Fragic Carpet

Maths is like a jungle: rich, teeming and full of surprises. A waterfall here, a glade of butterflies there, a bank of orchids yonder. There is always something new to see and a different route to try. But sometimes a different route will take you to the same place. I’ve already found two ways to reach this fractal (see Fingering the Frigit and Performativizing the Polygonic):

carpet2x2

Fractal Carpet


Now I’ve found a third way. You could call it the rep-tile route. Divide a square into four smaller squares:

square2x2

Add an extra square over the centre:

square2x2_1

Then keep dividing the squares in the same way:

carpet2x2_anim_1

Animated carpet (with coloured blocks)


carpet2x2_anim_2

Animated carpet (with empty blocks)


The colours of the fractal appear when the same pixel is covered repeatedly: first it’s red, then green, yellow, blue, purple, and so on. Because the colours and their order are arbitrary, you can use different colour schemes:

carpet2x2_col1

Colour scheme #1


carpet2x2_col2

Colour scheme #2


carpet2x2_col3

Colour scheme #3


Here are more colour-schemes in an animated gif:

carpet2x2_col

Various colour-schemes


Now try dividing the square into nine and sixteen, with an extra square over the centre:

carpet3x3

3×3 square + central square


carpet3x3_anim

3×3 square + central square (animated)


carpet4x4

4×4 square + central square


carpet4x4_anim

4×4 square + central square (animated)


You can also adjust the size of the square added to the 2×2 subdivision:

carpet2x2_1_2

2×2 square + 1/2-sized central square


carpet2x2_3_4

2×2 square + 3/4-sized central square


Elsewhere Other-Posted:

Fingering the Frigit
Performativizing the Polygonic

Boldly Breaking the Boundaries

In “M.I.P. Trip”, I looked at fractals like this, in which a square is divided repeatedly into a pattern of smaller squares:
2x2inner

2x2inner_static


3x3innera

3x3innera_static


3x3innerb

3x3innerb_static


As you can see, the sub-squares appear within the bounds of the original square. But what if some of the sub-squares appear beyond the bounds of the original square? Then a new family of fractals is born, the over-fractals:

fractal2x2a

fractal2x2a_static


fractal2x2b

fractal2x2b_static


fractal2x2c

fractal2x2c_static


fractal2x2d

fractal2x2d_static


fractal2x2e

fractal2x2e_static


fractal3x3a

fractal3x3a_static


fractal3x3b

fractal3x3b_static


fractal3x3c

fractal3x3c_static


fractal3x3d


fractal3x3e


fractal3x3f


fractal3x3g


fractal3x3h


fractal3x3i


fractal3x3j


fractal3x3k


fractal3x3l


fractal3x3m


fractal3x3n


fractal4x4a


fractal4x4c


fractal4x4b

M.i.P. Trip

The Latin phrase multum in parvo means “much in little”. It’s a good way of describing the construction of fractals, where the application of very simple rules can produce great complexity and beauty. For example, what could be simpler than dividing a square into smaller squares and discarding some of the smaller squares?

Yet repeated applications of divide-and-discard can produce complexity out of even a 2×2 square. Divide a square into four squares, discard one of the squares, then repeat with the smaller squares, like this:

2x2square2


2x2square3


Increase the sides of the square by a little and you increase the number of fractals by a lot. A 3×3 square yields these fractals:

3x3square2


3x3square3


3x3square6


3x3square7


3x3square8


3x3square9


3x3square10


And the 4×4 and 5×5 fractals yield more:
4x4square1


4x4square2



4x4square4


4x4square5


4x4square6


4x4square7


4x4square8


5x5square1


5x5square2


5x5square3


5x5square4


5x5square5


5x5square6


5x5square7