size(A,2)*size(B,2)) :: M
end function kron
subroutine write2file(M, filename)
integer, dimension(:,:), intent(in) :: M
character(*), intent(in) :: filename
end subroutine write2file
end interface
integer, parameter :: n = 4
integer, dimension(3,3) :: Vicsek, Sierpinski
integer, dimension(4,4) :: Hadamard
integer, dimension(3**n, 3**n) :: fracV, fracS
integer, dimension(4**n, 4**n) :: fracH
Vicsek = reshape( (/0, 1, 0,&
1, 1, 1,&
0, 1, 0/),&
(/3,3/) )
Sierpinski = reshape( (/1, 1, 1,&
1, 0, 1,&
1, 1, 1/),&
(/3,3/) )
Hadamard = transpose(reshape( (/ 1, 0, 1, 0,&
1, 0, 0, 1,&
1, 1, 0, 0,&
1, 1, 1, 1/),&
(/4,4/) ))
fracV = matkronpow(Vicsek, n)
fracS = matkronpow(Sierpinski, n)
fracH = matkronpow(Hadamard, n)
call write2file(fracV, 'Viczek.txt')
call write2file(fracS, 'Sierpinski.txt')
call write2file(fracH, 'Hadamard.txt')
end program
function matkronpow(M, n) result(Mpowern)
interface
function kron(A, B) result(M)
integer, dimension(:,:), intent(in) :: A, B
integer, dimension(size(A,1)*size(B,1), size(A,2)*size(B,2)) :: M
end function kron
end interface
integer, dimension(:,:), intent(in) :: M
integer, intent(in) :: n
integer, dimension(size(M, 1)**n, size(M,2)**n) :: Mpowern
integer, dimension(:,:), allocatable :: work1, work2
integer :: icount
if (n <= 1) then
Mpowern = M
else
allocate(work1(size(M,1), size(M,2)))
work1 = M
do icount = 2,n
allocate(work2(size(M,1)**icount, size(M,2)**icount))
work2 = kron(work1, M)
deallocate(work1)
allocate(work1(size(M,1)**icount, size(M,2)**icount))
work1 = work2
deallocate(work2)
end do
Mpowern = work1
deallocate(work1)
end if
end function matkronpow
function kron(A, B) result(M)
integer, dimension(:,:), intent(in) :: A, B
integer, dimension(size(A,1)*size(B,1), size(A,2)*size(B,2)) :: M
integer :: ia, ja, ib, jb, im, jm
do ja = 1, size(A, 2)
do ia = 1, size(A, 1)
do jb = 1, size(B, 2)
do ib = 1, size(B, 1)
im = (ia - 1)*size(B, 1) + ib
jm = (ja - 1)*size(B, 2) + jb
M(im, jm) = A(ia, ja) * B(ib, jb)
end do
end do
end do
end do
end function kron
subroutine write2file(M, filename)
integer, dimension(:,:), intent(in) :: M
character(*), intent(in) :: filename
integer :: ii, jj
integer, parameter :: fi = 10
open(fi, file=filename, status='replace')
do ii = 1,size(M, 1)
do jj = 1,size(M,2)
if (M(ii,jj) == 0) then
write(fi, '(A)', advance='no') ' '
else
write(fi, '(A)', advance='no') '*'
end if
end do
write(fi, '(A)') ' '
end do
close(fi)
end subroutine write2file
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* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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******************************** ******************************** ******************************** ********************************
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** **
****************************************************************************************************************************************************************************************************************************************************************
FreeBASIC
Type Matrix
As Integer x
As Integer y
As Integer Ptr Dato
End Type
Function kroneckerProduct(a As Matrix, b As Matrix) As Matrix
Dim As Integer m = a.x, n = a.y
Dim As Integer p = b.x, q = b.y
Dim As Matrix r
r.x = m * p
r.y = n * q
r.dato = Callocate(r.x * r.y, Sizeof(Integer))
Dim As Integer i, j, k, l
For i = 0 To m - 1
For j = 0 To n - 1
For k = 0 To p - 1
For l = 0 To q - 1
r.dato[(p * i + k) * r.y + (q * j + l)] = a.dato[i * a.y + j] * b.dato[k * b.y + l]
Next
Next
Next
Next
Return r
End Function
Function kroneckerPower(a As Matrix, n As Integer) As Matrix
Dim As Matrix pow = a
For i As Integer = 1 To n - 1
pow = kroneckerProduct(pow, a)
Next
Return pow
End Function
Sub printMatrix(text As String, m As Matrix)
Dim As Integer i, j
Print text & " fractal:"
For i = 0 To m.x - 1
For j = 0 To m.y - 1
Print Iif(m.dato[i * m.y + j] = 1, "*", " ");
Next
Print
Next
Print
End Sub
Dim As Matrix a = Type(3, 3, Callocate(9, Sizeof(Integer)))
a.dato[0] = 0: a.dato[1] = 1: a.dato[2] = 0
a.dato[3] = 1: a.dato[4] = 1: a.dato[5] = 1
a.dato[6] = 0: a.dato[7] = 1: a.dato[8] = 0
printMatrix("Vicsek", kroneckerPower(a, 4))
a.dato[0] = 1: a.dato[1] = 1: a.dato[2] = 1
a.dato[3] = 1: a.dato[4] = 0: a.dato[5] = 1
a.dato[6] = 1: a.dato[7] = 1: a.dato[8] = 1
printMatrix("Sierpinski carpet", kroneckerPower(a, 4))
Sleep
- Output:
Same as Kotlin entry.
gnuplot
File for the load command is the only possible imitation of the fine function in the gnuplot.
- Note
- Find plotff.gp here on RC
- dat-files are PARI/GP generated output files. They are too big to post them here on RC.
## KPF.gp 4/8/17 aev
## Plotting 3 KPF pictures.
## dat-files are PARI/GP generated output files:
#cd 'C:\gnupData'
##PKF1 from PARI/GP created file pkf1.dat
ttl = "Vicsec fractal"; clr = '"blue"'; filename = "pkf1";
load "plotff.gp"
##PKF2 from PARI/GP created file pkf2.dat
ttl = "Sierpinski carpet fractal"; clr = '"navy"';filename = "pkf2";
load "plotff.gp"
##PKF3 from PARI/GP created file pkf3.dat
ttl = "Sierpinski triangle fractal"; clr = '"dark-green"'; filename = "pkf3";
load "plotff.gp"
- Output:
3 plotted files: pkf1.png, pkf2.png and pkf3.png.
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
Test case 1. Vicsek fractal
Cross form
Saltire form
Test case 2. Sierpiński carpet fractal
Test case 3. Sierpiński triangle fractal
Test case 3. Other cases
Test case 4. Numbers between 0 and 1 can be used, to produce greyscale shades
(click or tap to see in real size)
Go
package main
import "fmt"
type matrix [][]int
func (m1 matrix) kroneckerProduct(m2 matrix) matrix {
m := len(m1)
n := len(m1[0])
p := len(m2)
q := len(m2[0])
rtn := m * p
ctn := n * q
r := make(matrix, rtn)
for i := range r {
r[i] = make([]int, ctn) // all elements zero by default
}
for i := 0; i < m; i++ {
for j := 0; j < n; j++ {
for k := 0; k < p; k++ {
for l := 0; l < q; l++ {
r[p*i+k][q*j+l] = m1[i][j] * m2[k][l]
}
}
}
}
return r
}
func (m matrix) kroneckerPower(n int) matrix {
pow := m
for i := 1; i < n; i++ {
pow = pow.kroneckerProduct(m)
}
return pow
}
func (m matrix) print(text string) {
fmt.Println(text, "fractal :\n")
for i := range m {
for j := range m[0] {
if m[i][j] == 1 {
fmt.Print("*")
} else {
fmt.Print(" ")
}
}
fmt.Println()
}
fmt.Println()
}
func main() {
m1 := matrix{{0, 1, 0}, {1, 1, 1}, {0, 1, 0}}
m1.kroneckerPower(4).print("Vivsek")
m2 := matrix{{1, 1, 1}, {1, 0, 1}, {1, 1, 1}}
m2.kroneckerPower(4).print("Sierpinski carpet")
}
- Output:
Same as Kotlin entry.
Haskell
This implementation compiles to javascript that runs in the browser using the ghcjs compiler . The reflex-dom library is used to help with svg rendering.
{-# LANGUAGE OverloadedStrings #-}
import Reflex
import Reflex.Dom
import Data.Map as DM (Map, fromList)
import Data.Text (Text, pack)
import Data.List (transpose)
-- Show Vicsek and Sierpinski Carpet fractals
main :: IO ()
main = mainWidget $ do
elAttr "h1" ("style" =: "color:black") $ text "Kroneker Product Based Fractals"
elAttr "a" ("href" =: "http://rosettacode.org/wiki/Kronecker_product_based_fractals#Haskell") $ text "Rosetta Code / Kroneker product based fractals / Haskell"
-- Show a Vicsek fractal
el "br" $ return ()
elAttr "h2" ("style" =: "color:brown") $ text "Vicsek Fractal"
showFractal [[0, 1, 0] ,[1, 1, 1] ,[0, 1, 0] ]
-- Show a Sierpinski Carpet fractal
el "br" $ return ()
elAttr "h2" ("style" =: "color:brown") $ text "Sierpinski Carpet Fractal"
showFractal [[1, 1, 1] ,[1, 0, 1] ,[1, 1, 1] ]
-- Size in pixels of an individual cell
cellSize :: Int
cellSize = 8
-- Given a "seed" matrix, generate and display a fractal.
showFractal :: MonadWidget t m => [[Int]] -> m ()
showFractal seed = do
let boardAttrs w h =
fromList [ ("width" , pack $ show $ w * cellSize)
, ("height", pack $ show $ h * cellSize)
]
fractals = iterate (kronekerProduct seed) seed
shown = fractals !! 3 -- the fourth fractal (starting from 0)
w = length $ head shown
h = length shown
elSvgns "svg" (constDyn $ boardAttrs w h) $ showMatrix shown
-- Compute the Kroneker product of two matrices.
kronekerProduct :: Num a => [[a]] -> [[a]] -> [[a]]
kronekerProduct xs ys =
let m0 = flip $ fmap.fmap.(*)
m1 = flip $ fmap.fmap.m0
in concat $ fmap (fmap concat.transpose) $ m1 xs ys
-- Show an entire matrix
showMatrix :: MonadWidget t m => [[Int]] -> m ()
showMatrix m = mapM_ showRow $ zip [0..] m
-- Show a single horizontal row of a matrix
showRow :: MonadWidget t m => (Int,[Int]) -> m ()
showRow (x,r) = mapM_ (showCell x) $ zip [0..] r
-- Show a circle in a box moved to the correct location on screen
showCell :: MonadWidget t m => Int -> (Int,Int) -> m ()
showCell x (y,on) =
let boxAttrs (x,y) = -- Place box on screen
fromList [ ("transform",
pack $ "scale (" ++ show cellSize ++ ", " ++ show cellSize ++ ") "
++ "translate (" ++ show x ++ ", " ++ show y ++ ")"
)
]
cellAttrs = -- Draw circle in box.
fromList [ ( "cx", "0.5")
, ( "cy", "0.5")
, ( "r", "0.45")
, ( "style", "fill:green")
]
in if (on==1) then -- Only draw circle for elements containing 1
elSvgns "g" (constDyn $ boxAttrs (x,y)) $
elSvgns "circle" (constDyn $ cellAttrs) $
return ()
else
return ()
-- Wrapper around elDynAttrNS'
elSvgns :: MonadWidget t m => Text -> Dynamic t (Map Text Text) -> m a -> m a
elSvgns t m ma = do
(el, val) <- elDynAttrNS' (Just "http://www.w3.org/2000/svg") t m ma
return val
Link to live demo: https://dc25.github.io/rosettaCode__Kronecker_product_based_fractals/ ( a little slow to load ).
J
Implementation:
V=: -.0 2 6 8 e.~i.3 3
S=: 4 ~:i.3 3
KP=: 1 3 ,/"2@(,/)@|: */
ascii_art=: ' *'{~]
KPfractal=:dyad def 'x&KP^:y,.1'
Task examples (order 4, 25% font size):
ascii_art S KPfractal 4
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*** *** *** *** *** *** *** ***
* * * * * * * * * * * * * * * *
*** *** *** *** *** *** *** ***
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
ascii_art V KPfractal 4
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
Java
This implementation does not use sparse matrices since the powers involved do not exceed 4.
package kronecker;
/**
* Uses the Kronecker product powers of two rectangular matrices
* to generate fractals and tests it with three examples.
*/
public class ProductFractals {
/**
* Find the Kronecker product of the arguments.
* @param a The first matrix to multiply.
* @param b The second matrix to multiply.
* @return A new matrix: the Kronecker product of the arguments.
*/
public static int[][] product(final int[][] a, final int[][] b) {
// Create matrix c as the matrix to fill and return.
// The length of a matrix is its number of rows.
final int[][] c = new int[a.length*b.length][];
// Fill in the (empty) rows of c.
// The length of each row is the number of columns.
for (int ix = 0; ix < c.length; ix++) {
final int num_cols = a[0].length*b[0].length;
c[ix] = new int[num_cols];
}
// Now fill in the values: the products of each pair.
// Go through all the elements of a.
for (int ia = 0; ia < a.length; ia++) {
for (int ja = 0; ja < a[ia].length; ja++) {
// For each element of a, multiply it by all the elements of b.
for (int ib = 0; ib < b.length; ib++) {
for (int jb = 0; jb < b[ib].length; jb++) {
c[b.length*ia+ib][b[ib].length*ja+jb] = a[ia][ja] * b[ib][jb];
}
}
}
}
// Return the completed product matrix c.
return c;
}
/**
* Print an image obtained from an integer matrix, using the specified
* characters to indicate non-zero and zero elements.
* @param m The matrix to print.
* @param nz The character to print for a non-zero element.
* @param z The character to print for a zero element.
*/
public static void show_matrix(final int[][] m, final char nz, final char z) {
for (int im = 0; im < m.length; im++) {
for (int jm = 0; jm < m[im].length; jm++) {
System.out.print(m[im][jm] == 0 ? z : nz);
}
System.out.println();
}
}
/**
* Compute the specified Kronecker product power
* of the matrix and return it.
* @param m The matrix to raise to the power.
* @param n The power to which to raise the matrix.
* @return A new matrix containing the resulting power.
*/
public static int[][] power(final int[][] m, final int n) {
// Start with m itself as the first power.
int[][] m_pow = m;
// Start the iteration with 1, not 0,
// since we already have the first power.
for (int ix = 1; ix < n; ix++) {
m_pow = product(m, m_pow);
}
return m_pow;
}
/**
* Run a test by computing the specified Kronecker product power
* of the matrix and printing matrix and power.
* @param m The base matrix raise to the power.
* @param n The power to which to raise the matrix.
*/
private static void test(final int[][] m, final int n) {
System.out.println("Test matrix");
show_matrix(m, '*', ' ');
final int[][] m_pow = power(m, n);
System.out.println("Matrix power " + n);
show_matrix(m_pow, '*', ' ');
}
/**
* Create the matrix for the first test and run the test.
*/
private static void test1() {
// Create the matrix.
final int[][] m = {{0, 1, 0},
{1, 1, 1},
{0, 1, 0}};
// Run the test.
test(m, 4);
}
/**
* Create the matrix for the second test and run the test.
*/
private static void test2() {
// Create the matrix.
final int[][] m = {{1, 1, 1},
{1, 0, 1},
{1, 1, 1}};
// Run the test.
test(m, 4);
}
/**
* Create the matrix for the second test and run the test.
*/
private static void test3() {
// Create the matrix.
final int[][] m = {{1, 0, 1},
{1, 0, 1},
{0, 1, 0}};
// Run the test.
test(m, 4);
}
/**
* Run the program to run the three tests.
* @param args Command line arguments (not used).
*/
public static void main(final String[] args) {
// Test the product fractals.
test1();
test2();
test3();
}
}
- Output:
This output uses 50% font size. Of course, it shows ASCII height distortion.
Test matrix
*
***
*
Matrix power 4
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
Test matrix
***
* *
***
Matrix power 4
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*** *** *** *** *** *** *** ***
* * * * * * * * * * * * * * * *
*** *** *** *** *** *** *** ***
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
Test matrix
* *
* *
*
Matrix power 4
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * *
* * * *
* *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * *
* * * *
* *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * *
* * * *
* *
* * * * * * * *
* * * * * * * *
* * * *
* * * * * * * *
* * * * * * * *
* * * *
* * * *
* * * *
* *
* * * *
* * * *
* *
* * * *
* * * *
* *
* *
* *
*
JavaScript
Using Version #1 of Kronecker product in JavaScript.
// KPF.js 6/23/16 aev
// HFJS: Plot any matrix mat (filled with 0,1)
function pmat01(mat, color) {
// DCLs
var cvs = document.getElementById('canvId');
var ctx = cvs.getContext("2d");
var w = cvs.width; var h = cvs.height;
var m = mat[0].length; var n = mat.length;
// Cleaning canvas and setting plotting color
ctx.fillStyle="white"; ctx.fillRect(0,0,w,h);
ctx.fillStyle=color;
// MAIN LOOP
for(var i=0; i<m; i++) {
for(var j=0; j<n; j++) {
if(mat[i][j]==1) { ctx.fillRect(i,j,1,1)};
}//fend j
}//fend i
}//func end
// Prime functions:
// Create Kronecker product based fractal matrix rm from matrix m (order=ord)
function ckpbfmat(m,ord) {
var rm=m;
for(var i=1; i<ord; i++) {rm=mkp(rm,m)};
//matpp2doc('R 4 ordd',rm,'*'); // ASCII "plotting" - if you wish to try.
return(rm);
}
// Create and plot Kronecker product based fractal from matrix m (filled with 0/1)
function cpmat(m,ord,color) {
var kpr;
kpr=ckpbfmat(m,ord);
pmat01(kpr,color);
}
// Fractal matrix "pretty" printing to document.
// mat should be filled with 0 and 1; chr is a char substituting 1.
function matpp2doc(title,mat,chr) {
var i,j,re='',e; var m=mat.length; var n=mat[0].length;
document.write(' <b>'+title+'</b>:<pre>');
for(var i=0; i<m; i++) {
for(var j=0; j<n; j++) {
e=' '; if(mat[i][j]==1) {e=chr}; re+=e;
}//fend j
document.write(' '+re+'<br />'); re='';
}//fend i
document.write('</pre>');
}
// mkp function (exotic arrow function): Return the Kronecker product
// of the a and b matrices
mkp=(a,b)=>a.map(a=>b.map(b=>a.map(y=>b.map(x=>r.push(y*x)),t.push(r=[]))),t=[])&&t;
- Required tests
<!-- VicsekFractal.html -->
<html>
<head>
<title>Vicsek fractal</title>
<script src="KPF.js"></script>
</head>
<body onload="cpmat([[0,1,0],[1,1,1],[0,1,0]],6,'navy')">
<h3>Vicsek fractal</h3>
<a href="SierpCarpetFractal.html"> Next: Sierpinski carpet fractal</a><br />
<canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>
</body></html>
<!-- SierpCarpetFractal.html -->
<html>
<head>
<title>Sierpinski carpet fractal</title>
<script src="KPF.js"></script>
</head>
<body onload="cpmat([[1,1,1],[1,0,1],[1,1,1]],6,'brown')">
<h3>Sierpinski carpet fractal</h3>
<a href="Checkerboard.html"/> Next: Checkerboard </a><br />
<canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>
</body></html>
<!-- Checkerboard.html -->
<html>
<head>
<title>Checkerboard</title>
<script src="KPF.js"></script>
</head>
<body onload="cpmat([[0,1,0,1],[1,0,1,0],[0,1,0,1],[1,0,1,0]],5,'black')">
<h3>Checkerboard</h3>
<a href="VicsekFractal.html"/> Next: Vicsek fractal </a><br />
<canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>
</body></html>
- Output:
Page VicsekFractal.html with VicsekFractaljs.png Page SierpCarpetFractal.html with SierpCarpetFractaljs.png Page Checkerboard.html with CheckbrdFractaljs.png
Julia
Julia has a builtin function `kron`:
function matkronpow(M::Matrix, n::Int)
P = copy(M)
for i in 1:n P = kron(P, M) end
return P
end
function fracprint(M::Matrix)
for i in 1:size(M, 1)
for j in 1:size(M, 2)
print(M[i, j] == 1 ? '*' : ' ')
end
println()
end
end
M = [0 1 0; 1 1 1; 0 1 0]
matkronpow(M, 3) |> fracprint
M = [1 1 1; 1 0 1; 1 1 1]
matkronpow(M, 3) |> fracprint
- Output:
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*** *** *** *** *** *** *** ***
* * * * * * * * * * * * * * * *
*** *** *** *** *** *** *** ***
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
Kotlin
This reuses code from the Kronecker_product#Kotlin task.
// version 1.2.31
typealias Matrix = Array<IntArray>
fun kroneckerProduct(a: Matrix, b: Matrix): Matrix {
val m = a.size
val n = a[0].size
val p = b.size
val q = b[0].size
val rtn = m * p
val ctn = n * q
val r: Matrix = Array(rtn) { IntArray(ctn) } // all elements zero by default
for (i in 0 until m)
for (j in 0 until n)
for (k in 0 until p)
for (l in 0 until q)
r[p * i + k][q * j + l] = a[i][j] * b[k][l]
return r
}
fun kroneckerPower(a: Matrix, n: Int): Matrix {
var pow = a.copyOf()
for (i in 1 until n) pow = kroneckerProduct(pow, a)
return pow
}
fun printMatrix(text: String, m: Matrix) {
println("$text fractal :\n")
for (i in 0 until m.size) {
for (j in 0 until m[0].size) {
print(if (m[i][j] == 1) "*" else " ")
}
println()
}
println()
}
fun main(args: Array<String>) {
var a = arrayOf(
intArrayOf(0, 1, 0),
intArrayOf(1, 1, 1),
intArrayOf(0, 1, 0)
)
printMatrix("Vicsek", kroneckerPower(a, 4))
a = arrayOf(
intArrayOf(1, 1, 1),
intArrayOf(1, 0, 1),
intArrayOf(1, 1, 1)
)
printMatrix("Sierpinski carpet", kroneckerPower(a, 4))
}
- Output:
Vicsek fractal :
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
Sierpinski carpet fractal :
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*** *** *** *** *** *** *** ***
* * * * * * * * * * * * * * * *
*** *** *** *** *** *** *** ***
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
Lua
Needs LÖVE 2D Engine
function prod( a, b )
local rt, l = {}, 1
for m = 1, #a do
for p = 1, #b do
rt[l] = {}
for n = 1, #a[m] do
for q = 1, #b[p] do
table.insert( rt[l], a[m][n] * b[p][q] )
end
end
l = l + 1
end
end
return rt
end
function love.load()
wid, hei = love.graphics.getWidth(), love.graphics.getHeight()
canvas = love.graphics.newCanvas( wid, hei )
mA = { {0,1,0}, {1,1,1}, {0,1,0} }; mB = { {1,0,1}, {0,1,0}, {1,0,1} }
mC = { {1,1,1}, {1,0,1}, {1,1,1} }; mD = { {1,1,1}, {0,1,0}, {1,1,1} }
end
function drawFractals( m )
love.graphics.setCanvas( canvas )
love.graphics.clear()
love.graphics.setColor( 255, 255, 255 )
for j = 1, #m do
for i = 1, #m[j] do
if m[i][j] == 1 then
love.graphics.points( i * .1, j * .1 )
end
end
end
love.graphics.setCanvas()
end
function love.keypressed( key, scancode, isrepeat )
local t = {}
if key == "a" then
print( "Build Vicsek fractal I" ); t = mA
elseif key == "b" then
print( "Build Vicsek fractal II" ); t = mB
elseif key == "c" then
print( "Sierpinski carpet fractal" ); t = mC
elseif key == "d" then
print( "Build 'H' fractal" ); t = mD
else return
end
for i = 1, 3 do t = prod( t, t ) end
drawFractals( t )
end
function love.draw()
love.graphics.draw( canvas )
end
Mathematica /Wolfram Language
m = {{0, 1, 0}, {1, 1, 1}, {0, 1, 0}};
ArrayPlot[KroneckerProduct[m, m, m, m]]
m = {{1, 1, 1}, {1, 0, 1}, {1, 1, 1}};
ArrayPlot[KroneckerProduct[m, m, m, m]]
m = {{0, 1, 1}, {1, 0, 1}, {1, 1, 0}};
ArrayPlot[KroneckerProduct[m, m, m, m]]
- Output:
Outputs three graphical visualisations of the three 4th order products.
Maxima
Using function defined in Kronecker product task page. [Kronecker Product]
pow_kron(matr,n):=block(MATR:copymatrix(matr),
for i from 1 thru n do MATR:altern_kronecker(matr,MATR),
MATR);
/* Examples (images are shown in format png)*/
/* A to generate Vicsek fractal */
/* B to generate Sierpinski carpet fractal */
A:matrix([0,1,0],[1,1,1],[0,1,0])$
B:matrix([1,1,1],[1,0,1],[1,1,1])$
/* Vicsek */
pow_kron(A,3)$
at(%,[0="",1="x"]);
/* Sierpinski carpet */
pow_kron(B,3)$
at(%,[0="",1="x"]);
Nim
import sequtils
type Matrix[T] = seq[seq[T]]
func kroneckerProduct[T](a, b: Matrix[T]): Matrix[T] =
result = newSeqWith(a.len * b.len, newSeq[T](a[0].len * b[0].len))
let m = a.len
let n = a[0].len
let p = b.len
let q = b[0].len
for i in 0..<m:
for j in 0..<n:
for k in 0..<p:
for l in 0..<q:
result[i * p + k][j * q + l] = a[i][j] * b[k][l]
func kroneckerPower(m: Matrix; n: int): Matrix =
result = m
for i in 2..n:
result = kroneckerProduct(result, m)
func `$`(m: Matrix): string =
for row in m:
for val in row:
result.add if val == 0: " " else: " *"
result.add '\n'
type B = range[0..1]
const A1: Matrix[B] = @[@[B 0, 1, 0], @[B 1, 1, 1], @[B 0, 1, 0]]
echo "Vicsek fractal:\n", A1.kroneckerPower(4)
echo ""
const A2: Matrix[B] = @[@[B 1, 1, 1], @[B 1, 0, 1], @[B 1, 1, 1]]
echo "Sierpinski carpet fractal:\n", A2.kroneckerPower(4)
- Output:
Vicsek fractal:
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Sierpinski carpet fractal:
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PARI/GP
- Note
- Find iPlotmat() here on Helper Functions page.
- Find matkronprod() here on Kronecker product page.
\\ Build block matrix applying Kronecker product to the special matrix m
\\ (n times to itself). Then plot Kronecker fractal. 4/25/2016 aev
pkronfractal(m,n=2,clr)={
my(r=m);
for(i=1,n, r=matkronprod(r,m));
iPlotmat(r,clr);
}
\\Requireq tests:
{\\ Vicsek fractal: VicsekFractalgp.png
my(M=[0,1,0;1,1,1;0,1,0]); print(" *** Vicsek fractal, order 4:");
pkronfractal(M,4,6);
}
{\\ Sierpinski carpet fractal: SierpCarpetFractalgp.png
my(M=[1,1,1;1,0,1;1,1,1]); print(" *** Sierpinski carpet fractal, order 4:");
pkronfractal(M,4,5);
}
{\\ Sierpinski triangle fractal: SierpTriFractalgp.png
my(M=[1,1;0,1]); print(" *** Sierpinski triangle fractal, order 7:");
pkronfractal(M,7,6);
}- Output:
*** Vicsek fractal, order 4: *** matrix(243x243) 3125 DOTS *** Sierpinski carpet fractal, order 4: *** matrix(243x243) 32768 DOTS *** Sierpinski triangle fractal, order 7: *** matrix: 256x256, 6561 DOTS
Perl
use Imager;
use Math::Cartesian::Product;
sub kronecker_product {
our @a; local *a = shift;
our @b; local *b = shift;
my @c;
cartesian {
my @cc;
cartesian {
push @cc, $_[0] * $_[1];
} [@{$_[0]}], [@{$_[1]}];
push @c, [@cc];
} [@a], [@b];
@c
}
sub kronecker_fractal {
my($order, @pattern) = @_;
my @kronecker = @pattern;
@kronecker = kronecker_product(\@kronecker, \@pattern) for 0..$order-1;
@kronecker
}
@vicsek = ( [0, 1, 0], [1, 1, 1], [0, 1, 0] );
@carpet = ( [1, 1, 1], [1, 0, 1], [1, 1, 1] );
@six = ( [0,1,1,1,0], [1,0,0,0,1], [1,0,0,0,0], [1,1,1,1,0], [1,0,0,0,1], [1,0,0,0,1], [0,1,1,1,0] );
for (['vicsek', \@vicsek, 4],
['carpet', \@carpet, 4],
['six', \@six, 3]) {
($name, $shape, $order) = @$_;
@img = kronecker_fractal( $order, @$shape );
$png = Imager->new(xsize => 1+@{$img[0]}, ysize => 1+@img);
cartesian {
$png->setpixel(x => $_[0], y => $_[1], color => $img[$_[1]][$_[0]] ? [255, 255, 32] : [16, 16, 16]);
} [0..@{$img[0]}-1], [0..$#img];
$png->write(file => "run/kronecker-$name-perl6.png");
}
See Kronecker-Vicsek, Kronecker-Carpet and Kronecker-Six images.
Phix
function kronecker(sequence a, b) integer ar = length(a), ac = length(a[1]), br = length(b), bc = length(b[1]) sequence res = repeat(repeat(0,ac*bc),ar*br) for ia=1 to ar do integer i0 = (ia-1)*br for ja=1 to ac do integer j0 = (ja-1)*bc for ib=1 to br do integer i = i0+ib for jb=1 to bc do integer j = j0+jb res[i,j] = a[ia,ja]*b[ib,jb] end for end for end for end for return res end function function kroneckern(sequence m, integer n) sequence res = m for i=2 to n do res = kronecker(res,m) end for return res end function procedure show(sequence m) for i=1 to length(m) do string s = repeat(' ',length(m[i])) for j=1 to length(s) do if m[i][j] then s[j] = '#' end if end for puts(1,s&"\n") end for puts(1,"\n") end procedure constant vicsek = {{0,1,0}, {1,1,1}, {0,1,0}}, siercp = {{1,1,1}, {1,0,1}, {1,1,1}}, xxxxxx = {{0,1,1}, {0,1,0}, {1,1,0}} show(kroneckern(vicsek,4)) show(kroneckern(siercp,4)) show(kroneckern(xxxxxx,4))
Output same as Julia/Kotlin/Factor
Python
Generate images of the fractals using PIL.
Using only python lists
import os
from PIL import Image
def imgsave(path, arr):
w, h = len(arr), len(arr[0])
img = Image.new('1', (w, h))
for x in range(w):
for y in range(h):
img.putpixel((x, y), arr[x][y])
img.save(path)
def get_shape(mat):
return len(mat), len(mat[0])
def kron(matrix1, matrix2):
"""
Calculate the kronecker product of two matrices
"""
final_list = []
count = len(matrix2)
for elem1 in matrix1:
for i in range(count):
sub_list = []
for num1 in elem1:
for num2 in matrix2[i]:
sub_list.append(num1 * num2)
final_list.append(sub_list)
return final_list
def kronpow(mat):
"""
Generate an arbitrary number of kronecker powers
"""
matrix = mat
while True:
yield matrix
matrix = kron(mat, matrix)
def fractal(name, mat, order=6):
"""
Save fractal as jpg to 'fractals/name'
"""
path = os.path.join('fractals', name)
os.makedirs(path, exist_ok=True)
fgen = kronpow(mat)
print(name)
for i in range(order):
p = os.path.join(path, f'{i}.jpg')
print('Calculating n =', i, end='\t', flush=True)
mat = next(fgen)
imgsave(p, mat)
x, y = get_shape(mat)
print('Saved as', x, 'x', y, 'image', p)
test1 = [
[0, 1, 0],
[1, 1, 1],
[0, 1, 0]
]
test2 = [
[1, 1, 1],
[1, 0, 1],
[1, 1, 1]
]
test3 = [
[1, 0, 1],
[0, 1, 0],
[1, 0, 1]
]
fractal('test1', test1)
fractal('test2', test2)
fractal('test3', test3)
Because this is not very efficent/fast you should use scipy sparse matrices instead
import os
import numpy as np
from scipy.sparse import csc_matrix, kron
from scipy.misc import imsave
def imgsave(name, arr, *args):
imsave(name, arr.toarray(), *args)
def get_shape(mat):
return mat.shape
def kronpow(mat):
"""
Generate an arbitrary number of kronecker powers
"""
matrix = mat
while True:
yield matrix
matrix = kron(mat, matrix)
def fractal(name, mat, order=6):
"""
Save fractal as jpg to 'fractals/name'
"""
path = os.path.join('fractals', name)
os.makedirs(path, exist_ok=True)
fgen = kronpow(mat)
print(name)
for i in range(order):
p = os.path.join(path, f'{i}.jpg')
print('Calculating n =', i, end='\t', flush=True)
mat = next(fgen)
imgsave(p, mat)
x, y = get_shape(mat)
print('Saved as', x, 'x', y, 'image', p)
test1 = [
[0, 1, 0],
[1, 1, 1],
[0, 1, 0]
]
test2 = [
[1, 1, 1],
[1, 0, 1],
[1, 1, 1]
]
test3 = [
[1, 0, 1],
[0, 1, 0],
[1, 0, 1]
]
test1 = np.array(test1, dtype='int8')
test1 = csc_matrix(test1)
test2 = np.array(test2, dtype='int8')
test2 = csc_matrix(test2)
test3 = np.array(test3, dtype='int8')
test3 = csc_matrix(test3)
fractal('test1', test1)
fractal('test2', test2)
fractal('test3', test3)
R
Generate and plot 3 Kronecker product based fractals.
Note: Find plotmat() and plotv2() here on Helper Functions page.
## Generate and plot Kronecker product based fractals. aev 8/12/16
## gpKronFractal(m, n, pf, clr, ttl, dflg=0, psz=600):
## Where: m - initial matrix (filled with 0/1); n - order of the fractal;
## pf - plot file name (without extension); clr - color; ttl - plot title;
## dflg - writing dump file flag (0/1); psz - picture size.
gpKronFractal <- function(m, n, pf, clr, ttl, dflg=0, psz=600) {
cat(" *** START:", date(), "n=", n, "clr=", clr, "psz=", psz, "\n");
cat(" *** Plot file -", pf, "\n");
r <- m;
for(i in 1:n) {r = r%x%m};
plotmat(r, pf, clr, ttl, dflg, psz);
cat(" *** END:", date(), "\n");
}
## Required tests:
# 1. Vicsek Fractal
M <- matrix(c(0,1,0,1,1,1,0,1,0), ncol=3, nrow=3, byrow=TRUE);
gpKronFractal(M, 4, "VicsekFractalR","red", "Vicsek Fractal n=4")
# 2. Sierpinski carpet fractal
M <- matrix(c(1,1,1,1,0,1,1,1,1), ncol=3, nrow=3, byrow=TRUE);
gpKronFractal(M, 4, "SierpinskiCarpetFR", "maroon", "Sierpinski carpet fractal n=4")
# 3. Plus sign fractal
M <- matrix(c(1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,1,1,1,
+0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1), ncol=7, nrow=7, byrow=TRUE);
gpKronFractal(M, 3, "PlusSignFR", "maroon", "Plus sign fractal, n=3")
# Also, try these 3. I bet you've never seen them before.
# 4. Wider Sierpinski carpet fractal (a.k.a. Sierpinski carpet mutant)
# Note: If your computer is not super fast it could take a lot of time.
# Use dump flag = 1, to save generated fractal.
#M <- matrix(c(1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,1,1,1), ncol=5,
#+nrow=5, byrow=TRUE);
#gpKronFractal(M, 4, "SierpinskiCarpetFw", "brown", "Wider Sierpinski carpet fractal n=4", 1)
# 5. "H" fractal (Try all other letters in the alphabet...)
#M <- matrix(c(1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,
#+0,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1), ncol=7, nrow=7, byrow=TRUE);
#gpKronFractal(M, 3, "HFR", "maroon", "'H' fractal n=3", 1)
# 6. Chessboard fractal.
#M <- matrix(c(1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,
# 0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1), ncol=8, nrow=8, byrow=TRUE);
#gpKronFractal(M, 2, "ChessBrdFractalR","black", "Chessboard Fractal, n=2")
- Output:
> M <- matrix(c(0,1,0,1,1,1,0,1,0), ncol=3, nrow=3, byrow=TRUE); > gpKronFractal(M, 4, "VicsekFractalR", "red", "Vicsek Fractal n=4") *** START: Mon Aug 29 16:14:14 2016 n= 4 clr= red *** Plot file - VicsekFractalR *** Matrix( 243 x 243 ) 3125 DOTS *** END: Mon Aug 29 16:14:14 2016 > M <- matrix(c(1,1,1,1,0,1,1,1,1), ncol=3, nrow=3, byrow=TRUE); > gpKronFractal(M, 4, "SierpinskiCarpetFR", "maroon", "Sierpinski carpet fractal n=4") *** START: Mon Aug 29 16:16:14 2016 n= 4 clr= maroon *** Plot file - SierpinskiCarpetFR *** Matrix( 243 x 243 ) 32768 DOTS *** END: Mon Aug 29 16:16:32 2016 > M <- matrix(c(1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,1,1,1, +0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1), ncol=7, nrow=7, byrow=TRUE); > gpKronFractal(M, 3, "PlusSignFR", "maroon", "Plus sign fractal, n=3") *** START: Thu Apr 06 21:45:33 2017 n= 3 clr= maroon psz= 600 *** Plot file - PlusSignFR *** Matrix( 2401 x 2401 ) 2560000 DOTS *** END: Fri Apr 07 09:31:07 2017
Raku
(formerly Perl 6)
sub kronecker-product ( @a, @b ) { (@a X @b).map: { (.[0].list X* .[1].list).Array } }
sub kronecker-fractal ( @pattern, $order = 4 ) {
my @kronecker = @pattern;
@kronecker = kronecker-product(@kronecker, @pattern) for ^$order;
@kronecker
}
use Image::PNG::Portable;
#Task requirements
my @vicsek = ( [0, 1, 0], [1, 1, 1], [0, 1, 0] );
my @carpet = ( [1, 1, 1], [1, 0, 1], [1, 1, 1] );
my @six = ( [0,1,1,1,0], [1,0,0,0,1], [1,0,0,0,0], [1,1,1,1,0], [1,0,0,0,1], [1,0,0,0,1], [0,1,1,1,0] );
for 'vicsek', @vicsek, 4,
'carpet', @carpet, 4,
'six', @six, 3
-> $name, @shape, $order {
my @img = kronecker-fractal( @shape, $order );
my $png = Image::PNG::Portable.new: :width(@img[0].elems), :height(@img.elems);
(^@img[0]).race(:12batch).map: -> $x {
for ^@img -> $y {
$png.set: $x, $y, |( @img[$y;$x] ?? <255 255 32> !! <16 16 16> );
}
}
$png.write: "kronecker-{$name}-perl6.png";
}
See Kronecker-Vicsek, Kronecker-Carpet and Kronecker-Six images.
REXX
This is a work-in-progress, this version shows the 1st order.
/*REXX program calculates the Kronecker product of two arbitrary size matrices. */
parse arg pGlyph . /*obtain optional argument from the CL.*/
if pGlyph=='' | pGlyph=="," then pGlyph= '█' /*Not specified? Then use the default.*/
if length(pGlyph)==2 then pGlyph= x2c(pGlyph) /*Plot glyph is 2 chars? Hexadecimal.*/
if length(pGlyph)==3 then pGlyph= d2c(pGlyph) /* " " " 3 " Decimal. */
aMat= 3x3 0 1 0 1 1 1 0 1 0 /*define A matrix size and elements.*/
bMat= 3x3 1 1 1 1 0 1 1 1 1 /* " B " " " " */
call makeMat 'A', aMat /*construct A matrix from elements.*/
call makeMat 'B', bMat /* " B " " " */
call KronMat 'Kronecker product' /*calculate the Kronecker product. */
call showMat 'Kronecker product', result /*display the Kronecker product. */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
KronMat: parse arg what; #= 0; parse var @.a.shape aRows aCols
parse var @.b.shape bRows bCols
do rA=1 for aRows
do rB=1 for bRows; #= # + 1; ##= 0; _=
do cA=1 for aCols; x= @.a.rA.cA
do cB=1 for bCols; y= @.b.rB.cB; ##= ## + 1; xy= x * y; _= _ xy
@.what.#.##= xy
end /*cB*/
end /*cA*/
end /*rB*/
end /*rA*/; return aRows * aRows || 'X' || bRows * bRows
/*──────────────────────────────────────────────────────────────────────────────────────*/
makeMat: parse arg what, size elements; arg , row 'X' col .; @.what.shape= row col
#=0; do r=1 for row /* [↓] bump item#; get item; max width*/
do c=1 for col; #= # + 1; @.what.r.c= word(elements, #)
end /*c*/ /* [↑] define an element of WHAT matrix*/
end /*r*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
showMat: parse arg what, size .; parse var size row 'X' col /*obtain mat name, sz*/
do r=1 for row; $= /*build row by row. */
do c=1 for col; $= $ || @.what.r.c /* " col " col. */
end /*c*/
$= translate($, pGlyph, 10) /*change──►plot glyph*/
say strip($, 'T') /*display line──►term*/
end /*r*/; return
- output when using the default input:
███ █ █ ███ █████████ █ ██ ██ █ █████████ ███ █ █ ███
Rust
Because Rust lacks support for images, this sample contains a simple implementation of writing PPM files.
use std::{
fmt::{Debug, Display, Write},
ops::Mul,
};
// Rust has (almost) no built-in support for multi-dimensional arrays or so.
// Let's make a basic one ourselves for our use cases.
#[derive(Clone, Debug)]
pub struct Mat<T> {
col_count: usize,
row_count: usize,
items: Vec<T>,
}
impl<T> Mat<T> {
pub fn from_vec(items: Vec<T>, col_count: usize, row_count: usize) -> Self {
assert_eq!(items.len(), col_count * row_count, "mismatching dimensions");
Self {
col_count,
row_count,
items,
}
}
pub fn row_count(&self) -> usize {
self.row_count
}
pub fn col_count(&self) -> usize {
self.col_count
}
pub fn iter(&self) -> impl Iterator<Item = &T> {
self.items.iter()
}
pub fn row_iter(&self, row: usize) -> impl Iterator<Item = &T> {
assert!(row < self.row_count, "index out of bounds");
let start = row * self.col_count;
self.items[start..start + self.col_count].iter()
}
pub fn col_iter(&self, col: usize) -> impl Iterator<Item = &T> {
assert!(col < self.col_count, "index out of bounds");
self.items.iter().skip(col).step_by(self.col_count)
}
}
impl<T: Display> Display for Mat<T> {
fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
// Compute the width of the widest item first
let mut len = 0usize;
let mut buf = String::new();
for item in (0..self.row_count).flat_map(|row| self.row_iter(row)) {
buf.clear();
write!(buf, "{}", item)?;
len = std::cmp::max(len, buf.chars().count());
}
// Then render the matrix with proper padding
len += 1; // To separate cells
let width = len * self.col_count + 1;
writeln!(f, "┌{:width$}┐", "", width = width)?;
for row in (0..self.row_count).map(|row| self.row_iter(row)) {
write!(f, "│")?;
for item in row {
write!(f, "{:>width$}", item, width = len)?;
}
writeln!(f, " │")?;
}
write!(f, "└{:width$}┘", "", width = width)
}
}
// Rust standard libraries have no graphics support. If we want to render
// an image, we can write, e.g., a PPM file.
impl<T> Mat<T> {
pub fn write_ppm(
&self,
f: &mut dyn std::io::Write,
rgb: impl Fn(&T) -> (u8, u8, u8),
) -> std::io::Result<()> {
let bytes = self
.iter()
.map(rgb)
.flat_map(|(r, g, b)| {
use std::iter::once;
once(r).chain(once(g)).chain(once(b))
})
.collect::<Vec<u8>>();
write!(f, "P6\n{} {}\n255\n", self.col_count, self.row_count)?;
f.write_all(&bytes)
}
}
mod kronecker {
use super::Mat;
use std::ops::Mul;
// Look ma, no numbers! We can combine anything with Mul (see later)
pub fn product<T, U>(a: &Mat<T>, b: &Mat<U>) -> Mat<<T as Mul<U>>::Output>
where
T: Clone + Mul<U>,
U: Clone,
{
let row_count = a.row_count() * b.row_count();
let col_count = a.col_count() * b.col_count();
let mut items = Vec::with_capacity(row_count * col_count);
for i in 0..a.row_count() {
for k in 0..b.row_count() {
for a_x in a.row_iter(i) {
for b_x in b.row_iter(k) {
items.push(a_x.clone() * b_x.clone());
}
}
}
}
Mat::from_vec(items, col_count, row_count)
}
pub fn power<T>(m: &Mat<T>, n: u32) -> Mat<T>
where
T: Clone + Mul<T, Output = T>,
{
match n {
0 => m.clone(),
_ => (1..n).fold(product(&m, &m), |result, _| product(&result, &m)),
}
}
}
// Here we make a char-like type with Mul implementation.
// We can do fancy things with that later.
#[derive(Clone, Copy, Debug, PartialEq, Eq, PartialOrd, Ord)]
struct Char(char);
impl Char {
fn space() -> Self {
Char(' ')
}
fn is_space(&self) -> bool {
self.0 == ' '
}
}
impl Display for Char {
fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
Display::fmt(&self.0, f)
}
}
impl Mul for Char {
type Output = Self;
#[allow(clippy::suspicious_arithmetic_impl)]
fn mul(self, rhs: Self) -> Self {
if self.is_space() || rhs.is_space() {
Char(' ')
} else {
self
}
}
}
fn main() -> std::io::Result<()> {
// Vicsek rendered in numbers
#[rustfmt::skip]
let vicsek = Mat::<u8>::from_vec(vec![
0, 1, 0,
1, 1, 1,
0, 1, 0,
], 3, 3);
println!("{}", vicsek);
println!("{}", kronecker::power(&vicsek, 3));
// We could render something by mapping the numbers to
// something else. But we could compute with something
// else directly, right?
let s = Char::space();
let b = Char('\u{2588}');
#[rustfmt::skip]
let sierpienski = Mat::from_vec(vec![
b, b, b,
b, s, b,
b, b, b,
], 3, 3);
println!("{}", sierpienski);
println!("{}", kronecker::power(&sierpienski, 3));
#[rustfmt::skip]
let matrix = Mat::from_vec(vec![
s, s, b, s, s,
s, b, b, b, s,
b, s, b, s, b,
s, s, b, s, s,
s, b, s, b, s,
], 5, 5,);
println!("{}", kronecker::power(&matrix, 1));
// This is nicer as an actual image
kronecker::power(&matrix, 4).write_ppm(
&mut std::fs::OpenOptions::new()
.write(true)
.create(true)
.truncate(true)
.open("kronecker_power.ppm")?,
|&item| {
if item.is_space() {
(0, 0, 32)
} else {
(192, 192, 0)
}
},
)
}
Sidef
func kronecker_product (a, b) { a ~X b -> map { _[0] ~X* _[1] } }
func kronecker_fractal(pattern, order=4) {
var kronecker = pattern
{ kronecker = kronecker_product(kronecker, pattern) } * order
return kronecker
}
var vicsek = [[0,1,0], [1,1,1], [0,1,0]]
var carpet = [[1,1,1], [1,0,1], [1,1,1]]
var six = [[0,1,1,1,0], [1,0,0,0,1], [1,0,0,0,0],
[1,1,1,1,0], [1,0,0,0,1], [1,0,0,0,1], [0,1,1,1,0]]
require("Imager")
for name,shape,order in [
[:vicsek, vicsek, 4],
[:carpet, carpet, 4],
[:six, six, 3],
] {
var pat = kronecker_fractal(shape, order)
var img = %O<Imager>.new(xsize => pat[0].len, ysize => pat.len)
for x,y in (^pat[0].len ~X ^pat.len) {
img.setpixel(x => x, y => y, color => (pat[y][x] ? <255 255 32> : <16 16 16>))
}
img.write(file => "kronecker-#{name}-sidef.png")
}
Output images: Kronecker Carpet, Kronecker Vicsek and Kronecker Six
Wren
import "./matrix" for Matrix
var kroneckerPower = Fn.new { |m, n|
var pow = m.copy()
for (i in 1...n) pow = pow.kronecker(m)
return pow
}
var printMatrix = Fn.new { |text, m|
System.print("%(text) fractal :\n")
for (i in 0...m.numRows) {
for (j in 0...m.numCols) {
System.write((m[i][j] == 1) ? "*" : " ")
}
System.print()
}
System.print()
}
var m = Matrix.new([ [0, 1, 0], [1, 1, 1], [0, 1, 0] ])
printMatrix.call("Vicsek", kroneckerPower.call(m, 4))
m = Matrix.new([ [1, 1, 1], [1, 0, 1], [1, 1, 1] ])
printMatrix.call("Sierpinski carpet", kroneckerPower.call(m, 4))
- Output:
Same as Kotlin entry.
zkl
Uses Image Magick and the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl
var [const] GSL=Import.lib("zklGSL"); // libGSL (GNU Scientific Library)
fcn kronecker(A,B){ //--> new Matrix
m,n, p,q := A.rows,A.cols, B.rows,B.cols;
r:=GSL.Matrix(m*p, n*q);
foreach i,j,k,l in (m,n,p,q){ r[p*i + k, q*j + l]=A[i,j]*B[k,l] }
r
}
fcn kfractal(M,n,fname){
R:=M;
do(n){ R=kronecker(R,M) }
r,c,img := R.rows, R.cols, PPM(r,c,0xFFFFFF); // white canvas
foreach i,j in (r,c){ if(R[i,j]) img[i,j]=0x00FF00 } // green dots
println("%s: %dx%d with %,d points".fmt(fname,R.rows,R.cols,
R.pump(0,Ref(0).inc,Void.Filter).value)); // count 1s in fractal matrix
img.writeJPGFile(fname);
}var [const] A=GSL.Matrix(3,3).set(0,1,0, 1,1,1, 0,1,0),
B=GSL.Matrix(3,3).set(1,1,1, 1,0,1, 1,1,1);
kfractal(A,4,"vicsek_k.jpg");
kfractal(B,4,"sierpinskiCarpet_k.jpg");- Output:
vicsek_k.jpg: 243x243 with 3,125 points sierpinskiCarpet_k.jpg: 243x243 with 32,768 points
Images at Vicsek fractal and Sierpinski Carpet fractal.
