How to visualize multi-dimensions in topological periodic table? This is a question for those who are familiar with topological periodic table.  The first row and right side columns represents dimensions of topological materials in periodic table.   I know that for $d=1$ we are talking about nanowire, for $d=2$ we are talking about film or coating on the surface, for $d=3$ we are talking about bulk material.  What $d=4$,$d=5$.....$d=8$ represents?  How to visualize it?
 A: For $d=4$, an example would be this research on Four-Dimensional Quantum Hall Effect with Ultracold Atoms, with the 4th dimension being only synthetic and not spatial.
A: The part of the table that goes beyond dimension $3$ can be experimentally realized with some "synthetic degrees of freedom", but it is also important to understand the context: this table was not "invented" for topological insulators by Kitaev. It was rather invented in a completely different setting in pure mathematics by Atiyah, Bott and coworkers for K-Theory and Bott periodicity. Later on Kitaev recognized the structure that Ryu, Schneider and coworkers saw in topological insulators with various symmetry classes actually fits perfectly with the Bott table. Since the Bott table has 8 dimensions (as its main point is an 8-periodicity of homotopy groups of certain spaces), our table also has 8 dimensions.
By the way note that the first two symmetry classes (no symmetry and chiral symmetry) have periodicity 2 in dimension so that actually for these cases, the table should only have two columns.
