Fractals: The Inter-dimensional Journey
What if I tell you the Romanesco Broccoli is coming from another dimension?

If you tried to eat it before, you would probably believe me right away. But we are here to explore some other properties rather than their exceptional taste!. It has a form of natural approximation of a 'fractal'. Each conic section is composed of a series of smaller cones, all arranged in a spiral.
 
Although its self-similar pattern continues at smaller levels, the Romanesco Broccoli is only an approximate fractal since the pattern eventually ends when the size becomes very very small. But in fractal geometry, we can repeat a particular pattern or a rule infinitely many times to create smaller and smaller copies of themselves.
 
And apparently, natural selection prefers fractal-form structures so that we can see them everywhere in nature.
But why are fractals spooky?
In geometry, we know that a line segment has "1" dimension. When we double its scale, its length doubles itself.
A square has "2" dimensions. It has a length and a width, so it covers a surface, and when we double its scale, we see four of the initial square.

A cube has "3" dimensions. It has a length, width, and height, so it has a volume, and when we double its scale, we see eight of the initial cube.

So all the dimensions we know (or are aware of) are integers. 

Can something have a dimension somewhere in 1and 2, or between 2 and 3?

Can a shape have a 1.5 dimension?

Spooky fractals are here to answer these questions. Let's see what happens if we use the same logic to find their dimensions.

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Image by Martin Rancourt
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Sierpinski Triangle

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Sierpinski Carpet

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Menger Sponge

So fractals do have non-integer dimensions. That is really scary for the Flatland community. 

There are more surprising facts about their inter-dimensional journey.

Let's start with a line to create the Peano Curve or the Hilbert Curve. Since they cover an entire plane, they are 2 dimensional. Amazing right?

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Peano Curve

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Hilbert Curve

Check out the Wikipedia page about the Hausdorff dimensions of fractals.

Fractal Geometry is a great place where you can find many things to surprise you. You may want to check out a whole unit of tasks, activities and lesson plans to explore more about fractals at the Tasks page of Polypad.

There are amazing videos about fractals. Here is a playlist  to have a general ideas as well as the specifics of the Fractal Geometry.