Basalt · mud · Euler · counting corners

The Six Comes Free

The Giant's Causeway is sold as a field of hexagons, but only about half its columns have six sides. The average of six is not the lava's doing: a network of cracks that meet three at a time, in Ys, averages six corners a cell whatever it looks like, and one whose cracks stop against each other in Ts, like a crazed glaze, averages four. Crack a square and watch the average sit at exactly four, pull the Ts into Ys and watch it climb, then scatter points at random and get six again. What the rock earned is the narrow spread, and that is measured here against 3,000 columns.

Not hexagons: six on average

Look at a photograph of the Giant's Causeway and count the sides of a few columns. Some have six. Many have five or seven, and a few have four or eight. Denis Weaire and Nicolas Rivier, reviewing the physics of random cellular patterns in 1984, put it bluntly:

“many writers have not actually examined the structure and fall easily into the habit of referring to it as ‘hexagonal’, which it is not”

D. Weaire and N. Rivier, Soap, cells and statistics: random patterns in two dimensions, Contemporary Physics 25 (1984), p. 85

The counts agree with them. J. P. O'Reilly published a survey of a patch of the Causeway in 1879; in the 206 columns of his measured area, as Robert Sosman tabulated them in 1916, 50.5% have six sides, 24.8% five, 19.2% seven, 3.5% four and 2.0% eight. In 2012 György Hetényi and colleagues counted 3,033 columns at 32 of the 50 sites they visited in France, Hungary and Iceland: 1,502 hexagons, 1,028 pentagons, 352 heptagons, 128 quadrilaterals and 23 octagons. About half are hexagons, in both. The average is a little under six, 5.91 for O'Reilly's area and 5.71 for Hetényi's sites, and both numbers turn out to mean something.

So the honest question is not why the columns are hexagons, since most of them are not. It is why the average is six, and why the spread around it is so narrow. The first half has an answer that has nothing to do with lava, and you can see it for yourself in a square of paper.

Crack a square

Drying mud and a crazed pottery glaze crack one crack at a time. A new crack runs across one piece and stops where it meets older cracks, usually square-on. Each end makes a T: the older crack runs straight through, and the new one stops against it.

Count what one such crack does to the corners. It cuts one cell into two. At each of its ends it makes a corner on either side of itself, which is four new corners. The cell on the far side of the older crack gains a point on its boundary, but its side runs straight through that point, so it gains no corner. One more cell, four more corners, every time, whatever the shapes involved. A square starts as one cell with four corners, so after any number of cracks the average cell has exactly four.

Steffen Bohn, Stéphane Douady and Yves Couder found the same thing in the cracks of ceramic glazes in 2005, and saw that it does not depend on the details:

“It is shown that, on the average, the number of sides of these domains is four.”

S. Bohn, S. Douady and Y. Couder, Four sided domains in hierarchical space dividing patterns, Physical Review Letters 94 (2005) 054503, abstract

Pull the Ts into Ys

Now press pull 20 Ts into Ys. Each press takes a T and nudges its meeting point a little way off the straight line, so that the older crack bends there. Nothing is added and nothing is removed: the same cracks, the same cells, the same junctions. But the cell that used to see a straight side now sees a corner, and the average climbs.

The count is simple enough to do by hand. At a Y, three cracks meet and each of the three cells there has a corner: three corners. At a T, one cell sees a straight line: two corners. Where a crack meets the edge of the square, two corners; and the square's own four corners are one each. The readout does that sum live and checks it against the corners counted from the geometry. It never fails, before or after pulling.

corners = 3·Y + 2·T + 2·(edge) + 4

In a large network the edge stops mattering. Every junction has three cells round it and every cell has corners, so, with t the share of junctions that are Ts, the average number of corners per cell comes to 6 − 2t: four when every junction is a T, six when every junction is a Y. The formula was printed in 1976 by Gray, Anderson, Devine and Kwasnik for the crack patterns of mud, frozen ground and basalt, in a form that also allows four cracks to cross at an X:

“The mean number of sides, [g], to the polygonal areas in such nets is [g] = 2(2J_T + 3J_Y + 4J_X)/(J_T + J_Y + 2J_X) where J_T, J_Y, and J_X are the proportions of T, Y, and X junctions, respectively.”

N. H. Gray, J. B. Anderson, J. D. Devine and J. M. Kwasnik, Topological properties of random crack networks, Journal of the International Association for Mathematical Geology 8 (1976) 617, abstract

With no Xs, J_Y = 1 − J_T, and their fraction is 2(3 − J_T), which is 6 − 2t. (Cracked 300 times, into 301 cells, and with every T pulled, the instrument's square comes out at 5.79 rather than 6, because the cracks that end on the square's edge make only two corners each. Crack it more and it creeps closer.)

Look at what the argument did not need. It never asked whether the Y's angles were 120°. A junction pulled only slightly off straight, at 170°, counts exactly as a perfect Y does. The six is a count of how cracks meet, three at a time with nothing running straight through. It is not a statement about hexagons at all.

Random points average six too

Here is the proof that the six costs nothing. Scatter points at random, with no order whatever, and give each point the territory that is closer to it than to any other point. The territories are the cells of a Voronoi diagram, and three of them meet at every corner. So Euler's count applies, and the average cell has six sides.

The square here wraps round, top to bottom and side to side, like the screen of an old arcade game, so no cell touches an edge and the average is not near six but exactly six, 6.000, on every scatter. With the points thrown at random only about three cells in ten are hexagons: Pierre Calka's table for an infinite random scatter gives 0.29473 for six sides, and pentagons (0.25946) are nearly as common. That is what six on average looks like with no order at all.

Now move the slider. Each point is given a disc it keeps to itself, and discs are thrown at random and kept only if they overlap nothing already down, until there are 400. The more of the floor the discs must cover, the more evenly the points space themselves, and the more hexagons appear. The average never moves off 6.000. Only the spread narrows. The gain slows as the floor fills: in our runs of 3,000 points the hexagons reach 47.8% at 45% and 49.9% at 50%, and the slider stops there, because a few points further on this way of throwing can no longer fit all the discs.

What the lava earned

So the stone's achievement is not the average but the spread. Use the menu under instrument 2 to lay a real count over the random one; the gold marks are the rock. With the slider at none, random points give too few hexagons and too many of everything else. Slide it up and the random pattern narrows toward the rock. At 50%, where the slider stops, the discs give hexagons in 49.9% of cells and a spread (the standard deviation of the number of sides) of 0.78.

The Causeway, Hetényi's sites and the discs at 50% sit together: about half hexagons, a spread near 0.8. Random points without the discs are nowhere near. Lucas Goehring and Stephen Morris, using a slightly different measure (the spread in the number of neighbours rather than sides; the two differ only at Ts, where the cell the older crack runs past has a neighbour it has no corner for, which is why neighbours average six in both kinds of network), calculated 0.75 ± 0.06 for the Causeway from O'Reilly's survey and 0.86 ± 0.05 for columns they grew in the laboratory from drying cornstarch, and found that their starch, once it stopped ordering, matched the Causeway's statistics. They drew the conclusion that the leftover disorder belongs to the pattern:

“This is contrary to the frequently encountered assumption that columnar jointing tends towards a perfect hexagonal lattice in cross-section”

L. Goehring and S. W. Morris, Order and disorder in columnar joints, Europhysics Letters 69 (2005) 739

The thrown discs are only a yardstick: a way of putting a number on how orderly the rock is, against something with no physics in it. Nobody claims lava throws discs, and the match could be a coincidence of two different processes that both stop short of a lattice.

Reading the junctions off a count

Gray's formula also runs backwards. If a large patch were counted cell by cell, and every junction were a T or a Y, the share of Ts would be (6 − mean) ÷ 2. By that arithmetic, which is ours and not the authors', Hetényi's average of 5.71 would mean about one junction in seven (14.6%) still a T, and O'Reilly's 5.91 about one in twenty-three (4.3%). Hetényi and colleagues report the same direction of travel as a view held in the earlier literature:

“It is also believed that Navg tends towards but never reaches 6 as the system “matures”; that is, the network of fractures tends towards a hexagonal shape, and the angle at vertices increases with system maturity from 90° to 120°”

G. Hetényi and colleagues, Scales of columnar jointing in igneous rocks, Bulletin of Volcanology 74 (2012), p. 473

The last two rows of the table read the same way: 199 columns near Garibaldi in British Columbia, counted by Lee in 1988, average 4.86 sides, and a columnar dike Sosman took from Geikie averages 4.87. Patterns like those are, by the same reckoning, more than half Ts, and nearer to cracked mud than to the Causeway.

That reading is only as good as the counting, and the counting is not as solid as a table makes it look. Hetényi's team had several people count the same redrawn map of O'Reilly's Causeway, and the six counts gave averages from 5.92 to 6.15: one counter found 134 hexagons among 199 columns, another 93 among 174. A corner of 170° is a corner to one eye and a straight side to another. The formula is exact about whatever the counter decided, and no more.

How the Ts become Ys in rock

The first cracks in a cooling lava are like the first cracks in mud. Goehring's thesis notes that the top surface of a lava flow is

“known to have mostly rectilinear joints [79], rather than hexagonal ones”

L. Goehring, On the scaling and ordering of columnar joints (PhD thesis, University of Toronto, 2008), p. 4

The difference is that the cracks in a thick flow keep going. As the cooling front moves down into the lava, the cracks grow down with it in small steps, and each step can start a little away from where the last one ended. The steps are the bands, called striae, that you can see across the sides of the columns. Over many steps the pattern can rearrange itself, and the angles at its junctions open from about 90° toward 120°: Ts turn into Ys. In mud the same thing can be watched in a dish: Goehring and colleagues wetted and dried clay again and again, and

“The angles between cracks were found to approach 120[°], with a relaxation time of approximately 4 generations.”

L. Goehring, R. Conroy, A. Akhter, W. J. Clegg and A. F. Routh, Evolution of mud-crack patterns during repeated drying cycles, Soft Matter 6 (2010) 3562

In the counting of instrument 1 that is exactly the button: the same network, its junctions pulled from T toward Y, and its average rising from four toward six. Real dried mud sits in between: in two photographs of real mud cracks measured by Ruhul Haque and colleagues in 2023, the cells averaged 5.42 and 5.60 corners.

What this page does not show

Neither instrument is a model of cooling lava. The cracks in instrument 1 are placed by a simple rule (bigger pieces crack first, across their long direction), and the pull from T to Y is a geometric nudge, not a mechanical one; the counts proved on the page hold for any placement and any nudge, which is why they are shown this way. The thrown discs are a yardstick for order, not a mechanism. The Causeway's counts are O'Reilly's as Sosman tabulated them in 1916, and its spread in neighbours is Goehring and Morris's calculation from the same survey; neither is a new count, and the 1976 formula is quoted from the paper's abstract, since the full text was not read. The page does not treat the size of the columns, which is set by how fast the lava cooled and is a separate story.

Sources

1. D. Weaire and N. Rivier, “Soap, cells and statistics: random patterns in two dimensions”, Contemporary Physics 25 (1984) 59 to 99, p. 85. doi:10.1080/00107518408210979.
2. S. Bohn, S. Douady and Y. Couder, “Four sided domains in hierarchical space dividing patterns”, Physical Review Letters 94 (2005) 054503, abstract (PubMed). doi:10.1103/PhysRevLett.94.054503.
3. N. H. Gray, J. B. Anderson, J. D. Devine and J. M. Kwasnik, “Topological properties of random crack networks”, Journal of the International Association for Mathematical Geology 8 (1976) 617 to 626, abstract. doi:10.1007/BF01031092.
4. P. Calka, “An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell”, Advances in Applied Probability 35 (2003) 863 to 870, Table 1 (author's preprint). doi:10.1239/aap/1067436323.
5. L. Goehring and S. W. Morris, “Order and disorder in columnar joints”, Europhysics Letters 69 (2005) 739 to 745 (arXiv:cond-mat/0501015). doi:10.1209/epl/i2004-10408-x.
6. L. Goehring and S. W. Morris, “Scaling of columnar joints in basalt”, Journal of Geophysical Research 113 (2008) B10203, Table 1. doi:10.1029/2007JB005018.
7. L. Goehring, On the scaling and ordering of columnar joints, PhD thesis, University of Toronto (2008), p. 4 (copy).
8. L. Goehring, R. Conroy, A. Akhter, W. J. Clegg and A. F. Routh, “Evolution of mud-crack patterns during repeated drying cycles”, Soft Matter 6 (2010) 3562 to 3567 (open copy). doi:10.1039/B922206E.
9. L. Goehring, “Evolving fracture patterns: columnar joints, mud cracks and polygonal terrain”, Philosophical Transactions of the Royal Society A 371 (2013) 20120353 (arXiv:1211.6762). doi:10.1098/rsta.2012.0353.
10. R. A. I. Haque, A. J. Mitra, S. Tarafdar and T. Dutta, “Evolution of polygonal crack patterns in mud when subjected to repeated wetting-drying cycles”, preprint (2023), section 4.2 and Table 1 (arXiv:2305.01991).
11. J. P. O'Reilly, “Explanatory notes and discussion on the nature of the prismatic forms of a group of columnar basalts, Giant's Causeway”, Transactions of the Royal Irish Academy 26 (1879) 641 to 734, not read here; its counts as tabulated in R. B. Sosman, “Types of prismatic structure in igneous rocks”, Journal of Geology 24 (1916) 215 to 234, Table I (Internet Archive).
12. G. Hetényi, B. Taisne, F. Garel, É. Médard, S. Bosshard and H. B. Mattsson, “Scales of columnar jointing in igneous rocks: field measurements and controlling factors”, Bulletin of Volcanology 74 (2012) 457 to 482, Fig. 7, Tables 3 and 4 and p. 473 (ETH open copy). doi:10.1007/s00445-011-0534-4.
13. L. J. Lee, Origin of columnar jointing in recent basaltic flows, Garibaldi area, southwest British Columbia, MSc thesis, University of Calgary (1988), Table 4 (PRISM). Its table prints a total of 200; its rows add to 199.