Integer and Non-Integer Dimensions

A couple of ideas from the last few weeks had me thinking about fractals. The first was Dan Anderson’s Gosper Island pictures:

(and there are *many* more pictures if you look at Dan’s twitter feed +/- a few days from this picture)

One of Mandelbrot’s finance books was on my mind, too:

 

So, when my son said that he wanted to do a geometry project this morning, I suppose that I was basically primed to think of something relating to fractals. The idea that I chose was a dimension.

I had the boys make a few simple shapes from our Zometool set and then we talked about what they thought about dimension:

 

In the last video we talked about 1 dimensional objects, but it wasn’t at all clear where the “1” was. Next we talked about 2 dimensional objects. That helped my younger son see the the “1”. Yay!

 

Having talked about the 1st and 2nd dimension, the third dimension was a piece of cake – well . . . other than a tiny bit of confusion about area and volume.

 

Now we got to the punch line of the project – shapes that don’t scale with integer powers. We used our Gosper Island pieces that we printed from Thingiverse:

Gosper Curve Coasters by simcop2387 on Thingiverse

These shapes have the unexpected property that when you scale lengths by a factor of 3, the area scales by a factor of 7. This idea was quite a surprise to my younger son – “I did not know that there were half dimensions.”

 

We have looked at this shape previously. It was almost a year ago, though, so I’m not all that surprised that it wasn’t fresh in their mind:

A fun fractal project – exploring the Gosper curve

I also wish that the shapes fit together just a bit better than they do so that the scaling by a factor of 7 was clearer. Still, though, I thin that this is a fun project and a neat way to introduce kids to the idea that not everything is a perfect shape from Euclidean geometry.

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