Geoff and The Seven Hills of Sheffield

Alternative title: The Sheffielder who went up a hill and came down a Geoff (not like that)

The unit of topographic prominence in which Sheffield has exactly seven hills.

The city of Sheffield is said to be built on seven hills, just like Rome. It has been repeated for at least ninety years:

The town is very hilly (said to be built on seven hills, like Rome)

George Orwell, diary, 3–4 March 1936

Note that even Orwell put it in brackets and hedged it with said to be. Nobody names the same seven, and careful attempts to count them arrive at five... or eight.

So let's be clear: Sheffield has exactly seven hills. (Although they're not the ones popularly named.)

1. The count is not the problem

Hills have hills. A rise on the flank of a larger rise is a hill by eye and not by any definition that also excludes the doorstep, so how many hills is not vague — it is ill-posed until you say how much of a rise counts as one. So what counts as a hill?

The standard measure is topographic prominence: the height of a summit above the lowest contour that encircles it and encircles nothing higher. Equivalently, how far you must descend from it before you can climb something taller. Britain already relies on this and publishes the thresholds — a Marilyn has at least 150 m of prominence, a Hewitt 30 m, a Tump 30 m — so “how many hills” already has several correct answers, one per threshold.

So the number of hills in a place is not a number. It is a function of the threshold — write it N(t), the count of summits with at least t of prominence — and it only ever goes down as t goes up.

2. The Geoff

Definition. Let p₁ ≥ p₂ ≥ … be the prominences of Sheffield in decreasing order, taken within its boundary as drawn on a stated date. One Geoff is
1 G = sup { t > 0 : N_Sheffield(t) ≥ 7 } = p₇

Sheffield defines the unit. Not each region its own — that would give every region seven hills by construction, and a unit under which every measurement comes out the same measures nothing. The Geoff is one length, fixed by one city, and everywhere else is held against it.

Read plainly, the supremum says something stronger than it looks. One Geoff is the most demanding standard under which Sheffield still has seven hills. Set the bar any lower and Sheffield has more than seven; set it any higher and it has six. “Sheffield has seven hills” is true at a great many thresholds, and this is the hardest of them — which is the whole reason the unit is worth defining rather than simply picking a round number that happens to work.

It works out to Sheffield’s seventh-largest prominence, which is what “choose seven, then find a threshold to suit” looks like once written down.

Definition. A Geoff is a summit of prominence of at least one Geoff.

British hill classifications already work this way — a Marilyn is a summit with 150m of prominence, a Tump 30m. The Geoff is only the first whose threshold is fixed by the count rather than the count by the threshold.

3. What the unit is pinned to

Corollary. Sheffield has seven Geoffs, and cannot have any other number of them. □

That is true by construction and carries no information about Sheffield. The content is in what happens when the boundary moves. The Geoff is fixed to Sheffield as drawn on a given date — the way a unit is fixed to a bar in a vault, and for the same reason. Sheffield’s boundaries have changed and will change again; when they do, its summits change, its prominences change, p₇ changes, and a new Geoff is defined. Sheffield still has seven, because it always will.

Corollary. Everywhere else does not. A region has however many summits clear one Geoff, which may be more than seven, fewer, or none — and since the unit is Sheffield’s, redrawing Sheffield’s boundary changes the number of hills in Rome.

A metre was once the distance between two scratches on a bar in Sèvres, and every length on earth was defined by it. The Geoff differs only in being pinned to a city council’s boundary review.

4. The current value

The Database of British and Irish Hills gives every summit in Britain its drop, which is prominence under the name Britain uses, and the ONS publishes the boundaries. Taking Sheffield as currently drawn:

1 Geoff = 40 metres

Here p₇ = 40 m and p₈ = 38 m, so there is no tie and Sheffield has exactly seven Geoffs. The database is complete to 30 m of drop and the unit sits above that, so nothing is hidden beneath its floor.

The boundary is not a formality. Sheffield’s two largest hills sit on the line — High Neb 30 m inside it, Back Tor 20 m — while Wharncliffe Chase, with more prominence than every hill in the city but one, lies 700 m outside. Moving the border a few hundred metres moves the unit and nothing else:

5. The seven are not the seven

The names that circulate locally — Stannington, Netherthorpe, Upperthorpe, Heeley, Norton, Wincobank, Crookesmoor, though the lists vary — have no summit in common with the seven above. Six of them do not appear in the database at any prominence, being districts on rising ground rather than summits. The one that does is Wincobank Hill at 38 m, which makes it eighth: the first hill excluded by the unit invented to make the count come out at seven.

6. Other UK cities

Held against Sheffield’s 40 m, the 361 local authorities of the United Kingdom come out as follows: 95 have more than seven Geoffs, 150 have between one and six, 108 have none at all, and 8 have exactly seven. This is a real league table, because the unit no longer moves to flatter whoever is being measured.

A local authority is not a city, though. Highland has 2899 Geoffs and is not a town. Restricting it to 47 places that actually are cities, and whose council covers roughly the city rather than half a county:

Sheffield is 5thth of 47. Four cities are built on more hills than Sheffield — Swansea, Edinburgh, Wrexham, Newport — and not one of them has ever mentioned it, which is either restraint or an oversight. Sheffield shares its seven with Bradford, and with six places that are not cities: Barnsley, the Isles of Scilly, Pendle, Renfrewshire, Stafford and Winchester.

20 of the cities have no Geoffs at all: Birmingham, Bristol, Cambridge, Colchester, Coventry, Derby, Kingston upon Hull, Leicester, Manchester, Norwich, Nottingham, Oxford, Peterborough, Portsmouth, Salford, Southampton, Southend-on-Sea, St Albans, Wolverhampton, York.

Among English cities Sheffield is 1st of 38:

Somewhere else:

7. A proposal

Sheffield ties with Bradford, and a tie can be broken. Bradford’s seventh hill has 41 m of prominence, so any unit above 41 m puts Bradford below seven — while Sheffield, which defines the unit, has seven whatever the unit turns out to be.

But the unit cannot simply be raised. One Geoff is the largest threshold at which Sheffield has seven hills, so anything above 40 m is a threshold at which Sheffield has six — and is therefore not a Geoff. The seven is fixed by the definition, the unit by Sheffield’s terrain, and the only free variable in the whole apparatus is where the line goes. It moves one way: raising the Geoff means taking in a hill bigger than the seventh Sheffield already has. Not a redefinition, not a re-survey, not a better map.

Sheffield does not need another hill. It needs a bigger one, and it needs to annex it.

The nearest hill outside the city is Wharncliffe Chase — 102.6 m of prominence, 318.9 m high, and 700 m beyond the line, in Barnsley. Taking it in makes p₇ the old p₆, so one Geoff becomes 49 m, and the standing of every city in England is revised accordingly:

First outright, and Bradford falls to 3. The table above is not a fudge and is not offered as one. It is a costing.

The cost falls on Barnsley, and it should be stated plainly rather than buried: Barnsley is today one of the 8 authorities with exactly seven Geoffs, and the transfer takes it to 3 — losing the hill itself, and then facing the higher bar the hill creates. That is a real loss to a neighbour, incurred so that Sheffield need not share first place. The authors consider it proportionate.

This is a matter for the Mayor.

8. The neighbours

Leeds has 2 Geoffs. It holds 8 summits, but only The Chevin and Tinshill Moor clear the unit; the rest fall between 35 m and one metre. Leeds is therefore two sevenths of Sheffield, and there is no arrangement of the data under which it is more. Rubbish.

Manchester has 0 Geoffs, from 1 summit. Nothing in this paper can be said about Manchester, and that is a result rather than an omission. Pathetic.

9. Bristol

Bristol also claims to be built on seven hills. Bristol has 0 Geoffs, from 3 summits.

This is a stronger result than the one about Sheffield, and worth stating precisely. Sheffield’s seven is unverified: the number was borrowed, never checked, and the hills it names are not the hills it has. Bristol’s is false. Since N is non-increasing and Bristol’s summit count is 3 in total, N(t) ≤ 3 for every t > 0 — so there is no threshold, at any scale, however small, at which Bristol has seven hills. The claim is not unsupported. It is refuted by the definition, on every possible reading of it. Suck it Bristol.

10. Rome

Which leaves the comparison the whole thing rests on. The database stops at the Irish Sea, so Rome was computed from scratch, from a satellite elevation model of the city and 30 km around it, by the same definition as everything above.

By this measure Rome has five Geoffs.

And none of the five is Monte Mario. The highest hill in Rome is a ridge running north off the right bank of the Tiber, and it does not fall forty metres before the ground rises again, so it does not clear one Geoff. It was already excluded from the canonical seven for standing outside the ancient walls; it is now excluded from the five as well, which leaves Rome’s most famous hill on no list of Rome’s hills at all.

On the instrument. Britain’s figures come from surveyed ground heights; Rome’s come from a radar surface model, which sees rooftops and trees — Monte Mario reads 149 m against a surveyed 139. The definition is identical and the instrument is not, so five is a firm answer to a slightly different question, and the honest margin on it is a summit either way.

11. Could Rome have had seven?

Rome’s boundary has moved more than Sheffield’s, and always outwards, so the question has a direction: every historical Rome was smaller, and a smaller region holds fewer summits. Approximating each by a circle of equal area about the Capitoline:

So the answer is no, and then yes, and the yes is the better one.

No historical Rome had seven Geoffs. Ancient Rome — the Rome of the Seven Hills, the Rome the comparison actually invokes — has none at all. The canonical seven are spurs of one plateau divided by shallow valleys, and not one of them falls forty metres before the ground rises again. Under the unit that exists because of them, they are not hills.

And the modern comune is exactly the right size. Seven Geoffs fall within 20.1–21.7 km of the Capitoline, which is an area of roughly 1,273–1,474 km²; Rome’s comune is 1,285 km², squarely inside that band. A circular Rome of precisely the area Rome has would have seven hills.

Rome is the right size to have seven hills and the wrong shape.

12. Conclusion

Sheffield has, has always had, and will always have, exactly seven hills.

So what about Rome? Rome^* has exactly seven Geoffs, just like Sheffield.

* drawn as a circle of its own area.

Method and sources

Summits and their drop from The Database of British and Irish Hills v18.5 (CC BY 4.0); boundaries from the ONS Open Geography Portal, ONS Local Authority Districts (December 2024) UK BFC, at full resolution — the generalised outlines move the line further than the margin that decides Sheffield. Contains OS data © Crown copyright and database right; ONS data under the Open Government Licence. Rome from the SRTM one-arcsecond model (public domain) with its boundary from OpenStreetMap (ODbL).

This page is short and has gaps in it on purpose. The proofs, the judgement calls, the caveats and the things that would need defending are written up properly elsewhere, along with everything needed to reproduce both figures.