Diagram of map

The action starts at C, the initial position. Because this is the first time around the loop, this is also Z, the current position.

The first thing we do is to multiply Z by itself to get Z2.

You can compute Z2 using a calculator, but we will understand much better if we think about it graphically. Z is represented on the graph as a value by the line A-B. The value of a complex number can be thought of as the combination of the length of a line plus the direction that the line is going in.

So we are about to multiply the line A-B by itself. How do we do this in complex arithmetic?

Well, you draw a line from 1 to B, and you now have a triangle whose corners are A, 1 and B. Just make sure you can see that...

Then you find a new point, D, which makes another triangle whose corners are A, B, D. Hopefully you can see this OK on the diagram. The trick is that the new triangle has to have the same proportions as the first triangle, i.e. it is the same shape but in this case it is smaller than the first triangle.

It is as if the first triangle A, 1, B has been rotated around the 'A' corner to become the new triangle A, B, D. Because the side of the old triangle going from A to 1 is longer than the side of the new triangle going from A to B, we have to shrink the new triangle in order to keep it the same shape. If you are mathematically inclined you can read more about this kind of multiplication here.

The only other thing we have to do is to add on the "value" of C. Remember that the value is the length of a line plus the direction the line is going in. The value of C is represented by the green line from A to B. So when we add C to Z2, we just draw a similar line D-E, the same length and the same direction as the line A-B.

And that's it. We have made the first Mandelbrot Jump, and we are now at E. The next diagram will take us through the second jump (quite easy now that we have done the first).

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