To investigate I'll have to create some definitions.

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. 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.

As I mentioned, these are very similar to the researched are of self tiling tile sets or setisets for short. These tile sets have any number of tiles, but they are restricted to just dense tiles. Their work presents a solution for the two tile case but it requires reflection so the shapes aren't symmetric through a similarity transform.

The search space is roughly 10 dimensional, we can reduce it to about 8 dimensional by looking at only the shapes that touch but don't overlap. My first pass at this still has little gaps sometimes, but here are some of the results:

An interesting question is whether any dense shapes exist in this search space. The closest I have found randomly is:

Which, with a bit of manipulation we can bring closer together:

and perhaps even a bit closer still:

Parameters are given in the code here, and here are some parameters from previous pictures:

While it is not enough searching to be sure, my hunch is that there doesn't exist a dense set of twin tiles. However the space of twin gnomons is 1DOF larger so it would no doubt get closer.

It should be mentioned that one can get arbitrarily close to a dense shape just by getting really close to the two trivial (but invalid) solutions because shape A and B are the same:

- Two A5 sheets making an A4 sheet (i.e. rectangles of aspect ratio sqrt 2).
- Two 45 degree right-angles triangles.

If we are willing to go to 3D space then there is a known dense twin-tile pair called the triangle duo, these are two flat (2D) triangles in 3D space.

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