The GWB Lower to NY is closed for roadwork. All vehicles use the Upper Level. There are no longer delays at the GWB Upper to NY. The GWB Upper to NY has a delay of 30 minutes due to volume.

Contents

- 1 What is the GW bridge problem?
- 2 What is the busiest time on the George Washington Bridge?
- 3 How much does it cost to go over the George Washington bridge in New York?
- 4 How long does it take to walk across the George Washington bridge?
- 5 What is the busiest bridge in the world?
- 6 What is the strongest bridge in the world?
- 7 What is the widest bridge in the world?
- 8 How much is toll from NJ to NY?
- 9 Can you cross every bridge once?
- 10 Is it possible to cross all 7 bridges?
- 11 Why did the West Seattle Bridge crack?
- 12 Does Seattle still have a floating bridge?
- 13 What was the main problem with iron bridges?

## What is the GW bridge problem?

External links –

- Visual aids
- Composite aerial images of GW Bridge traffic flow around the upper-level toll plaza (illustrations with one and three toll lanes opened at the affected Fort Lee entrance)
- NJ DOT map showing jurisdictions of highways leading to the bridge (affected Fort Lee entrance to upper-level toll plaza shown at “Kelby St” label on the map)

- News coverage about GW Bridge lane closure scandal from:
- NJ.com (especially latest news from The Star-Ledger and NJ Advance Media)
- NJ Spotlight “Christie Page”, not limited to Bridgegate
- NorthJersey.com (especially latest news from The Record ) plus a ten-minute documentary video of their coverage, and their interactive The GWB Files presentation

- Key players, according to:
- The New York Times with graph depicting relationships
- NJ.com (NJ Advance Media)
- NJ Spotlight with interactive graph depicting relationships
- NorthJersey.com
- WNYC-AM/FM (New York) and New Jersey Public Radio with one-year audio retrospective

- Timeline of events, according to:
- Associated Press
- David Wildstein’s testimony as reported by NJ Advance Media, September 2016
- Milowent (includes background history and detailed Bridgegate chronology. references news reports and documents released by New Jersey Transportation Committee.)
- NJ Spotlight, as of May 2015 (interactive)
- “A Timeline for the George Washington Bridge Scandal”, The New York Times, August 31, 2016, Retrieved September 7, 2016,
- The Record, plus their updated graphical timeline
- The Star-Ledger or NJ Advance Media, as of March 2014, and updated in September 2014 and May 2015 and September 2016,
- USA Today
- WNBC-TV4 (New York)
- WNYC-AM/FM (New York) and New Jersey Public Radio

- Official documents
- Transcript of Christie’s January 9, 2014, press conference
- Mastro’s investigation report commissioned by the Office of the Governor of New Jersey (released on March 27, 2014, and updated on April 14 with lawyers’ memoranda summarizing each of 75 interviews)
- Interim Report to the New Jersey Legislature by the committee’s attorney, Reid Schar, December 8, 2014
- Subpoenaed documents released by New Jersey Legislature in all of 2014
- Wildstein plea agreement (signed January 21, 2015, released May 1, 2015)
- Indictment of Baroni and Kelly (filed April 23, 2015, unsealed May 1, 2015)
- Juror questionnaire in Baroni-Kelly trial including list of potential witnesses and/or subjects of testimony (Attachment A), September 2016
- Brennan complaint against Christie, September 2016

Coordinates : 40°51′14″N 73°58′01″W / 40.85389°N 73.96694°W

## What is the busiest time on the George Washington Bridge?

What time is rush hour in NYC? – Rush hour in New York City is typically considered to be between 7-10am and 4-7pm Monday through Friday. This is when the majority of commuters are traveling to and from work, leading to traffic congestion on the roads and public transportation systems.

During rush hour, it can take much longer than normal to travel around the city due to the increased volume of traffic. Public transportation users may experience overcrowded trains and buses as well as longer wait times for service. Drivers will find themselves stuck in slow-moving or stopped traffic on major highways, bridges, and tunnels throughout the city.

It’s best to plan ahead if you’re trying to get somewhere during rush hour – leave extra time for your commute or consider taking alternate routes if possible.

#### How busy is the George Washington Bridge?

Lower deck – USS Nautilus passes under the bridge in 1956, when the bridge had only a single deck The completion of the George Washington Bridge’s lower deck, as well as the construction of a new bus terminal and other highway connections near the bridge, were recommended in a 1955 study that suggested improvements to the New York City area’s highway system.

- The lower deck was approved by the U.S.
- Army Corps of Engineers,
- A Bergen County leader voted against the construction of the lower level in 1956, temporarily delaying construction plans.
- The New York City Planning Commission approved the George Washington Bridge improvement in June 1957, and the Port Authority allocated funds to the improvement that July.

The $183 million project included the construction of the lower deck; the George Washington Bridge Expressway, a 12-lane expressway connecting to the Alexander Hamilton Bridge and the Cross Bronx Expressway (later I-95 and US 9 ); the George Washington Bridge Bus Station above the expressway; and a series of new ramps to and from the Henry Hudson Parkway.

On the New Jersey side, two depressed toll plazas, one in each direction, were to be constructed for lower level traffic. Highway connections were also being built on the New Jersey side, including a direct approach from I-95, Construction of the approaches started in September 1958. Work on the lower level itself started on June 2, 1959, but work was briefly halted later that year because of a lack of steel.

By February 1960, construction was underway on the lower level; the supporting steelwork for the future deck had been completed, and the sections for the lower deck were being installed. The George Washington Bridge’s lower deck would comprise 75 steel slabs; each slab weighed 220 tons and measured 108 feet (33 m) wide by 90 feet (27 m) feet long, with a thickness of 30 feet (9.1 m).

- The construction of the slabs proceeded from either side of the bridge.
- The right-of-way for the George Washington Bridge Expressway had been almost entirely cleared except for the ventilation buildings for the 178th-179th Street Tunnels.
- The segments of the lower deck had been laid completely by September 1960, at which point workers started pouring the concrete for the deck’s roadway, a process that took five weeks.

The layer of concrete measured 4 inches (10 cm) thick. Finally, the deck was paved over with a 2.5-inch (6.4 cm) layer of asphalt. New ramps to the George Washington Bridge in New Jersey, including from the newly completed I-95, opened in mid-1962. The lower deck was opened to the public on August 29, 1962.

- The lower level, nicknamed “Martha” after George’s wife Martha Washington, increased the capacity of the bridge by 75 percent, and simultaneously made the George Washington Bridge the world’s only 14-lane suspension bridge.
- In addition to providing extra capacity, the lower level served to stiffen the bridge in high winds; before the lower deck was constructed, the George Washington Bridge was known to swing up to 30 inches (76 cm).

The George Washington Bridge Bus Station opened on January 17, 1963 and the Alexander Hamilton Bridge opened on January 15, 1963, thus allowing more traffic to use the George Washington Bridge. In the first year after the lower level’s opening, the expanded bridge had carried 44 million vehicles.

### How long will the George Washington Bridge last?

November 14, 2018 / 6:05 PM / CBS New York NEW YORK (CBSNewYork) – If you’ve crossed the George Washington Bridge lately you may have noticed a lot of construction on the north side. The bridge is getting a major facelift to increase its lifespan. CBS2’s Jenna DeAngelis went to the top of the bridge on Wednesday for a behind-the-scenes look at what’s going on. File – A view of the George Washington Bridge, Sept.7, 2016 in New York City. (Photo by Drew Angerer/Getty Images) So for its birthday, the GWB is getting the gift of a longer life. It is undergoing a rehab project to keep it strong and sturdy. MORE : Traffic At GWB Toll Plaza Is The Second Worst In The Country, Report Says Rogers allowed CBS2 to step inside her office for a closer look at the work being done to replace the 592 suspender ropes on the bridge.

- That’s every one of them.
- The suspender ropes actually support the roadway beneath us,” Rogers said.
- The main cables carry most of the load and the suspenders drop from there and support the roadway.” While the work is being done a platform is placed under the main cables, which are being repaired.
- A special system will also be added to prevent corrosion.

CBS2’s DeAngelis walked along those large cables for an overview. Beneath her, was the busiest bridge, which more than 300,000 cars cross a day. When asked if traffic will be impacted at any point during the overhaul, Rogers said, “No more than our normal construction.

You know, we study the traffic patterns. We do some daytime closures during non-rush hour and we do a lot of night work to try and not impact the public.” This is part of the “Restoring the George” program, a nearly $2 billion investment made up of 11 projects. Among them, expanding the sidewalks. “When we’re done with this group of projects we’re gonna have done something for the cyclists, for the pedestrians, for the motoring public,” GWB general manager Ken Sagrestano said.

This is the first time the bridge is getting new ropes and it may not happen again for a very long time. “We expect them to last well into the next century – a once-in-a-lifetime kinda thing,” said Roger Prince, the Port Authority’s deputy director of operations and capital program delivery.

- On Wednesday afternoon, workers were busy on the north side of the bridge.
- Eventually, they’ll move over to the south side.
- The project’s completion date is expected to be some time in 2025, and through the entire project the bridge will remain open.
- The good news is project leaders say the project is moving on schedule.

For a look at how weekly construction could impact traffic on the bridge, please click here,

In: George Washington Bridge

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### Is the 7 bridge problem possible?

Euler’s Proof – On August 26, 1735, Euler presents a paper containing the solution to the Konigsberg bridge problem. He addresses both this specific problem, as well as a general solution with any number of landmasses and any number of bridges. This paper, called ‘Solutio problematis ad geometriam situs pertinentis,’ was later published in 1741,

Euler’s paper is divided into twenty-one numbered paragraphs, and in what follows, a simplified version of Euler’s paragraphs will be presented. In the first two paragraphs of Euler’s proof, he introduces the Konigsberg Bridge problem. In Paragraph 1, Euler states that he believes this problem concerns geometry, but not the geometry well known by his contemporaries, that involves measurements and calculations, but instead a new kind of Geometry, which Leibniz referred to as Geometry of Position.

Then in Paragraph 2, Euler explains to his audience how the Konigsberg problem works. Euler provided a sketch of the problem (see Euler’s Figure 1 ), and called the seven distinct bridges: a, b, c, d, e, f, and, g. In this paragraph he states the general question of the problem, “Can one find out whether or not it is possible to cross each bridge exactly once?” Euler’s Figure 1 from ‘Solutio problematis ad geometriam situs pertinentis,’ Eneström 53 After stating the general question he is trying to solve, Euler begins to explore different methods of finding a solution. In Paragraph 3, Euler tells the reader that to solve this specific problem, he could write down all possible paths, but this technique would take a great deal of time, and would not work for larger configurations with more bridges and land masses.

- Because of these issues, Euler decided to choose a different method for solving this problem.
- In Paragraph 4, he begins simplifying the problem by inventing a convenient system to represent the crossing of a bridge.
- Euler decides that instead of using the lowercase letters to represent the crossing of a bridge he would write the capital letters representing the landmasses.

For instance, referencing his Figure 1, AB would signify a journey that started in landmass A, and ended in B. Furthermore, if after traveling from landmass A to B, someone decides to move to landmass D, this would simply be denoted, ABD. In Paragraph 5, Euler continues his discussion on this process explaining that in ABDC, although there are four capital letters, only three bridges were crossed.

- Euler explains that no matter how many how many bridges there are, there will be one more letter to represent the necessary crossing.
- Because of this, the whole of the Königsberg Bridge problem required seven bridges to be crossed, and therefore eight capital letters.
- In Paragraph 6, Euler continues explaining the details of his method.

He tells the reader that if there is more than one bridge that can be crossed when going from one landmass to the other, it does not matter which bridge is used. For example, even though there are two bridges, a and b, that can take a traveler from A to B, it does not matter with Euler’s notation which bridge is taken. Euler’s Figures 2 and 3 from ‘Solutio problematis ad geometriam situs pertinentis,’ Eneström 53 In Paragraph 7, Euler informs the reader that either he needs to find an eight-letter sequence that satisfies the problem, or he needs to prove that no such sequence exists.

- Before he does this for the Königsberg Bridge problem, he decides to find a rule to discover whether a path exists for a more general problem.
- He does this in Paragraph 8 by looking at much simpler example of landmasses and bridges.
- Euler draws Figure 2, and he begins to assess the situations where region A is traveled through.

Euler states that if bridge a is traveled once, A was either where the journey began or ended, and therefore was only used once. If bridges a, b, and c are all traveled once, A is used exactly twice, no matter if it is the starting or ending place. Similarly, if five bridges lead to A, the landmass A would occur exactly three times in the journey.

Euler states that, “In general, if the number of bridges is any odd number, and if it is increased by one, then the number of occurrences of A is half of the result.” In other words, if there is an odd number of bridges connecting A to other landmasses, add one to the number of bridges, and divide it by two, to find out how many total times A must be used in the path, where each bridge is used once and only once (i.e.

Total Occurrences of A where A has an odd # of bridges = (# of Bridges + 1) / 2 ). Using this fact Euler solves the Königsberg bridge problem in Paragraph 9. In that case, since there are five bridges that lead to A, it must occur three times (see his Figure 1, above).

Similarly, B, C, and D must appear twice since they all have three bridges that lead to them. Therefore 3(for A) + 2(for B) + 2(for C) + 2(for D) = 9, but Euler already stated that there must only be eight occurrences for the seven bridges. This is a contradiction! Therefore, it is impossible to travel the bridges in the city of Königsberg once and only once.

The end, or is it? While the people of Königsberg may be happy with this solution, the great mathematician Leonhard Euler was not satisfied. Euler further continues his proof to deal with more general situations.

### What is up with the West Seattle Bridge?

West Seattle Bridge is open – The West Seattle Bridge reopened on September 17, 2022 after 2 ½ years of closure, repairs, and comprehensive testing, All people driving and taking the bus are welcome on the bridge. Additionally, all restrictions on the Spokane Street Swing Bridge (low bridge) have ended. The low bridge is now open for all to use at any time.

#### Can you walk on the George Washington bridge at night?

The GWB Sidewalk is open from 6 a.m. until 11:59 p.m. daily. Hours of Operation are subject to change and may be impacted by events such as inclement weather, maintenance, or construction. Sign up for George Washington Bridge Sidewalk Alerts for changes in schedule.

## How much does it cost to go over the George Washington bridge in New York?

How much is the George Washington Bridge toll in 2021

Vehicle type | Toll Tag Rate (off-peak hours) | Cash Rate |
---|---|---|

Car, SUV, Pickup truck | $11.75 | $16 |

Carpool | $11.75 | $16 |

2-axle Truck | $36 | $44 |

5-axle Truck | $90 | $110 |

## How long does it take to walk across the George Washington bridge?

If you would like to walk above the Hudson River and you are in the New York City area or in Fort Lee, New Jersey take a walk, if you are not afraid of heights, on the George Washington Bridge. It takes about 20 minutes to leisurely walk across. The view is spectacular and breathtaking to say the least.

## What is the busiest bridge in the world?

New York City’s own George Washington Bridge is not only the busiest bridge in the U.S. but the busiest bridge in the world, with more than a quarter of a million motor vehicles passing over the bridge every single day.

## What is the strongest bridge in the world?

2. The Millau Viaduct, France | Eiffage – The Millau Viaduct in France is considered, as of September 2020, the tallest bridge in the world with, a height that surpasses even the Eiffel Tower. It is a multi-span cable-stayed bridge that crosses the Gorge valley of the river Tarn.

- The viaduct is composed of 127,000 cubic meters of concrete and 26,200 tons of reinforcing steel.
- Furthermore, 5,000 tons of pre-stressed steel are used for the cables and shrouds.
- One of the major structural demands regarding the design of the bridge was to balance the unsymmetrical live loads in the different spans of the bridge.

Furthermore, due to its immense height, it was subjected to high bending moments, so the piers had to be designed as wide and strong as the box sections. The bridge has so far helped with congestion of traffic because it has shortened the travel time between Clermont-Ferrand and Beziers. About company | Eiffage Eiffage, a French civil engineering company and the fifth largest in all Europe, is responsible for building the tallest bridge in the world. The company works in several sectors including construction, infrastructure, energy systems, concessions, etc.

## What is the widest bridge in the world?

Sydney Harbor Bridge, Australia – The Sydney Harbor Bridge is the widest in the world. This bridge not only carries traffic, it also has two railroad tracks, a bicycle path, and a pedestrian walkway running across it. That’s why it was necessary to design it as the world’s widest bridge, at 160 feet, a record it has held since 1932, when it was completed.

- This suspension bridge connects Sydney’s bustling business district with a residential area.
- Interesting fact: The Sydney Harbor Bridge has a vantage point with unique views of the city and its harbor.
- It’s called the Pylon Lookout and can be accessed off the walkway near the eastern side of the bridge.

Be warned: There is a cost of admission, and you will need to climb about 200 steps to reach the vantage point.

## How much is toll from NJ to NY?

Drivers traveling between New York and New Jersey can expect to see higher E-ZPass charges starting Sunday, Jan.8. Last month, the Port Authority of New York and New Jersey voted to approve toll increases on the agency’s six interstate crossings as part of the agency’s $8.3 billion 2023 budget.

- Those increases will affect the Lincoln Tunnel, Holland Tunnel and George Washington Bridge, as well as Staten Island’s three New Jersey-bound bridges, the Goethals Bridge, Bayonne Bridge and Outerbridge Crossing.
- The increases were determined via an inflation-based schedule that was mandated by the Port Authority board in 2008.

This schedule triggers automatic increases whenever the cumulative impact of inflation on existing toll rates reaches $1 as measured from the last increase. For most drivers, the cash-by-mail toll will increase from $16 to $17, with the the E-ZPass peak hours toll increasing from $13.75 to $14.75 and the E-ZPass non-peak hours toll increasing from $11.75 to $12.75.

However, not everyone pays the same rate to cross the Port Authority’s bridges and tunnels. Toll rates vary based on E-ZPass enrollment, vehicle type and time of day, with higher rates during peak hours, for larger vehicles and for those using Tolls-by-Mail. Weekday peak hours are defined as 6 a.m. to 10 a.m.

and 4 p.m. to 8 p.m., with weekend peak hours ranging from 11 a.m. to 9 p.m. Off-peak hours are defined as all hours outside of the aforementioned peak hours. Overnight hours for trucks are defined as Sundays through Thursdays, from 10 p.m. until 6 a.m. the following morning.

Off-peak E-ZPass: $12.75 Peak E-ZPass: $14.75 Tolls-by-Mail All Hours: $17

Class 2 (Vehicles with two axles and dual rear wheels)

Off-peak E-ZPass: $38 Peak E-ZPass: $40 Truck Weekday Overnight Hours: $35 Tolls-by-Mail All Hours: $46

Class 3 (Vehicles with three axles)

Off-peak E-ZPass: $57 Peak E-ZPass: $60 Truck Weekday Overnight Hours: $52.50 Tolls-by-Mail All Hours: $69

Class 4 (Vehicles with four axles)

Off-peak E-ZPass: $76 Peak E-ZPass: $80 Truck Weekday Overnight Hours: $70 Tolls-by-Mail All Hours: $92

Class 5 (Vehicles with five axles)

Off-peak E-ZPass: $95 Peak E-ZPass: $100 Truck Weekday Overnight Hours: $87.50 Tolls-by-Mail All Hours: $115

Class 6 (Vehicles with at least six axles)

Off-peak E-ZPass: $114 (additional axles $19 each) Peak E-ZPass: $120 (additional axles $20 each) Truck Weekday Overnight Hours: $105 (additional axles $17.50 each) Tolls-by-Mail All Hours: $138 (additional axles 23 each)

Class 7 (Class 1 or 11 vehicles with trailers)

Off-peak E-ZPass: $24.25 (additional axles $11.50 each) Peak E-ZPass: $26.25 (additional axles $11.50 each) Tolls-by-Mail All Hours: $36 (additional axles $19 each)

Class 8 (Two-axle buses and mini buses)

Off-peak E-ZPass: $14.50 Peak E-ZPass: $17 Tolls-by-Mail All Hours: $27

Class 9 (Three-axle buses and mini buses)

Off-peak E-ZPass: $14.50 Peak E-ZPass: $17 Tolls-by-Mail All Hours: $27

Class 11 (Motorcycle)

Off-peak E-ZPass: $11.75 Peak E-ZPass: $13.75 Tolls-by-Mail All Hours: $17

Green Pass (Class 1 vehicles)

Off-peak E-ZPass: $9.25 (additional axles $11.50 each) Peak E-ZPass: $14.75 (additional axles $11.50 each)

Green Pass (Class 7 vehicles)

Off-peak E-ZPass: $20.75 (additional axles $11.50 each) Peak E-ZPass: $26.25 (additional axles $11.50 each)

Staten Island Bridges Plan

Off-peak E-ZPass: $7.38 (3 or more trips taken within a month) Peak E-ZPass: $7.38 (3 or more trips taken within a month)

ABOUT STATEN ISLAND BRIDGES PLAN The Staten Island Bridges Plan offers half-priced E-ZPass tolls at the Bayonne Bridge, Goethals Bridge or Outerbridge Crossing for those making at least three Staten Island-bound trips per month that are not included under other Port Authority discount plans.

Staten Island residents must enroll in the Staten Island Bridges Plan in order to receive the discounted rates, and the plan only applies to non-commercial vehicles. Accounts that only register one or two trips in a given month are charged the standard E-ZPass tolls. Those that register three or more trips are charged the discounted rate for all trips during the month, including the first two.

All E-ZPass tags on the account qualify, making it easy for multi-driver households to save. To enroll in the Staten Island Bridges Plan (Plan Code: PASI ), contact the E-ZPass Customer Service Center at www.ezpassny.com or 1-800-333-8655. If you purchase a product or register for an account through one of the links on our site, we may receive compensation.

### How long until West Seattle Bridge is repaired?

The West Seattle Bridge will reopen to traffic on Sept.18, 2022.

## Can you cross every bridge once?

Königsberg bridge problem, a recreational mathematical puzzle, set in the old Prussian city of Königsberg (now Kaliningrad, Russia), that led to the development of the branches of mathematics known as topology and graph theory, In the early 18th century, the citizens of Königsberg spent their days walking on the intricate arrangement of bridges across the waters of the Pregel (Pregolya) River, which surrounded two central landmasses connected by a bridge (3).

Additionally, the first landmass (an island) was connected by two bridges (5 and 6) to the lower bank of the Pregel and also by two bridges (1 and 2) to the upper bank, while the other landmass (which split the Pregel into two branches) was connected to the lower bank by one bridge (7) and to the upper bank by one bridge (4), for a total of seven bridges.

According to folklore, the question arose of whether a citizen could take a walk through the town in such a way that each bridge would be crossed exactly once. In 1735 the Swiss mathematician Leonhard Euler presented a solution to this problem, concluding that such a walk was impossible.

To confirm this, suppose that such a walk is possible. In a single encounter with a specific landmass, other than the initial or terminal one, two different bridges must be accounted for: one for entering the landmass and one for leaving it. Thus, each such landmass must serve as an endpoint of a number of bridges equaling twice the number of times it is encountered during the walk.

Therefore, each landmass, with the possible exception of the initial and terminal ones if they are not identical, must serve as an endpoint of an even number of bridges. However, for the landmasses of Königsberg, A is an endpoint of five bridges, and B, C, and D are endpoints of three bridges. More From Britannica number game: Graphs and networks It would be nearly 150 years before mathematicians would picture the Königsberg bridge problem as a graph consisting of nodes (vertices) representing the landmasses and arcs (edges) representing the bridges.

- The degree of a vertex of a graph specifies the number of edges incident to it.
- In modern graph theory, an Eulerian path traverses each edge of a graph once and only once.
- Thus, Euler’s assertion that a graph possessing such a path has at most two vertices of odd degree was the first theorem in graph theory.

Euler described his work as geometria situs —the “geometry of position.” His work on this problem and some of his later work led directly to the fundamental ideas of combinatorial topology, which 19th-century mathematicians referred to as analysis situs —the “analysis of position.” Graph theory and topology, both born in the work of Euler, are now major areas of mathematical research.

### Can you cross all 7 bridges once?

If you love computers — particularly if you love programming computers — then surely, you know the value of grit. You know how important it is to be resilient, to keep going, to forge forward, and convince yourself that there must be a (better) way to do something.

- But there’s another trait that’s equally important, if not more important that this: the ability to shift perspective.
- Repositioning one’s perspective is certainly a hallmark characteristic of a great programmer.
- All of the developers that I admire and respect exemplify this trait above all others.
- However, it’s not just engineers who have benefited from this — all problem solvers and creative thinkers seem to do this on a daily basis, too.

In fact, I’d venture to say that it was their ability to see things with another angle and in a different light that made them the innovators they were. There is perhaps no better example of this than Leonhard Euler, the mathematician, astronomer, and engineer (just a few of his many titles), who made significant contributions to calculus and the dude who actually created graph theory that we depend on every single day.

- But how did he even do this? Well, for one thing, he definitely didn’t do it by waking up one day and thinking: okay you guys, I’m going to make a new branch of mathematics that’ll eventually be the cornerstone of how the web works in like, 300 years! Nope.
- Instead, here’s the truth of what happened: he was faced with a simple problem.

And the story of how Euler solved this problem centers around how he shifted his perspective, and approached it in a way that no one else ever had before. Way back in 1735, Euler heard about an interesting problem that the town of Königsberg was facing.

At the time, Königsberg was a city in Germany, and the city was built around a river called the Pregel River. This city thrived with its merchant economy and part of the reason that it did so well was because it was structured in a particularly interesting way. There were two large islands in the middle of the Pregel River, and they were each connected to one another, as well as to the two riverbanks on either side, which comprised the majority of the city.

And how were they connected? By bridges, of course! Seven of them, in fact. Here’s a beautiful print that illustrates the city of Königsberg much better than I ever could: The City of Königsberg, Historic Cities Research Project The citizens of Königsberg spent their Sundays walking around town, enjoying their beautiful city. In the process, they came up with a game — which, as it turned out, proved to be incredibly difficult to accomplish.

- The goal was to walk across all of the seven bridges crossing the islands only once, without ever repeating a single bridge in the course of one’s walk.
- At first, when people asked Euler to solve this problem, he brushed it off, thinking that it had nothing to do with mathematics, and therefore wasn’t really worth his time.

But the more that he thought about it, the more intrigued he became. In a letter to a mathematician friend of his, he wrote: This question is so banal, but seemed to me worthy of attention in that geometry, nor algebra, nor even the art of counting was sufficient to solve it.

- He was hooked.
- Euler was so entranced, in fact, that he ended up writing a paper later that year that would contain a solution to the bridge problem.
- But before we understand how Euler solved this problem, we need to cover a few basic graph theory rules first.
- We’ve already learned about some of the different types of graphs that are possible through graph theory, like directed and undirected graphs.

But, we’ve still only just scratched the surface of how graphs work. For example, just as edges can be classified as directed or undirected, vertices (or nodes, as they’re sometimes called) can also be categorized based upon how they link to other nodes in a graph. Degrees of vertices Let’s take a look at the graph to the side. From first glance, we can already glean a lot from this seemingly simple structure. We know that vertex C is connected to vertex D. And we also can tell that vertex B is not connected to vertex D.

Easy, right? Well, these connections end up being important as a graph grows, and it can be helpful to be able to easily identify which vertices are connected with other vertices. The term we use to describe two connected vertices is adjacent, For example, vertex F and E are adjacent because they have a common edge (the edge FE connects them).

Another useful (and important!) way of classifying vertices is by their degree. The degree of a vertex is the total number of vertices that are adjacent to that vertex. The tl;dr version of defining the degree of a vertex: how many neighbor vertices does this node have? Vertex D is only connected to one other node, so it has a degree of 1.

- Vertex C, on the other hand, has four other neighboring nodes, which means that it has a degree of 4.
- Important thing to note: a vertex could have an even or an odd degree — but we’ll dive into that more a bit later.
- Okay, so we can classify edges based on whether they are directed or undirected.
- We can also classify vertices based on how many other vertices they are connected to.

Cool. But wait: there’s more! (Obviously). There’s yet another way to identify graphs, and it has to do with how we traverse through a graph — that is to say, the way that we get from one vertex to another. Generally, when we use graphs to implement a data structure, we’re concerned with how our data relates to one another.

Which is to say, we care about how nodes are connected to other nodes because we usually want to get from one node to another. This is where graph traversal comes in. There are different algorithms that can come in handy for traversal, depending upon what type of graph you have. But we won’t get into those today.

Instead, let’s start simple and talk about the words that we use when we talk about walking through a graph. Most of the time, when people refer to traversing through a graph, they use the term “path”. A path in a graph just means the way that you can get from one vertex (the origin) to another (the destination).

- In order to get from one vertex to another, we have to traverse through some edges in the graph.
- A path represents which node we’re starting from, which edges we’re passing through, and which node we’re ending at.
- However, many times people mix up the term “path” with the term “simple path”.
- They’re actually a little bit different, and it’s important to acknowledge what sets the two terms apart.

A path doesn’t have to follow any rules — we can start wherever you like and traverse through the graph in whichever way we’d like, just as long as we end up at the destination node. Simple paths have a lot more rules that must be followed. A simple path is a unique type of path that is far more constricted: in a simple path, no nodes or edges can be repeated while traversing the graph. Paths and cycles within a graph If you ever find yourself in front of a graph problem, be sure to clarify and understand if you are dealing with finding a path through the graph, or a simple path — because they can profoundly impact the way that you deal with the data structure.

- Finally, there’s also the cycle, which is a simple path, except that we must end our traversal at the same node that we started off at.
- That is to say, the starting “origin” node is the exact same as the ending “destination” node, and we have traversed all the nodes and edges of the graph in the process.

We already know that the formal definition for a graph is an ordered pair, containing a set of vertices and a set of edges. We can define a path (or a simple path, or a cycle) in a similar way: as an ordered list of directed edges, The directed edges here is important since we need to be able to show which node is the origin, which node is the destination, and which edge we’re traversing as we cross from one node to another.

Okay, okay—we’ve covered all these terms and rules and stuff, but what I want to know iswhat happened to Euler?! Well, it’s time to (finally) find out. When Euler was solving his seven bridge problem, he broke it down into smaller, bite-sized pieces. He simplified the problem into parts, and visualized the bridges of Königsberg in a different way.

Ultimately, he solved this problem by approaching it with a different perspective. Instead of writing out every single possible path crossing (which Euler noted would be unsustainable/unscalable, and inefficient), he simplified the problem by visualizing it. The Seven Bridges of Königsberg, in Euler’s perspective According to analysis of Euler’s notebooks conducted by the Mathematical Association of America, Euler represented the landmasses as capital letters — A, B, C, D — and decided to track a bridge “crossing” by the landmasses that one started at and ended at.

- For example, to cross from landmass A to B, the trip on the bridge would be referred to as AB.
- And if one were to then cross from landmass B to D, the entire journey would be visualized as: ABD.
- This would represent crossing two bridges, and touching three landmasses.
- In the process of doing this exercise, Euler realized that in order to cross seven bridges — as was the case in the city of Königsberg — the problem needed at least eight “landmasses”, or letter sequences in order to solve the problem.

Euler realized that it was impossible to cross each of the seven bridges of Königsberg only once! The way that Euler solved this problem was by changing his approach, and creating a kind of representation that no one had really done before. He visualized the seven bridges problem as a network, which eventually became the basis for the graph structure that we know today. The Seven Bridges of Königsberg, in graph format Even though Euler solved the puzzle and proved that the walk through Königsberg wasn’t possible, he wasn’t entirely satisfied. So he kept going and found that, given certain situations, it is completely possible to cross the bridges of “network” only once.

- Today, we often refer to this type of graph traversal as a graph with a Eulerian path or with a Eulerian circuit,
- A Eulerian path is a path wherein we only visit each edge in the graph once, while a Eulerian circuit is a Eulerian path that is a cycle — we only visit every edge once, and we end on the exact same node that we started off on.

Euler determined that, given a certain set of circumstances in a graph/network, a simple path can be possible to find. And because of him, we have yet another way to classify graphs: based on whether they are Eulerian or not. When is something Eulerian? Euler’s mathematical proof determined that there are certain conditions that a graph must meet if we are to traverse through its edges only once. For example, in order for an undirected graph to have a Eulerian cycle, all of the vertices with a degree in the graph must be connected — that is to say, all of the vertices that are connected in the graph must have a degree greater than zero — and all of the vertices in the graph must be of an even degree.

Similarly, in order for an undirected graph to have a Eulerian path (but not a cycle), all of the vertices with a non-zero degree must be connected, and one of these two things must happen: 1) two vertices must have an odd degree OR 2) all of the vertices in the connected graph must be of an even degree.

This can be a little bit confusing unless we really stop and think about it. I love the way that professors of The NRICH Project at the University of Cambridge explain why this bridge problem isn’t possible based on Euler’s proof: Euler proved it couldn’t be done because he worked out that to have an odd vertex you would have to begin or end the trip at that vertex.

- Think about it).
- Since there can only be one beginning and one end, there can only be two odd vertices if you’re going to be able to trace over each arc only once.
- Since the bridge problem has 4 odd vertices, it just isn’t possible to do! Ultimately, it all came down to the degrees of the vertices in the end! (Remember I said that was going to be important?) It isn’t possible to solve the bridge problem if there are four vertices with an odd degree.

According to Euler’s proof, we could only solve it if either all the vertices in the graph were even, or if only two of the vertices were odd. But four odd vertices? Nope. No way. In the graphs below left, you’ll notice some situations where Euler’s proof holds up to the test. Eulerian paths and cycles in action In Graph 1, there are two vertices (A and E) with an odd degree, and so it is possible to traverse each edge once. However, we won’t end up at the same vertex that we started on. In Graph 2, all of the vertices have an even degree, so we could not just traverse each edge only once, but we could end up at the same place that we started! Graph 3 though — sadly, there’s just no possible way to make this work on a Eulerian level.

This graph has the exact same issue as the Königsberg problem: there are four vertices that are odd, and since we know we can never have more than two odd degree vertices, we can be sure that this graph isn’t Eulerian, in the slightest! The story of what happened to Königsberg after Euler put it on the map is interesting (and also a little bit sad!).

In 1875, the city built another bridge between the nodes B and C. This resulted in only two vertices with an odd degree, which solved the impossible walking problem! Euler’s original drawing (“Solutio problematis ad geometriam situs pertinenti”), MAA Euler Archive But, unfortunately, in the case of Königsberg, all did not end that well. When Prussia dissolved, Königsberg eventually became a part of Russia. Two of the original bridges were bombed by Allies in 1942, during WWII.

- Two other bridges were later demolished to make room for a highway.
- And, the city has since been renamed Kaliningrad.
- You can still visit Kaliningrad today, and walk your own Eulerian path across the five bridges that remain.
- Because, guess what — now, it’s possible! I like to think that Euler would probably get a good laugh out of that.

There is a copious amount of research and writing around Euler’s approach to the Seven Bridges problem. I couldn’t get into the details of all of it, but the details are out there! If you want to cross all seven bridges on your own, I suggest starting with this handy resources.

Early Writings on Graph Theory: Euler Circuits and The Königsberg Bridge Problem, Professor Janet Heine Barnett Eulerian Path and Circuit for Undirected Graph, GeeksForGeeks The Seven Bridges of Königsberg, Professor Jeremy Martin Leonard Eulers Solution to the Königsberg Bridge Problem, Teo Paoletti Graph Theory, WebWhompers

## Is it possible to cross all 7 bridges?

This month’s math puzzle dates back to 1735 when it was first solved by Leonhard Euler, a Swiss mathematician and physicist. The puzzle is called The Seven Bridges of Königsberg. It’s based on an actual city, then in Prussia, now Kaliningrad in Russia.

- The city is divided by a river with two islands in between and, further downstream, the river splits the city again.
- The problem is deceptively simple: there are (or were, in Euler’s time) seven bridges to connect the two islands and the downstream parts of the town.
- Euler wondered if a person could walk across each of the seven bridges once and only once to touch every part of the town.

Starting and ending at the same spot was not a requirement. Here is a map you can use to try and solve the problem for yourself: Euler’s Drawing of Konigsberg Bridges Which do you think is more important to solve this problem: the number of bridges or the location of each bridge? Answer: the number of bridges. Euler proved the number of bridges must be an even number, for example, six bridges instead of seven, if you want to walk over each bridge once and travel to each part of Königsberg.

- The solution views each bridge as an endpoint, a vertex in mathematical terms, and the connections between each bridge (vertex).
- Euler realized only an even number of bridges yielded the correct result of being able to touch every part of the town without crossing a bridge twice.
- Euler used math to prove it was impossible to cross all seven bridges only once and visit every part of Königsberg.

By doing so, he set in motion a series of discoveries and insights about how space and intersecting spaces can be defined, as well as their properties. A detailed description of Euler’s solution in in the Wikipedia link below this article. If you’ve ever seen a mobius strip, for example, you’ve seen an example of topology, a mathematical field of study evolved from Euler’s solution to this problem.

- Topology is concerned with space and how things connect one to another, as well as continuity and boundaries of space.
- Topology also studies how properties of a space change and don’t change when the space is expanded or contracted.
- In computing, topology is useful in understanding the networks (paths) data can flow within any system, as well as how sets of data might relate to each other.

The Seven Bridges of Königsberg also is similar to another common computing problem called sometimes the Traveling Salesman Problem where you try to find the most efficient route given a set of restrictions like the seven bridges in Euler’s problem. Non-mathematicians (likely you, definitely me) experience the Traveling Salesman problem any time we get on a train or bus.

The Traveling Salesman Problem is figuring out the most efficient way to travel between pairs of cities of specified distances. Managing scarce resources (trains, buses) that travel along finite routes is a perfect problem for computing to solve because computers are faster and more efficient. But first we need Euler and others to state the problem and define solutions with math.

We then program our computers to do the math. Topology also deals with set theory, how groups of things can be sorted into sets to identify common elements with other groups as well as unique elements. A Venn diagram is a great example of a set. And programming sometimes has to sort data in different ways.

- Which sorting method works best for a situation can be determined by set theory.
- And what happened to the seven bridges from Euler’s time? Two did not survive World War II.
- Two bridges were demolished and replaced with a single highway.
- Of the three remaining bridges, one was rebuilt in 1935 while the other two remain intact as Euler knew them.

And, of course, Königsberg, Prussia has changed its name to Kaliningrad, Russia.

## Why did the West Seattle Bridge crack?

What cracked the West Seattle Bridge? Hidden design problem may have doomed it all along Published on May 13, 2020 The closed West Seattle Bridge undergoing repairs, as seen from 33rd Avenue Southwest, April 2020. Image Credit: SounderBruce. CC ASA 4.0 International

The West Seattle Bridge, closed in March because of excessive cracking, might have been doomed since the day it opened in 1984.City officials have listed several factors that could have contributed to the damage, including more and heavier buses and trucks, a seventh lane added years ago, a jammed rubber bearing that thwarts thermal expansion, and even the 2001 Nisqually earthquake.But a hidden problem in the bridge’s original design might also be to blame.

A leading theory says the 220,000-ton bridge was gradually weakened by long-term shrinking of concrete within the twin girders that support the mainspan above the Duwamish Waterway. Local experts have pointed to an innate behavior of concrete known as “creep” that causes some bridges worldwide to sag by middle age.

Volume loss from creep likely caused high-tension steel cables within the girders, which compress and strengthen the bridge, to slacken, University of Washington professor said this spring. That in turn weakened the concrete and made it vulnerable to crack, he said. In hindsight, consultant John Clark, who analyzed the cracks when the Seattle Department of Transportation (SDOT) discovered them in 2013, wishes he had pushed the city then to seal the cracks by re-tightening the mainspan with new steel.

Continue reading at The Seattle Times. Originally written by for, : What cracked the West Seattle Bridge? Hidden design problem may have doomed it all along

### Which floating bridge sank in Seattle?

Lacey V. Murrow Memorial Bridge sinks to the bottom of Lake Washington After a howling wind- and rainstorm on, state’s historic floating Lacey V. Murrow Memorial Bridge breaks apart and sinks to the bottom of Lake Washington, between Seattle and its suburbs to the east.

Because the bridge’s disintegration happened relatively slowly, news crews were able to capture the whole thing on camera, broadcasting it to a rapt audience across western Washington. “It looked like a big old battleship that had been hit by enemy fire and was sinking into the briny deep,” said one observer.

(He added: “It was awesome.”) The Murrow Bridge was the brainchild of engineer Homer Hadley, who in 1921 proposed a “floating concrete highway, permanent and indestructible, across Lake Washington.” Figuring out a way to cross that lake, between up-and-coming Seattle and its (at that time) sleepy small-town neighbors to the east, was a particular challenge because an ordinary “fixed-pier” bridge was out of the question: The lake was too deep, and its bottom was too mushy.

- Still, people scoffed at what they called “Hadley’s Folly” (one civic organization declared that his “chain of scows across Lake Washington would stand out as a municipal eyesore”), but eventually, mostly because they had no other options, they came around to his way of thinking.
- Construction began on the bridge, named after the state highways director (and brother of famous newsman Edward R.

Murrow), in 1939; it was completed 18 months later. In November 1990, the 6,600-foot-long bridge, made of 22 floating bolted-together pontoons, was in the process of being converted from a two-way road to a one-way road. (A parallel bridge had been completed the year before, effectively doubling the amount of traffic that could cross the lake.) The state highway department alleged that construction crews had left the pontoons’ hatches open, leaving them vulnerable to the weekend’s heavy rains and large waves.

(For its part, the construction company refused to accept responsibility for the disaster, countering that “the probable cause of the failure was progressive bond slip at lapped splices in the bottom slabdue to failure in bond.” It did eventually agree to pay the state $20 million, however.) For whatever reason, at midday on November 25, the center pontoons began to sink.

As they disappeared under the water, they pulled more and more of the crumbling roadway down with them. By the end of the day, the bridge was gone. Fortunately, no one was injured in the incident. The Murrow Bridge was soon rebuilt. READ MORE: : Lacey V.

## Does Seattle still have a floating bridge?

From Wikipedia, the free encyclopedia

Evergreen Point Floating Bridge (2016) | |
---|---|

Looking east towards Medina from the multi-use trail | |

Coordinates | 47°38′27″N 122°15′33″W / 47.64080°N 122.25926°W Coordinates : 47°38′27″N 122°15′33″W / 47.64080°N 122.25926°W |

Carries | SR 520 (6 lanes) |

Crosses | Lake Washington |

Locale | Seattle, Washington |

Official name | The SR 520 Albert D. Rosellini Evergreen Point Floating Bridge |

Named for | Albert Rosellini |

Owner | Washington State Department of Transportation (WSDOT) |

Characteristics | |

Design | Pontoon bridge |

Material | Precast concrete |

Total length | 7,710 feet (2,350 m) |

Width | 116 feet (35 m) (at midpoint) |

Design life | 75 years |

History | |

Construction cost | $4.56 billion (project budget) |

Opened | April 11–25, 2016 |

Dedicated | April 2, 2016 |

Replaces | Evergreen Point Floating Bridge (1963–2016) |

Statistics | |

Toll | $1.25–$6.30 |

Location | |

Wikimedia | © OpenStreetMap |

The Evergreen Point Floating Bridge, also known as the 520 Bridge and officially the Governor Albert D. Rosellini Bridge, carries Washington State Route 520 across Lake Washington from Seattle to its eastern suburbs, The 7,710-foot-long (2,350 m) floating span is the longest floating bridge in the world, as well as the world’s widest measuring 116 feet (35 m) at its midpoint.

### Why did the West Seattle Bridge fail?

The West Seattle Bridge, closed in March because of excessive cracking, might have been doomed since the day it opened in 1984. City officials have listed several factors that could have contributed to the damage, including more and heavier buses and trucks, a seventh lane added years ago, a jammed rubber bearing that thwarts thermal expansion, and even the 2001 Nisqually earthquake. Traffic Lab is a Seattle Times project that digs into the region’s thorny transportation issues, spotlights promising approaches to easing gridlock, and helps readers find the best ways to get around. It is funded with the help of community sponsors Madrona Venture Group and PEMCO Mutual Insurance Company.

- Seattle Times editors and reporters operate independently of our funders and maintain editorial control over Traffic Lab content.
- A leading theory says the 220,000-ton bridge was gradually weakened by long-term shrinking of concrete within the twin girders that support the mainspan above the Duwamish Waterway.

Local experts have pointed to an innate behavior of concrete known as “creep” that causes some bridges worldwide to sag by middle age. A crossing between islands in Palau even imploded after repairs. “It is possible that the amount of creep that has occurred is greater than the designers, and broader engineering community, would have anticipated,” said Seattle structures director Matt Donahue, who ordered the West Seattle Bridge barricaded when cracks accelerated 2 feet in two weeks.

Now, contractors are racing to shore up the span, to preserve it long enough to determine if it can be repaired. Volume loss from creep likely caused high-tension steel cables within the girders, which compress and strengthen the bridge, to slacken, University of Washington professor John Stanton said this spring.

That in turn weakened the concrete and made it vulnerable to crack, he said. In hindsight, consultant John Clark, who analyzed the cracks when the Seattle Department of Transportation (SDOT) discovered them in 2013, wishes he had pushed the city then to seal the cracks by re-tightening the mainspan with new steel.

#### What is the problem with the Roosevelt bridge?

STUART, Fla, — The Roosevelt Bridge has reopened in Stuart after Florida Department of Transportation officials determined there were no structural concerns with the bridge. Both the bridge and Old Dixie Highway lanes are now open in all directions. ****UPDATE ON ROOSEVELT BRIDGE**** FDOT has inspected the bridge and determined it is structurally sound and safe to open.

The Roosevelt Bridge is now open in all directions. Old Dixie Hwy is also open in all directions. — Stuart Police Dept. (@cityofstuart) October 30, 2021 The bridge had been closed in both directions after officials said there was a report of possible cracking in the bridge. Officials said someone who was walking by the bridge called police at about 7 p.m.

to report what appeared to be a crack in the northbound lanes and a chunk of concrete missing from the southbound lanes. This is not the first time the bridge has been shut down after cracking and falling concrete was reported. In June 2020, a large crack was seen along the south end of the bridge,

The bridge was shut down for five months and reopened ahead of schedule in November after crews worked to repair several cracks and corrosion issues that were found during a routine inspection. The old Roosevelt Bridge, which was scheduled to be closed Monday through Wednesday for repairs, remains open, a spokesman for Martin County Fire Rescue told WPTV.

Copyright 2021 Scripps Media, Inc. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed.

#### Why was the West Seattle Bridge shut down?

The West Seattle Bridge closed in March 2020 ‘in the interest of public safety’ after cracks in the bridge support structure – which were originally discovered in 2013 – had ‘rapidly accelerated.’ The approximately 1,300-foot-long concrete structure has been a major route for travelers since it was built in 1984.

## What was the main problem with iron bridges?

Catastrophic Failures – The combination of widespread use and weak strength led to many disastrous collapses of cast iron bridges. The Tay Bridge in Scotland in 1879 was one of the most serious examples of such a collapse. The center portion of the bridge collapsed taking a train with it during a violent storm.