Let us define self-similarity in a strict sense as:

*self-similarity:**shape A is composed of a finite number of smaller copies of shape A*

I'll then define cross-similarity as:

*cross-similarity*:*distinct shapes A,B,C,.. are composed of a finite number of smaller copies of each shape A,B,C,..*

A special case I'll label as:

*co-similarity*:*distinct shapes A,B,C,.. are composed of one smaller copy of each shape A,B,C,..*

So unlike self-similarity, co-similar shapes mutually nest inside each other. Like with self-similarity, these shapes typically have a fractal border.

The simplest non-trivial special case is for two shapes A,B. Let's label them twin-fractals.

**twin-fractals:***shape A is composed of a small copy of shape A and shape B*

*shape B is composed of a small copy of shape A and shape B*

*i.e.*

*A = aT1A + bT2B*

*B = cT3A + dT4B for some scalars 0 < a,b,c,d < 1 and Euclidean transforms*

*Ti*

*twin-gnomons*, these are twin fractals where shape A and shape B have the same relative scale when composing.

**twin-gnomons:***shape A and shape B compose to a larger copy of shape A*

*shape A and shape B compose to a larger copy of shape B*

*i.e.*

*A = aT1A + aT2B*

*B = cT3A + cT4B for scalars 0*

*< a,c < 1*

We can find these using an Iterated Function System or limit-sets approach. Here are some examples. The white is shape A, yellow and pink are shape B in its two locations:

A special case of twin-gnomons I'll call

*twin-tiles*, these are twin-gnomons where the same scale is used for composing shapes A and B:

**twin-tiles:***shape A and shape B compose to a larger copy of shape A*

*shape A and shape B compose to a larger copy of shape B, with the same enlargement factor*

*i.e.*

*A = aT1A + aT2B*

*B = aT3A + aT4B for scalar*

*0 < a < 1*

*Here are equivalent twin tiles:*

A feature of the twin tiles is that one can grow larger and larger versions of shape A and B by composing in the two ways, so it exhibits co-similarity.

I'm uncertain whether any of these can be made dense, but I am certain that they can be made to be connected. I think the best results come when the composition is just touching without overlap. This is close in the top image of the set above, but the yellow shape B doesn't quite touch at the bottom and the pink shape B overlaps a little. If I tweak the transforms we get something that is more connected:

I think this is an interesting area to explore. White and yellow are mutually similar, rather than self-similar shapes.