What Makes a Significant Hill?

Is it possible to define a globally applicable, height and prominence-free definition of a significant hill?

As a child I could reel off the highest points in each continent and in numerous countries around the world. Soon after we were married, my wife and I made a point of climbing the highest summits at least in England, Wales and Scotland, although tentative plans to work through the Munros ground to a halt after three Cairngorm summits plus Schiehallion.

All hills are worth climbing of course, because they're there, but are some hills more worth climbing than others? There are many lists of hills out there, of which the Munros are by far the most famous. But, stuck in East Anglia, with even the Peak District barely in range, and certainly nothing approaching 3000 feet, does that mean one should give up? Well, even in the South-East there are Marilyns.

The definitions of a [significant] hill seem so arbitrary though, 3000 feet for Munros, 2500 feet for Corbetts, 150m prominence for Marilyns, and so on. Is it possible to make a qualitative definition of a significant hill that is globally applicable? I suggest it is: a significant hill is one which overlooks all surrounding hills, or to put it more formally:

“A significant summit is one for which (owing to the curvature of the earth) all surrounding topography subtends a negative angle from the horizontal.”

Or in other words, a significant summit has a whole hemisphere of uninterrupted sky above it.

I term such a hill a vanity because it looks down on everything around it, and it also somehow seems to follow in sequence after Munro, Marilyn ...

Armed with the free-to-download NASA topographic radar data (then only available at 3-arc second resolution), the estimable British Hills database, and supplemented by a lot of eyeballing of OS maps for the flatter regions of the kingdom, I wrote a Perl script to locate all the vanities in the geographic British Isles, plus the bits of the UK not included in this, i.e. the Channel Islands.

The (provisional) results, listed at the end of this post are (in my opinion) fascinating. First, there aren't that many, only 121 in total. Wales has only three because of the prominence of Snowdon, and in fact summits of volcanic origin feature prominently in the list, as they both tend to protrude above the general level when formed, and are resistant to erosion. Geometry obviously dictates that there tend to be more vanities in lowland areas than upland ones, but then of course they are easier to climb (or 'bag' if you prefer the technical term). They can also occur in binary doublets when neither of two close summits quite overlooks the other (e.g. Buckden Pike and Great Whernside in the Pennines), or in chains, where the change in height of a range of hills is very gradual, such as the South Downs, and, much more subtly, the East Anglian plateau.

Wherever you live, there will be a vanity within reach (horizontally, if not vertically). And their summits have such wonderfully different characters - the sharp spike of Sgurr Alasdair in the Cuillins (so I'm told), the grand, solid summit of Snowdon (with café removed), the high rolling windswept heights (High Willhays on Dartmoor, Urra Moor in the North York Moors), the uninterrupted downland views of Swyre Head (Dorset) or Ditchling Beacon (South Downs), and the surprisingly airy views of Olmstead Green near Haverhill, Suffolk (128m, in the middle of a cornfield) and even the baby of them all, Winterton-on-Sea, Norfolk (22m, surrounded by wind turbines). Of course some are more ridiculous than sublime, such as Crowborough, Sussex with houses all around, Wendover Woods in the Chilterns where you can hardly see the downward slope for the trees, and to be savoured as the most un-summit like of them all, the one in Epping Forest, enclosed by trees, and even when standing clear on the tarmac road, no downward dip in sight. So, why not start vanity-bagging?

A few technical terms before the promised vanity list:

A virtual vanity is any area that is virtually a vanity and by virtue of the eye-height of the observer (which could be defined as 1.5m to standardize) looks down on all surrounding topography. Every vanity has a virtual vanity area around it, small for sharp summits, potentially huge for very flat plateaus.

Topographical smog is any area where human activity obscures the underlying natural topography. There is a thin N-S wedge of the fens which suffers from this. The original sandbanks and small islands have been flattened and overtopped by the sea and river banks. In this area all the sea banks (at 8m) and probably all the river banks (mostly at 7m) form linear virtual vanities.

A vanity domain is the area around a vanity that 'belongs' to it. To find which vanity a point belongs to, move from that point to the point overlooking it that subtends the largest positive angle, and repeat until you reach a vanity. All points leading to a given vanity form the vanity's domain. This gives a quantitative measure of the relative importance of different vanities, although I haven't calculated this because computationally neither I nor my computer were up to it, and also you have to decide how you are going to weight the sea areas overlooked by vanities.

In mountainous areas the idea of a sub-vanity makes sense, with hierarchies of summit children which are directly overlooked at the largest angle by a (parent) vanity, and grandchildren which are directly overlooked at the largest angle by a child. In flatter areas the convex nature of most hills rarely makes this meaningful.

A word of warning, this not being an ideal world, many of the lower vanities have no public access (e.g. Arbury Hill and Whatborough Hill in the south Midlands). Moreover when you get there, trees (Ditchling Beacon slightly, Wendover Woods completely), reservoirs (Therfield), buildings and hedges (Honey Hill) or fog (most anywhere) may intrude on the view. Also hill spot heights are notoriously subject to revision, and new vanities may appear and old ones disappear if anyone undertakes to do a more rigorous calculation of the list...

You can of course generate a list of vanities for any area in the world and seek to bag them, from the Himalayas to the Andes, though I would suggest that the British ones might provide a more gentle introduction, except for the (politically) contentious vanity that is Rockall...

Rank	Overall	Height	Name	Location	Grid Ref
Scotland
1	1	1344m	Ben Nevis	56.79686N, 5.00352W	21667712
2	2	1309m	Ben Macdui	57.07037N, 3.66910W	29887989
3	3	1214m	Ben Lawers	56.54491N, 4.22089W	26357414
4	4	1183m	Carn Eige	57.28770N, 5.11513W	21238261
5	5	1174m	Ben More	56.38595N, 4.54010W	24327244
6	6	1108m	Sgurr Mor	57.70013N, 5.01676W	22038718
7	9	998m	Ben More Assynt	58.13816N, 4.85823W	23189201
8	10	992m	Sgurr Alasdair	57.20662N, 6.22418W	14508207
9	12	966m	Ben More	56.42483N, 6.01403W	15257330
10	13	962m	Ben Klibreck - Meall nan Con	58.23517N, 4.41120W	25859299
11	15	927m	Ben Hope	58.41311N, 4.60786W	24779501
12	20	874m	Goatfell	55.62583N, 5.19187W	19916415
13	22	843m	Merrick	55.13927N, 4.46841W	24275855
14	23	840m	Broad Law	55.49814N, 3.35271W	31466235
15	27	799m	Clisham	57.96370N, 6.81254W	11549073
16	30	785m	Beinn an Oir	55.90241N, 6.00412W	14986749
17	49	481m	Ward Hill	58.90061N, 3.34053W	0322810022
18	51	450m	Ronas Hill	60.53387N, 1.44596W	0430511834
19	52	430m	Conachair	57.82058N, 8.57288W	01009002
20	53	418m	The Sneug	60.14012N, 2.09568W	0394711395
21	59	293m	Royl Field	60.03953N, 1.29110W	0439511285
22	61	285m	Saxa Vord	60.82747N, 0.84106W	0463112166
23	62	283m	Fitful Head	59.90557N, 1.38277W	0434611135
24	84	219m	Ward Hill	59.54592N, 1.63275W	0420810733
Wales
1	7	1085m	Snowdon - Yr Wyddfa	53.06850N, 4.07623W	26093543
2	19	886m	Pen y Fan	51.88402N, 3.43679W	30122215
3	46	536m	Foel Cwmcerwyn	51.94616N, 4.77458W	20942311
Ireland
1	8	1039m	Carrauntoohil	51.99966N, 9.74213W	V803844
2	14	952m	Brandon	52.23513N, 10.25446W	Q460116
3	16	925m	Lugnaquilla	52.96633N, 6.46390W	T032917
4	17	919m	Galtymore Mountain	52.36604N, 8.17838W	R878237
5	21	850m	Slieve Donard	54.18012N, 5.91982W	J358276
6	25	814m	Mweelrea	53.63707N, 9.82975W	L789668
7	26	806m	Nephin	54.01317N, 9.36775W	G103079
8	28	795m	Mount Leinster	52.61811N, 6.77945W	S826525
9	29	792m	Fauscoum	52.24591N, 7.53586W	S316105
10	31	751m	Errigal	55.03424N, 8.11222W	B928207
11	34	688m	Croaghaun	53.98389N, 10.19609W	F559060
12	35	678m	Sawel Mountain	54.81964N, 7.03853W	H617973
13	36	666m	Cuilcagh	54.20080N, 7.81070W	H123280
14	37	647m	Truskmore	54.37401N, 8.37085W	G759473
15	42	615m	Slieve Snaght	55.19643N, 7.33343W	C424390
16	43	595m	Slieve League	54.65068N, 8.70649W	G544783
17	44	550m	Trostan	55.04568N, 6.15441W	D179235
18	47	527m	Arderin	53.03953N, 7.65410W	S232988
19	54	391m	Slievecallan	52.84009N, 9.27042W	R144773
England
1	11	978m	Scafell Pike	54.45423N, 3.21151W	32155072
2	18	893m	Cross Fell	54.70307N, 2.48696W	36875343
3	24	815m	The Cheviot	55.47846N, 2.14546W	39096205
4	32	704m	Great Whernside	54.16098N, 1.99844W	40024739
5	33	702m	Buckden Pike	54.20486N, 2.06163W	39604787
6	38	636.3m	Kinder Scout	53.38481N, 1.87391W	40843875
7	39	633m	Bleaklow Head	53.46127N, 1.85897W	40943960
8	41	621m	High Willhays	50.68521N, 4.01117W	25800892
9	45	540m	Brown Clee Hill	52.47673N, 2.59960W	35932867
10	48	519m	Dunkery Beacon	51.16278N, 3.58675W	28911415
11	50	454m	Urra Moor - Round Hill	54.40637N, 1.08590W	45945015
12	55	297m	Walbury Hill	51.35249N, 1.46506W	43731616
13	56	295m	Leith Hill	51.17650N, 0.37147W	51391431
14	57	294.3m	Milk Hill	51.37773N, 1.85160W	41041643
15	58	294m	Tan Hill	51.38130N, 1.88342W	40821647
16	60	288m	Long Knoll	51.13775N, 2.30713W	37861376
17	63	280m	Black Down	51.05864N, 0.68935W	49191296
18	64	279m	Lewesdon Hill	50.80783N, 2.79956W	34371012
19	65	278m	Bardon Hill	52.71449N, 1.32059W	44593131
20	66	277m	Win Green	50.98562N, 2.10804W	39251206
21	67	277m	Liddington Castle	51.51631N, 1.69900W	42091797
22	68	274m	Bulbarrow Hill	50.84926N, 2.31714W	37771055
23	69	270m	Butser Hill	50.97785N, 0.98029W	47161203
24	70	269.6m	Botley Hill	51.27885N, 0.01263W	53871551
25	71	267m	Wendover Woods	51.77332N, 0.71043W	48902090
26	72	267m	Telegraph Hill	50.84341N, 2.50700W	36441050
27	73	257m	Bald Hill	51.65539N, 0.94901W	47281957
28	74	252m	Carnmenellis	50.18307N, 5.22931W	16950364
29	75	252m	Watch Croft	50.16505N, 5.61347W	14200357
30	76	248m	Ditchling Beacon	50.90173N, 0.10761W	53311130
31	77	242m	Crowborough	51.05524N, 0.15398E	55101306
32	78	241m	St Boniface Down	50.60383N, 1.19964W	45670785
33	79	240m	Chanctonbury Ring	50.89657N, 0.38223W	51381120
34	80	239m	St Catherine's Hill	50.59333N, 1.30365W	44930772
35	81	235m	Wrotham Hill	51.31685N, 0.28485E	55931600
36	82	230m	Whatborough Hill	52.64551N, 0.86720W	47673059
37	83	225m	Arbury Hill	52.22446N, 1.21066W	45402587
38	85	214m	Wilmington Hill	50.80981N, 0.19653E	55481034
39	86	214m	Honey Hill	52.38513N, 1.06249W	46382768
40	87	208m	Swyre Head	50.60583N, 2.09438W	39340784
41	88	197m	Hollingbourne	51.26958N, 0.66523E	58591557
42	89	195m	Hart Hill	51.22352N, 0.78863E	59471509
43	90	188m	Cheriton Hill	51.11407N, 1.13882E	61971396
44	91	185m	Hastingleigh	51.16810N, 0.99275E	60921453
45	92	175m	North's Seat	50.87740N, 0.61893E	58431119
46	93	168m	Normanby Top	53.45305N, 0.31316W	51213964
47	94	168m	Therfield	52.01677N, 0.05728W	53332372
48	95	151m	Stenigot	53.32602N, 0.11394W	52563827
49	96	149m	Scamblesby Thorpe	53.28898N, 0.04655W	53023787
50	98	128m	Great Wood	52.17205N, 0.61169E	57862558
51	99	128m	Olmstead Green	52.05097N, 0.38182E	56332419
52	100	124m	Weston Colville	52.14841N, 0.36655E	56192527
53	101	124m	Withersfield North Wood	52.11698N, 0.40871E	56492493
54	102	117m	Epping Forest	51.68277N, 0.06873E	54302003
55	105	110m	Danbury Church	51.71415N, 0.57402E	57782049
56	106	105m	Beacon Hill	52.92587N, 1.24709E	61833414
57	107	101m	Melton Constable	52.85594N, 1.01105E	60273330
58	108	95m	Brink Hill	52.75583N, 0.60538E	57583208
59	109	95m	Bradenham Hill	52.65675N, 0.84647E	59253104
60	110	95m	Little Dunham	52.68002N, 0.77548E	58763128
61	111	89m	Toftrees	52.80465N, 0.82942E	59073268
62	112	87m	Docking	52.89574N, 0.62269E	57643364
63	113	75m	Poringland	52.57463N, 1.33867E	62623027
64	114	71m	Carleton Rode	52.49939N, 1.09567E	61012936
65	115	64m	Bedfield Long Green	52.25868N, 1.25037E	62182673
66	116	63 m	Brundish	52.28072N, 1.29891E	62502699
67	117	56m	Metfield	52.36239N, 1.40050E	63152793
68	118	51m	Higher Newford	49.92822N, 6.30438W	09120120
69	119	30m	Lowestoft	52.48880N, 1.74182E	65402945
70	120	22m	Winterton-on-Sea	52.70874N, 1.67029E	64793187
Isle of Man
1	40	621m	Snaefell	54.26329N, 4.46165W	23974880
Channel Isles
1	97	143m	Jersey Les Platons	49.24778N, 2.10500W	-
2	103	114m	Sark Le Moulin	49.43250N, 2.36250W	-
3	104	111m	Guernsey Hautnez	49.43500N, 2.57500W	-
Outliers
1	121	21.4m	Rockall	57.59667N, 13.68861W	-