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

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