Dense Fractal Space Construction

Suppose we want to design a series of basic reusable modules that can serve as the basis of a design aesthetic for some spacefaring civilization.

Let's start with a sphere, for maximum strength and material economy. It should be as easy as possible to expand stations or ships built out of these modules over time, and we'd like subsequently-added sphere modules to pack together as tightly as possible to minimize travel distances within the structure. That means each sphere should be able to connect to 12 others (the 3 dimensional kissing number), so each basic module should have twelve cylindrical attachment points (unused ones can be capped off with windows or airlocks or sphere-section bulkheads). There are two ways to arrange those--with stacked layers either aligned with each second layer over, or shifted by one lattic position--but the aligned arrangements, corresponding to placing connections at the vertices of a cuboctahedron (or on the faces of a rhombic dodecahedron) maximizes symmetry and maximizes the number of straight line paths from one attachment point through the center of the sphere to an opposite attachment point, which makes it easy to build straight-line structures with individual modules in many different orientations.

Standardized modules and standardized attachment points, resulting in standard exterior spacing between components, also suggests that these modules should come with standardized exterior anchor points for EVA, which means nicely standardized greeblies completing the design aesthetic. While doing a high ropes course once, I came across an ingenious design for a passive Continuous Belay System (an system which allows you to move around a structure relatively freely without ever becoming disconnected from a safety line)--a special C-clip is used which has an opening large enough for flat plates to pass through, which allows passing by mounting points for cables, but small enough that it cannot come off the cable itself. Specially-designed intersection plates also allow sliding the clip from one cable to another without ever becoming detached. This is slightly more restrictive than two-carabiner systems, where you move one at a time while keeping the other attached, but also much more idiot-proof, and provides a functional set of greeblies in terms of the anchor and intersection plates. The details of how the plates are designed are not easily visible when zoomed out to look at a whole module, but they do provide nice greebling.

Now, you can just densely tile space with the standard minimum-sized modules. But a collection of 13 basic modules forming a "raspberry" has the 12 exterior modules arranged such that each one has one of its connection points exactly at the vertex of a larger cuboctahedron.

Thus, these raspberry units can be used to tile space as well, in exactly the same pattern as the base units. (Also note that the doubled anchor-rings around each intermodule connection provide multiple "lanes" for workers to pass each other on EVA.) One might choose to do so, leaving a lot of cluster-internal attachment points unused, so as to set up a recursive city-like structure of local neighborhoods, larger districts, and whole cities. But additionally, these can be used to define a scale for larger modules that will fit into the same system. For example, a large sphere can enclose a raspberry except where the 12 external attachments poke through, providing double-hull protection for the interior lower-scale modules, or permitting leaving out some members of the full cluster while maintaining a standard external interface. In this following image (which took forever to render), 13 raspberries are each encased in a glass secondary hull and used to form a second-scale raspberry structure.

And, once larger-scale modules are established as A Thing, you can just have monolithic modules of that size which have whatever internal configuration you want, for holding stuff that needs more space (say, gravity centrifuges, or nuclear reactors, or whatever), and smaller scale modules will fit nicely in between them.

Non-spherical modules can also be added to the system as long as they have attachment points that align with the lattice points for their specified scale. So, if you have, e.g., a long mass driver or something, you don't need to waste a ton of sphere space on it--you can make a cylindrical module with attachment points spaced appropriately to tesselate more standard modules around it. More prosaically, however, they may be some utility in having polyhedral modules.

This cuboctahedral modules, with connectors at each vertex, has less interior volume lower inherent pressure-hull capacity, but provides the convenience of flat surface for mounting equipment on the exterior, and more room between modules for maintenance access. As a raspberry cluster, it looks like this:

Meanwhile, this rhombic dodecahedral module, with connection points in the center of each face, does a better job of efficiently filling space, leaving less wasted space betwee modules. 

That's no good for external maintenance access, but when working with larger scale modules, where there would be more wasted space between spheres, this may be a good option, at least for low-pressure environments. As a raspberry cluster, it looks like this: