From T to Y

There are many fascinating geological structures around the globe, and to this author, those formed by rapid cooling of solidified lava (basalt) are especially so. Examples of these columns abound: in the Giant’s Causeway (Northern Ireland) (Fig. 1), Devil’s Tower (Wyoming), and the Devil’s Postpile (California) to name a few. Iceland has many such sites also (Fig. 2 was taken at one of these).

There has been much research on how such geological patterns arise, in this case from crack propagation, but some insights can be gained from geometrical considerations. At the beginning of the process, secondary cracks meet existing ones at nearly right angles (T-junctions), but as the pattern propagates inward, the junctions tend to evolve toward 120° angles (Y-junctions).¹ 

It is useful to introduce a dimensionless measure R (as defined below) as T-junctions evolve to Y-junctions. The area of the rectangular cell (shaded region in Fig. 3) is $A=3L2/4$⁠; it is essentially a constant of proportionality in what follows. The normalized rectangular cell (shaded) of width L/2 has been chosen for simplicity to accommodate the transition from a T-junction to a hexagonal Y-junction. In order to track R in a transition from a T-junction to a Y-junction, a “transition variable” x is introduced (Fig. 3). A T-junction corresponds to p ≡ x/L = 0. (Of course, both mathematically and physically, the variable x can increase toward the lower boundary of the shaded region, resulting in a steeper Y-junction.²)

Define R(p) = A/LL_c(p), where L_c(p) is the crack length within the cell. [According to fracture mechanics,²  R(p) is a measure of the crack energy release rate.]

Question 1: Show using elementary geometry that $R(p)=3(3−4p+21+16p2)−1$⁠.

Question 2(a): Show that R(p) is maximized when $p=3/12$

Question 2(b): Show that the result in part (a) above corresponds to a Y-junction for a regular hexagon.