According to Keller's conjecture:
In any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face.
Perhaps surprisingly, this is false for every dimension greater than 7. See here for 2D and 3D visuals.
Is there any intuitive way to "visualise" how dimension 8 and above manage to tile the space in such a way?
Whereas neural networks of our brains have some potential for this visualization (according to this article published exactly year ago, they can create structures in up to $11$ dimensions), I think this task is out of the present reach of our minds.
What we have to visualize? According to Wikipedia, “the disproof of Keller's conjecture, for sufficiently high dimensions, has progressed through a sequence of reductions that transform it from a problem in the geometry of tilings into a problem in group theory, and from there into a problem in graph theory”. Namely, in 1949 Hajós reformulated Keller's conjecture in terms of factorizations of Abelian groups, and in 1986 Szabó simplified this formulation. In 1990 Corrádi and Szabó (1990) reformulated Szabó's result as a condition about the existence of a large clique in a certain family of graphs, which subsequently became known as the Keller graphs. Corrádi and Szabó showed that the maximum clique in this graph has size at most $2^n$, and that if there is a clique of this size then Keller's conjecture is false, because, given such a clique, we can construct a required tiling covering space by cubes of side two whose centers have coordinates that, when taken modulo four, are vertices of the clique. (Also below in the article was written that this reformulation is not exact, because “the translation from cube tilings to graph theory can change the dimension of the problem”). But this observation concerns the clique in a concrete representation of a Keller graph, whereas a search for such clique is an abstract graph problem, which can be treated, for instance, by refined SAT-solvers (a usual tool in modern combinatorial graph theory) as did Mackey at al. Thus in $8$-dimensional space the required tiling is generated by transaltions on a basic (not necessarily connected) shape consisting of $2^8=256$ unit cubes. Moreover, I guess the shape can be rather irregular, as coming from a solution of an abstract graph problem.
I think it is very hard to visualize tilings by such shapes. For instance, recently I was dealing with this twice +500 bounty question on an existence of a special tiling of a cube by $2\times 2\times 1$ bricks. Since Carl Schildkraut showed here that if a required tiling exists then the edge of the cube is at least $24$, I think that a corresponding tiling should be rather irregular, so too hard to describe and too complicated to be dealt by hand. Thus I wrote an assisting program. I tried a bit to find a required tiling of a cube of edge $24$ in the program’s shell, but this still was a rather hard and long task, so I switched to other problems.
Finally I recall that Charles H. Hinton argued that we can develop a skill for high (four) -dimensional visualization by freeing our imagination of objects of “elements of the self”, related to our vision and location, in order to imagine the things as they are, for instance, to see not only their surfaces but also inner points. This aim can be achieved by special imagination exercises. This is one of branches of my project of investigation of a mind space, to which I hope to involve other contemplators from the Mind & Life Institute, founded by “Tenzin Gyatso, the 14th Dalai Lama—the spiritual leader of the Tibetan people and a global advocate for compassion; Francisco Varela, a scientist and philosopher; and Adam Engle, a lawyer and entrepreneur”. Unfortunately, I heard that “Hinton later introduced a system of coloured cubes by the study of which, he claimed, it was possible to learn to visualise four-dimensional space (Casting out the Self, 1904). Rumours subsequently arose that these cubes had driven more than one hopeful person insane”. For me, as a mad scientist, this is OK, but now I’m finishing a habilitation and this would decrease an important statistical parameter for my institute. So I have to postpone the project for a couple of years.
