3d object modeling conundrum (help)

Hi,
I am a total n00b so i was wondering if anyone had any ideas about how to make something for a project i am working on. I want to make a 3d geometrical model/object (think a web of intertwined lines (like a ball of wires interwoven but with space between them). This object will then be rotating or moving. I assume this is a relatively easy thing to create using 3d software. Now comes the tricky part. From one perspective looking at the ball (2dimensionally) i want to be able the spaces between enclosed regions created by overlapping lines to fill with a color(say black). Very other space is then left blank9 much like a skewed checkerboard but constantly changing. The image attached is an example of one still of the theoretical object. This checkerboard shapes would then be formed depending on which way you rotated the object.
Can any one help?!!!
Let me know if i can better clarify this issue. Thanks so much[/img]

I didn't understood clearly how you want it to be rendered.

"From one perspective looking at the ball (2dimensionally) i want to be able the spaces between enclosed regions created by overlapping lines to fill with a color. Very other space is then left blank9 much like a skewed checkerboard but constantly changing."

Can you clarify this?