Talk:Seven Bridges of Königsberg/Today
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Here we discuss not exactly the real city of Kaliningrad as it stands today but attempt to determine the current state of the bridges in the real city and adapt this into a problem in graph theory.
References[edit]
credit:NASA but no citation; uploaded 2004
for comparison
Locator map
- ScanEx Russia High resolution, clear photo but unfortunately not quite wide enough field of view.
- CityVision City map, wide field. There is clearly a bridge spanning the entire river Pregel some distance to the east. Two bridges or one?
- Google Maps Unfortunately, the portion of satellite database covering the old town is of very poor quality. Don't even bother with the map database. The far east bridge looks about 5 km east of the old town.
- Australian Mathematics Trust Some history, several maps and (non-satellite) photos, and an extensive discussion of history of the bridges. Of particular interest are the correct names of all bridges and landmasses, although I fear mention of colored edges.
Discussion[edit]
Photo recon[edit]
Annotated copy of Google satellite photo (current). Numbering is arbitrary.
The previous photo reduced to a schematic.
Okay, I'd like to start by noting the large cathedral on the central island, 400. Not directly relevant to bridges but useful in describing variations in terms of colored nodes.
That aside, there appear to be 11 structures spanning the Pregel near the old town. Not shown is the area far to the east, including another span (or spans). I think we have to limit this discussion to the old town area. I don't know whether the official current city limits extend that far east, nor do I really care. I don't see any other structures between the old town and the far east, nor any at all to the west before the Baltic.
401 appears to be a railway bridge; see the yards and the characteristically gentle curves of rail lines. Do we include this in any discussion of the problem? This depends on one's approach to practical mathematical games. Some hardy travelers have indeed walked the modern bridges -- successfully or not, it's not always clear. (In one case, the author admits to cheating on 2 bridges.) The first-class traveler strolling around town with a glass of wine, arm-in-arm with a fellow mathematician or Kant researcher, will no doubt scorn the railway bridge. The stringy backpacker will certainly scurry across it on the crossties. The middle-class middle-European resident probably drives back and forth on the highway and rarely takes any bridge that won't pass a car.
402 seems not to be a bridge at all but a lock. When closed, of course, the backpacker can certainly evade the local police and run across it, too.
403 and 405 are highway bridges. We need to see closeup photos to determine if foot travel is practical. For comparison, the Golden Gate Bridge is a traditional, if somewhat long and cool walk; the San Francisco-Oakland Bay Bridge is strictly vehicular and I doubt anyone has crossed it on foot and lived to tell the tale. These look much tamer and I'll accept them for the moment.
I can't tell what 404 may be. It doesn't seem reasonable to be an old footbridge; ship traffic wouldn't be able to get under it to go upriver. The angle is odd, too. It may be anything from a fallen I-beam to a bit of lint. I don't think it's walkable. I begin to think it's a high-tension power line.
406, 407, 408, and 409 look unquestionably to be bridges one might walk. Comparison with the 1613 map suggests that 407 may indeed be a surviving old bridge. Others appear in old locations but construction details lead me to think they're new. These issues are unimportant unless we want to reconstruct not only the current state but some historical ones as well.
410 certainly looks like a bridge but I have doubts. It doesn't seem to go anywhere at the south end; such a wide bridge should have wide roads passing over it, as do 403 and 405. So close to the obviously modern 409, too. I think it's another lock or perhaps a dam.
This makes me wonder also about 411. That looks like a bridge whose northern approach passes through a tunnel. The difficulty with this interpretation is the structure immediately to the north, spanning an east-west artery. If that's an overpass, where does it go? Straight into that apartment building? Perhaps 411 is yet another waterway control structure.
Of course, there's nothing to stop the backpacker from crossing these on foot, too. I suspect maintenance workers drive over them frequently but they may not be open to the public.
Graph of presumed bridges.
So, taking the most conservative estimate permitting foot traffic over the central highway bridges, let's say that 403, 405, 406, 407, 408, and 409 are open to our recreation. That is six bridges. If we include all but the possible power line 404, we have ten. I offer a graph. Probably passable bridges in green, questionable paths in red.
Considering the six-bridge problem, the cathedral has odd degree as does the south bank. The east bank (actually a rather long island extending far upriver) and the north bank have even degree. Therefore, an Euler walk is possible starting or ending at the cathedral with the other end on the south bank. No Euler cycle is possible.
Additional routes in the ten-bridge problem do not change the parity of the cathedral or east bank; they do however give the north bank odd degree and the south, even. Thus, the backpacker must begin or end on the north bank but he will still find the other end of his walk at the cathedral. John Reid ° 06:28, 18 November 2006 (UTC)