Mathematical Challenge (Concept only).
Posted: Sun Aug 17, 2008 8:40 pm
Hi everyone.
This is an idea that I had ages ago, but haven't had the time or inspiration to finish.
You know how to recursively quarter squares, right?
Now, if we move the centers to produce non-square quadrilaterals, we can get a non-affine warped grid/checkerboard effect.
I have the general outline of how to do it, but haven't thought of a good way to find a finite solution set.
My intuition is that there are solutions, but I have no proof.
Let me explain.
Within CF, we want a finite - preferably small - number of rules corresponding to different shapes and subdivisions.
Because of the way transforms are inherited, each rule need only deal with one canonical form, however distorted it appears on canvas.
The canonical forms of parent and child shapes with their corresponding subdivisions need to form a closed solution set to allow arbitrarily deep mutual recursion.
One possible canonical form is the union of a unit square and a triangle (a square with one side extended by some amount producing two non-right angles).
The inline transforms calling the main rules would use a combination of shear (sorry, 'skew') and scale to pass quarters in canonical form (plus any required re-alignment).
That's not a very explicit description, sorry, and maybe I forgot some details.
(But if your imagination doesn't fill in the blanks, you probably wouldn't have much luck solving it anyway.)
Any takers?
This is an idea that I had ages ago, but haven't had the time or inspiration to finish.
You know how to recursively quarter squares, right?
Now, if we move the centers to produce non-square quadrilaterals, we can get a non-affine warped grid/checkerboard effect.
I have the general outline of how to do it, but haven't thought of a good way to find a finite solution set.
My intuition is that there are solutions, but I have no proof.
Let me explain.
Within CF, we want a finite - preferably small - number of rules corresponding to different shapes and subdivisions.
Because of the way transforms are inherited, each rule need only deal with one canonical form, however distorted it appears on canvas.
The canonical forms of parent and child shapes with their corresponding subdivisions need to form a closed solution set to allow arbitrarily deep mutual recursion.
One possible canonical form is the union of a unit square and a triangle (a square with one side extended by some amount producing two non-right angles).
The inline transforms calling the main rules would use a combination of shear (sorry, 'skew') and scale to pass quarters in canonical form (plus any required re-alignment).
That's not a very explicit description, sorry, and maybe I forgot some details.
(But if your imagination doesn't fill in the blanks, you probably wouldn't have much luck solving it anyway.)
Any takers?