please empty your brain below

Everyone loves a colourful map!

(Small typo in penultimate paragraph :-) )
Oh good. Maps.

If you note that small boroughs tend to be near other small boroughs, then that would even out the numbers. Indeed large boroughs are not as dominant in the touching stakes as you might expect.
As well as touching. London boroughs do a lot of sharing. Take Sutton as an example.
Sutton and Kingston share one Highways department.
Sutton and Merton share one Parks contractor.
Sutton, Croydon and Bromley have recently had their Police forces merged.

Richmond and Wandsworth share nearly all services.
Looking all all those colours and lines makes me wonder where the borough’s got their boundaries in the first place...
too many boroughs. the fact that a few have (mostly on the quiet) combined services proves perhaps it time for a rethink. not that it will solve much in terms of (lack of) "affordable" housing (to rent), sustainable care services, dealing with homelessness, etc
The South London Waste Partnership is a service combination for four boroughs: Kingston, Croydon, Merton and Sutton
Wikipedia is the kind of pedant who believes the City of London isn't a London borough:

https://en.wikipedia.org/wiki/List_of_London_boroughs
Interesting post, thank you.

DG has used a different colour for each number on the map but it reminded me of the minimum number of colours needed to separate areas on map which most people now agree is four.

It was proved to be possible with five colours 200 years ago, but it is only 40 years since a proof was provided that only four colours are necessary. This could only be done using computers so there remain some doubters.

Full info here
Until 1965 the area now covered by Greater London was administered by:

- 39 Municipal Boroughs,
- 28 Metropolitan Boroughs,
- 16 urban districts,
- three County Boroughs,
- two complete county councils and parts of four others,
- the City of London Corporation.

Can you really say that Lewisham touches Tower Hamlets when there is not a single river crossing between them? Ditto Greenwich and Barking; or Bexley and Barking or Havering.

Or even the City and Lambeth? Though at least there you have the Waterloo & City line, and the river bus.
I am using the Ordnance Survey definition of touching. I am happy to believe them.
Looking at timbo's list - but actually not all of Middlesex is now in Greater London. Potters Bar was transferred to Hertfordshire County Council and the Staines-Sunbury-Shepperton area was hived off to Surrey County Council despite being on the "wrong" side of the river. So maybe that line should read "one complete county council, most of another and parts of four others"!
The average number of bordering boroughs being six is indeed a feature of all maps.

An idealized perfectly regular map would be a hexagonal grid with each area bordering 6 others, and the real world map can be considered a deformation of the hexagonal grid, with every new border created simultaneously removing another.
Gregg's theorem only works for a limited subset of maps. Consider a chessboard map. The average touching count is 4 - or slightly more if you bring in the surrounding sea.
& the London Libraries Consortium has at least a dozen councils as members
"where the boroughs got their boundaries in the first place"

The London boroughs as defined in the London Government Act 1963 mainly combined -- as said above -- existing local authorities, including county boroughs, district councils and metropolitan boroughs.

And they were mainly created in the 19th century from previous administrative areas -- right down to parishes at the smallest level, which were probably based on church parishes, though they were changed in the 18th and 19th century as the population grew and houses were built, requiring new churches.

And before that, parishes were probably based on individual land-holders' property boundaries, right back to Anglo-Saxon times.

That left some oddities. The metropolitan borough of Woolwich famously had a northern enclave on the Essex side of the river.

"right back to Anglo-Saxon times"

... or earlier. Look at the Edgware Road, founded on the line of a Roman Road and still an administrative boundary for much of its length -- possibly since the Romans divided up the countryside for administrative purposes.

London local govt needs reform desperately, a maximum of 24 boroughs of similar population and/or area
Dubibubno...
Pedantry can be good if it means accuracy. While the City of London Corporation carries out local government for its area, just as London boroughs do for theirs, the legal basis on which the City operates is significantly different from the boroughs.

The OS data set lists East Northamptonshire and South Kesteven as touching. That boundary is interesting.
if we're playing the pedantry game- does the "touching boroughs outside of London" map count simply "local authorities" or just those either styling themselves as Borough councils and/or consisting almost entirely of a ceremonial Borough? most are Non-Metropolitan District Councils within Non-Metropolitan Counties, with two Unitary Councils. London Borough Councils are a form of local authority unique to London in their powers and responsibilities (the Metropolitan Boroughs of the west midlands and north of England are different again).

A clear "not a Borough" is Epping Forest. Not called a Borough, does not contain any ceremonial Boroughs.

I need help.
ah, the return of the "fade-to-grey"
If they ever reform local govt in London hopefully we shall end up with no more than 20 boroughs each of about 500,000 souls. Which ones are ripe for merger or dismemberment? Would you add in any district council areas too. The only candidate I would nominate is Watford










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