How do I plot this red dot?

Question

Here is the dot

And here is how I arrived at it »

Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which lie on the dotted red line(pic 2), using an equation?

Answer 1

It's not very clear what you're trying to ask, but what you're describing looks like a region quadtree.

Each step in your IFS corresponds to a choice of child node (or subquadrant), and the tree path of any given point is the same as the list of numeric input steps in your question.

It does not really matter whether you store the point as an arbitrary-precision coordinate pair or tree traversal: the binary digits required to store its coordinates correspond exactly to its quadtree traversal path.

Answer 2

It seems like you are applying $f_1,f_2,f_3$ where 1,2,3 maps the entire square to down right, up right, and down left respectively, and your red dot is the solution to $f_1(f_2(f_3(x))) = x.$ Now, you just have to find these functions, which is rather easy and left as an exercise. Now, the composition of these 3 functions (in the order above) is:

$$f(x,y) = ((x+5)/8,(y-3)/8)$$

Now, solve $(x+5)/8 = x$, giving $x=5/7.$ Now, solve $(y-3)/8 = x$, giving $x=-3/7.$

The red dot should be $(5/7,-3/7).$

I have assumed that the corners of the original square is $(\pm 1,\pm 1)$.