This is the area inside which all TfL services run.
Specifically it's the smallest convex polygon encompassing all TfL-operated services. That means the boundary is all straight lines, and none of the angles bend inwards. All of the vertices are either stations or bus stops. You cannot catch a TfL service outside this shape. Here's the map.
This is an entirely unofficial thing, and also inherently meaningless. But I still think it's interesting, so indulge me for a moment.
The TfL Polygon has nine vertices. Clockwise from top left, they are Chesham, Cheshunt, Epping, Shenfield, North Stifford, Bluewater, Chartwell, Dorking and Slough.
The first four vertices in that list - the furthest north - are stations. All the other vertices are bus stops.
To make this exercise worthwhile, we need to look at the TfL Polygon in relation to the boundary of Greater London.
As you can see, the TfL Polygon is larger than Greater London. Specifically it's 70% larger, having an area of 2704km² whereas Greater London covers 1569km². On average, the TfL Polygon runs three or four miles outside Greater London. The greatest distance is seven miles, to Chesham - a legacy of the Metropolitan line stretching way out into Buckinghamshire. A tiny section of Havering, near North Ockendon, lies outside the TfL Polygon, but that's OK because almost nobody lives there, and it probably shouldn't be part of the capital anyway.
It's by no means the case that everywhere inside the TfL Polygon gets a TfL service. There are no buses in the northwest corner, only a tiny handful of stations. There are no TfL stations to the south of Greater London, only a tiny number of unusually long bus routes poking down into Surrey and Kent. In general TfL services creep minimally over the Greater London boundary, because TfL exists to serve its own home area, and the few outliers which define the vertices of the TfL Polygon are very much the exception.
Let's run round the boundary and see what's going on.
Chesham → Cheshunt
This is the longest side of the TfL Polygon, at 25 miles in length. It skirts Rickmansworth and Watford, then runs just north of the M25 from South Mimms eastwards. A year ago there would have been an extra vertex here in Potters Bar, nudging fractionally north, but when TfL cut back route 298 from the Cranborne Road Industrial Estate to the station, that disappeared.
Cheshunt → Epping
Before May 2015, when services to Cheshunt became part of the Overground, this vertex would have been a mile to the south at Waltham Cross bus station instead. Meanwhile the Epping vertex is another legacy of the outer suburban Underground. It would have been considerably further out in Ongar until 1994, and the extent of the TfL Polygon perhaps illustrates the sense in cutting back.
Epping → Shenfield
Shenfield is another May 2015 addition, courtesy of Crossrail. Prior to TfL Rail taking over this end of the line, the corresponding vertex was on Brentwood High Street, served by the number 498 bus. TfL aren't here to serve Hertfordshire and Essex, which is why no London bus routes cross the M25 between Potters Bar and Brentwood, other than dribbling a few hundred metres over the line in Waltham Cross.
Shenfield → North Stifford
Here's the only side of the polygon which passes (very) briefly through Greater London. The redundant part of the capital (near North Ockendon) is mostly fields and a golf course, and home to barely a dozen people. The next vertex at the Davy Down bus stop in North Stifford is only necessary because the 370 bus wiggles into Lakeside shopping centre from the east, otherwise a straight edge direct from Shenfield to Bluewater would have sufficed.
North Stifford → Bluewater
Here's why you shouldn't take the TfL Polygon too seriously. It may only be four miles direct between the bus stops at Davy Down and Bluewater, but to travel between them using TfL services requires heading at least 20 miles upriver to Woolwich and then back again. The journey can be done almost direct using an hourly Thurrock bus service, but crossing the Thames estuary by public transport is an ill-served afterthought.
Bluewater → Chartwell
Once again, this edge of the TfL Polygon closely follows the M25. Chartwell's bus service is a real oddity, a Sundays-only summer extension of route 246 to Winston Churchill's house, now a National Trust property. Once November rolls round, and until March, the Chartwell vertex reverts to the Kent town of Westerham. This reduces the area of the TfL Polygon by 50km².
Chartwell → Dorking
The southern edge of the TfL Polygon is a consequence of three particularly long bus routes stretching well out of London. As well as the 246 to Chartwell there's the 405 to Redhill and the 465 to Dorking. Redhill becomes a vertex of the TfL Polygon in the winter, narrowly, when the 246 retreats to Westerham. The 6½ mile elongation of route 465 into the depths of Surrey is a true anachronism, but Surrey help with funding, so who's complaining?
Dorking → Slough
This edge of the TfL Polygon is 20 miles long, skipping through rural and suburban Surrey. I had to check carefully to make sure we didn't need an extra vertex in Staines, but not quite - the bus station's half a mile inside the specified boundary.
Slough → Chesham
When I say Slough, I mean a bus stop on the High Street by the Queensmere shopping centre, served by route 81. It's TfL's westernmost bus stop, and we've also visited the southernmost (in Dorking), though not the northernmost nor easternmost. And finally it's back to Chesham, marginally skipping Amersham, through a non-TfL hinterland where public transport costs a lot more than £1.50 a journey.
When Crossrail is complete, on some belated unspecified date, the TfL Polygon will extend considerably to the west. Reading is 17 miles further out than Slough, which adds 750km² to the polygon's internal area, a 25% increase. Only five additional stations lie in the extension zone, again confirming that the TfL Polygon is practically irrelevant to those living within it. But as a measure of TfL's ambitions to extend its influence beyond Greater London, balanced against the realities of a budget forcing it to retreat, it's an intriguing indication of intent.