# 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):

Fractal Carpet

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

Add an extra square over the centre:

Then keep dividing the squares in the same way:

Animated carpet (with coloured blocks)

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:

Colour scheme #1

Colour scheme #2

Colour scheme #3

Here are more colour-schemes in an animated gif:

Various colour-schemes

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

3×3 square + central square

3×3 square + central square (animated)

4×4 square + central square

4×4 square + central square (animated)

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

2×2 square + 1/2-sized central square

2×2 square + 3/4-sized central square

Elsewhere Other-Posted:

# 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:

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:

# 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:

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:

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