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 n-th 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.
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. <lang C>
- 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; } </lang>
Factor
<lang factor>USING: io kernel math math.matrices sequences ;
- mat-kron-pow ( m n -- m' ) 1 - [ dup kron ] 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@</lang> Output shown at order 4 and 25% font size.
- Output:
*
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* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
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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.
<lang gnuplot>
- 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" </lang>
- Output:
3 plotted files: pkf1.png, pkf2.png and pkf3.png.
Go
<lang 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")
}</lang>
- 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.
<lang haskell>{-# 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</lang>
Link to live demo: https://dc25.github.io/rosettaCode__Kronecker_product_based_fractals/ ( a little slow to load ).
J
Implementation:
<lang J>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'</lang>
Task examples (order 4, 25% font size):
ascii_art S KPfractal 4
*********************************************************************************
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*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
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*** ****** ****** ****** ****** ****** ****** ****** ****** ***
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* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
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ascii_art V KPfractal 4
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Java
This implementation does not use sparse matrices since the powers involved do not exceed 4.
<lang Java> 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();
}
} </lang>
- 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.
<lang 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(' '+title+':
');
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('');
} // 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; </lang>
- Required tests
<lang 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')">
Vicsek fractal
<a href="SierpCarpetFractal.html"> Next: Sierpinski carpet fractal</a>
<canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>
</body></html> </lang>
<lang 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')">
Sierpinski carpet fractal
<a href="Checkerboard.html"/> Next: Checkerboard </a>
<canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>
</body></html>
<lang 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')">
Checkerboard
<a href="VicsekFractal.html"/> Next: Vicsek fractal </a>
<canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>
</body></html> </lang>
- 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`: <lang julia>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</lang>
- Output:
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*** *** *** *** *** *** *** ***
* * * * * * * * * * * * * * * *
*** *** *** *** *** *** *** ***
********* ********* ********* *********
* ** ** * * ** ** * * ** ** * * ** ** *
********* ********* ********* *********
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
*** ****** ****** *** *** ****** ****** ***
*************************** ***************************
* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
*************************** ***************************
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
Kotlin
This reuses code from the Kronecker_product#Kotlin task. <lang scala>// 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))
}</lang>
- Output:
Vicsek fractal :
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
*********************************************************************************
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * *
*** *** *** *** *** *** *** *** ***
* * * * * * * * *
* * *
*** *** ***
* * *
* * * * * * * * *
********* ********* *********
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
* * *
*** *** ***
* * *
* * * * * * * * *
***************************
* * * * * * * * *
* * *
*** *** ***
* * *
*
***
*
* * *
*********
* * *
*
***
*
Sierpinski carpet fractal :
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
*********************************************************************************
* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
*********************************************************************************
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*** *** *** ****** *** *** ****** *** *** ***
* * * * * * * ** * * * * * * ** * * * * * * *
*** *** *** ****** *** *** ****** *** *** ***
********* ****************** ****************** *********
* ** ** * * ** ** ** ** ** * * ** ** ** ** ** * * ** ** *
********* ****************** ****************** *********
*********************************************************************************
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*********************************************************************************
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
* * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * *
*** ****** ****** ****** ****** ****** ****** ****** ****** ***
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* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *
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* ** ** ** ** ** ** ** ** * * ** ** ** ** ** ** ** ** *
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*** ****** ****** *** *** ****** ****** ***
* * * ** * * ** * * * * * * ** * * ** * * *
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*** *** *** *** *** *** *** ***
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Lua
Needs LÖVE 2D Engine <lang lua> 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 </lang>
PARI/GP
- Note
- Find iPlotmat() here on Helper Functions page.
- Find matkronprod() here on Kronecker product page.
<lang parigp> \\ 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);
} </lang>
- 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 6
<lang perl6>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 $png = Image::PNG::Portable.new: :width(@img[0].elems), :height(@img.elems);
for ^@img[0] X ^@img -> ($x, $y) {
$png.set: $x, $y, |( @img[$y;$x] ?? <255 255 32> !! <16 16 16> );
}
$png.write: "kronecker-{$name}-perl6.png";
}</lang>
See Kronecker-Vicsek, Kronecker-Carpet and Kronecker-Six images.
Phix
<lang 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))</lang> Output same as Julia/Kotlin/Factor
Python
Generate images of the fractals using PIL.
Using only python lists <lang python>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) </lang>
Because this is not very efficent/fast you should use scipy sparse matrices instead <lang python>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)</lang>
R
Generate and plot 3 Kronecker product based fractals.
Note: Find plotmat() and plotv2() here on Helper Functions page.
<lang r>
- 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")
</lang>
- 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
REXX
This is a work-in-progress, this version shows the 1st order. <lang rexx>/*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*/</lang>
- Output:
███ █ █ ███ █████████ █ ██ ██ █ █████████ ███ █ █ ███
Sidef
<lang ruby>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")
}</lang> Output images: Kronecker Carpet, Kronecker Vicsek and Kronecker Six
zkl
Uses Image Magick and the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl <lang 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);
}</lang> <lang zkl>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");</lang>
- 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.
