Plausible universality of uniaxial order in self-assembly of cross junctions in space dimension $d \ge 3$

Abstract

We consider the self-assembly of cross junctions in a general space dimension ($d$) as an extension of the problem studied in a previous paper for $d = 3$. This problem is equivalent to constructing a $d$-dimensional hypercubic jungle gym, at all junctions of which $2d$ rods with different colours meet. The analysis reveals a unique feature of the $d = 3$ case: the forced presence of at least one perfectly-ordered (singly coloured) direction (axis), in contrast to the possible absence of such a direction in $d \ge 4$. However, we will show that the uniaxial order is overwhelming not only in $d = 3$ but also for $d \ge 4$ in a sufficiently large system.

Figures (2)

• Figure 1: Cross junctions and example of their self-assembled states in dimension two (a) and three (b). While there is no variety in dimension two except for exchanging colours, many possibilities exist in dimension three.
• Figure 2: Two examples of $2\times 2\times 2\times 2$ part of self-assembled states without completely ordered axes in dimension four. a) specified by eq. \ref{['splitcase']}, which decomposes the dimension into $2+2$; b) all axes are equivalent. Large (outer) and small (inner) cubes (of dimension three) indicate sectors with different coordinates in the fourth dimension ($w$), which is depicted along the body diagonals of the cubes, with the inside direction being positive. The half number of the arms of each junction, such as those at the outer/inner sides of hypercubes, is omitted for clarity. Different colours are distinguished using different symbols marked for most segments.