Continuing the architectural theme, Isabel at God Plays Dice has a post on the ultimate fate of the real world Königsberg bridge problem. Königsberg had seven bridges, and in 1736 Euler proved it was impossible to find a path that allowed you to cross each bridge exactly once.
In World War II, several of the bridges were bombed, and later some were replaced. In present-day Königsberg, now Kaliningrad, there are now only five bridges, and you can now find a path that allows you to cross each bridge exactly once.
Fascinating! My father perhaps had inadvertently contributed to the solution of the problem - in April 1945 he swam across Pregel pushing in front of him a radio transmitter precariously perched on an empty wooden box — to direct shell fire of his artillery battery. The use of bridges was out of question.
Wow, what a Great Post!…
[..]Today I saw this really great blog post, and i wanted to link to it. [..]…
Excerpt from my nearly completed (over 400 pages) Quantum Computing novel “Fermi’s Facebook” –
I know Konigsberg Bridges Fu!
The Königsberg bridge problem asks if the seven bridges of the city of Königsberg, formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. This is equivalent to asking if the multigraph on four nodes and seven edges (right figure) has an Eulerian cycle. This problem was answered in the negative by Euler (1736), and represented the beginning of graph theory.
I looked down to my feet, and ran across the southeastern-most bridge, which crumbled beneath my feet. The wings on my ankles got me across. The same happened as I dashed across the central of the three northern bridges.
Bratelli checked off two boxes on the clipboarded chart. “The brain must change state if the mind changes state. The subject’s lucid dreaming is his experience of achieving conscious awareness of dreaming while still asleep. Lucid dreams used to be generally thought to arise from non-lucid dreams in REM sleep. An obstacle to experimental studies of lucid dreams was that spontaneous lucidity is quite rare. However, subjects such as our Caltech gentleman from Altadena can be trained to become lucid via pre-sleep autosuggestion. Look at the big screen: we can see that he sees himself to be in an augmented version of Russia.”
On a practical note, Jan Kåhre observed that the two bridges over which I’d Mercury-winged no longer exist and that the southwest and northwest bridges are now a single bridge passing above, with a stairway in the middle leading down to the western island. Even so, there was still no Eulerian cycle on the north and south mainland nodes nor the two island nodes using the modern Königsberg bridges, although there is an Eulerian path.
“He’s got significant cross-talk to the Math centers of his brain and extended sensorium now,” said Alaric.
I noticed with my vastened access to Math literature and topographic maps, that the bridge linking the two mid-river islands was originally constructed to be lifted. Thankfully, today it was in no condition to be lifted. Otherwise, the tour problem with bridge lifted would have presented a new challenge.
I could see, ashore, a big but abandoned building at the north shore. Could be the remains of a gigantic effort to solve the ancient geometric problem of doubling the cube, I thought.
“The Subject has succeeded in becoming lucid because we told him to tell himself, before going to sleep,” said Cardoza, “to recognize that he was dreaming by noticing the bizarre events of the dream. An experimental advantage is that subjects can signal that they have become lucid by making a sequence of voluntary eye movements. In combination with retrospective reports confirming that lucidity was attained and that the eye movement signals were executed, these voluntary eye movements can be used as behavioral indication of lucidity in the sleeping, dreaming subject, as evidenced by EEG and EMG tracings of sleep.”