First, I wish to tip my cap to [xenzag] and the "One Tube Deserves Another" idea (see link below).
In reading that idea, I decided that toilet paper rolls should not only be able to connect to each other in a linear fashion, but should have the opportunity to branch off into other dimensions. However,
I failed to come up with a smooth, easy to make variation on the tube itself to allow any sort of side branching.
So, my idea doesn't call for any change to the standard roll core. It's just a bag of plastic connectors, with protrusions onto which cores can be fitted. And more connectors! and more cores! until you have a giant viola! You can use similar diametered longer tubes from your paper towel rolls and Christmas wrapping! Build a model of the Leaning Eiffel Tower! (No, the original doesn't do that when it gets rained on.) Geodesics! Why, it can fail to support all the kids in the neighborhood!
All right, I'll go clean up.
But I'm keeping the plastic parts.