I haven’t written about hill lists for a while, and after writing about the classic Scottish hill lists, and dealing in separate posts with the Corbetts and the Donalds, I’m overdue to write about the third (and original) classic, the Munros. But instead, I’m veering off into the long grass with this one, which deals with a list covering the whole world, featuring 6464 separate peaks, all of which place a summit observer “on top of the world”, by strict geometric criteria.

The list is an offshoot of the work of Kai Xu, at Yale University, which he described in a paper entitled Beyond Elevation: New Metrics to Quantify the Relief of Mountains and Surfaces of Any Terrestrial Body. The paper offers four new descriptors for the way in which mountain peaks relate to the surrounding terrain: dominance, jut, submission, and rut, which together sound like a firm of sadomasochistic lawyers. You can find details of jut on Xu’s website devoted to the topic, but the On Top Of The World (hereafter, OTOTW) list is derived from the measure Xu calls submission.

Submission is defined in Xu’s paper as follows:

The submission of point p is the maximum height of any point on the planetary surface above the horizontal plane of p:
[…]
Submission measures how high the surroundings of a point rise above the point itself, yielding a value greater than or equal to 0 for any point on the planetary surface. As with dominance, submission only considers points within a local vicinity, as points very far away from p correspond to negative height values irrelevant to the calculation of submission.
[…]
A point with a submission equal to (or less than) 0 is known as a dominant point. A person standing at a dominant point is “on top of the world,” as no point rises above their horizontal plane.

The OTOTW list includes all those summits that are also dominant points, under Xu’s definition. Time for a diagram:

The summit in the middle of my diagram above (the one with the little observer perched on its top), is associated with a local horizontal plane that I’ve sketched in blue. Nearby hills fail to pierce this horizontal plane because they are too low. A higher peak at left is sufficiently far away that the curvature of the Earth prevents its summit piercing the horizontal plane. My little observer is therefore “on top of the world”.

Coming up with an exhaustive list of such summits requires the processing of a shed-load of topographic data, and also factoring in the lumpy shape of the geoid, the true shape of the Earth at sea level. You can find a nice map of Xu’s entire collection of OTOTW summits here.

It’s a fine thing to contemplate, but I thought I’d simplify the contemplation a little by honing down, very parochially, on the hills I know well—the twenty OTOTW summits in Scotland, shown on my map at the head of this post.

The first thing to notice is that the big hills drive out the small—the northern mainland of Scotland is dominated by eleven high summits, all of them of Munro status—that is, higher than 3000 feet (914 metres). Two of these Munros lie offshore, the highest points on the islands of Skye and Mull, but they’re near enough to the mainland to suppress the OTOTW aspirations of many west-coast hills.

The Southern Uplands, meanwhile, are dominated by the two highest hills in that region—Merrick in the west and Broad Law in the east.

The outlying islands are far enough from the Highland giants to generate their own OTOTW summits—Goatfell on Arran, Beinn an Oir in the Paps of Jura, An Cliseam on Harris, and Ward Hill on the island of Hoy, in the Orkneys. Even farther out, we get our final three summits—all low, but far enough from everything else to still reach OTOTW status—Ronas Hill in Shetland, Conachair on St Kilda, and Da Sneug on Foula.

On the mainland, some summits seem oddly close together—the Ben More / Ben Lawers pair; the trio of Ben Hope, Ben Klibreck and Ben More Assynt. These groupings are made possible by the fact that the hills involved have roughly similar heights. Lawers is just 40 metres higher than Ben More, and the 26-kilometre separation between the two is enough to drop Lawers (by my rough calculation) about 15 metres below the local horizontal plane drawn from Ben More’s summit. Ben Klibreck is 35 metres higher than Ben Hope, but 23 kilometres away, dropping it about six metres below Hope’s local horizontal.

And for those familiar with the Scottish hills and outlying islands, there are some surprising omissions. Ben Wyvis (1046m) stands in notable isolation, but doesn’t make OTOTW status—the summit of Sgurr Mor (1109m) is just high enough to break through Wyvis’s local horizontal. The little island of North Rona, 70 kilometres northwest of Cape Wrath, is low (just 108 metres), but also a long way from any high ground—surely it should qualify? But a distant glimpse of Foinaven (911m) on the mainland is enough to pierce Rona’s horizontal plane. (And Foinaven, in turn, falls victim to Ben More Assynt, farther to the south.) And the whole chain of islands of the Outer Hebrides is denied OTOTW status by sight of Sgurr Alasdair (and the other Skye Cuillins), until the terrain gets high enough, and far enough north, for An Cliseam to triumph.

Finally, there’s actually a twenty-first Scottish OTOTW summit that isn’t listed by Xu—the Atlantic islet of Rockall, which since 1972 has been officially (in the UK at least) part of Scotland. Over 300 kilometres from the nearest land, and just 17 metres high, absolutely nothing is visible above its sea horizon, making it an obvious shoo-in for On Top Of The World status. I suspect the omission from Xu’s list is because the topographic databases he processed in order to generate his data just don’t contain this tiny bit of remote real estate.

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Note: CCCP stalwarts Steve and Rod contributed significantly to the discussion of hills that have surprisingly failed OTOTW status, and it was Steve who spotted Rockall as a missing qualifier.