Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces; when done, these lines form i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply well to infinite tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes. Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex; the cut surface is the vertex figure. This is the most common approach, the most understood. Different authors make the slice in different places. Wenninger cuts each edge a unit distance from the vertex.

For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint. Other authors make the cut through the vertex at the other end of each edge. For an irregular polyhedron, cutting all edges incident to a given vertex at equal distances from the vertex may produce a figure that does not lie in a plane. A more general approach, valid for arbitrary convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary; this construction determines the combinatorial structure of the vertex figure, similar to a set of connected vertices, but not its precise geometry. However, for non-convex polyhedra, there may not exist a plane near the vertex that cuts all of the faces incident to the vertex. Cromwell forms the vertex figure by intersecting the polyhedron with a sphere centered at the vertex, small enough that it intersects only edges and faces incident to the vertex; this can be visualized as making a spherical scoop, centered on the vertex.

The cut surface or vertex figure is thus a spherical polygon marked on this sphere. One advantage of this method is that the shape of the vertex figure is fixed, whereas the method of intersecting with a plane can produce different shapes depending on the angle of the plane. Additionally, this method works for non-convex polyhedra. Many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of abstract polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex. More formally it is the - section Fn/V; this set of elements is elsewhere known as a vertex star. The geometrical vertex figure and the vertex star may be understood as distinct realizations of the same abstract section. A vertex figure of an n-polytope is an -polytope. For example, a vertex figure of a polyhedron is a polygon, the vertex figure for a 4-polytope is a polyhedron. In general a vertex figure need not be planar.

For nonconvex polyhedra, the vertex figure may be nonconvex. Uniform polytopes, for instance, can have star polygons for faces and/or for vertex figures. Vertex figures are significant for uniforms and other isogonal polytopes because one vertex figure can define the entire polytope. For polyhedra with regular faces, a vertex figure can be represented in vertex configuration notation, by listing the faces in sequence around the vertex. For example is a vertex with one triangle and three squares, it defines the uniform rhombicuboctahedron. If the polytope is isogonal, the vertex figure will exist in a hyperplane surface of the n-space. By considering the connectivity of these neighboring vertices, a vertex figure can be constructed for each vertex of a polytope: Each vertex of the vertex figure coincides with a vertex of the original polytope; each edge of the vertex figure exists on or inside of a face of the original polytope connecting two alternate vertices from an original face. Each face of the vertex figure exists on or inside a cell of the original n-polytope.... and so on to higher order elements in higher order polytopes.

For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the "Dorman Luke" construction. If a polytope is regular, it can be represented by a Schläfli symbol and both the cell and the vertex figure can be trivially extracted from this notation. In general a regular polytope with Schläfli symbol has cells as, vertex figures as. For a regular polyhedron, the vertex figure is, a q-gon. Example, the vertex figure for a cube, is the triangle. For a regular 4-polytope or space-filling tessellation, the vertex figure is. Example, the vertex figure for a hypercube, the vertex figure is a regular tetrahedron; the vertex figure for a cubic honeycomb, the vertex figure is a regular octahedron. Since the dual polytope of a regular polytope is regular and represented by the Schläfli symbol indices reversed, it is easy to see the dual of the vertex figure is the cell of the dual polytope. For regular polyhedra, this is a special case of the Dorman Luke construction.

The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid. One octahedron and four truncated

June 2009 Washington Metro train collision

During the afternoon rush hour of June 22, 2009, a subway train-on-train collision occurred between two southbound Red Line Washington Metro trains in Northeast, Washington, D. C. United States. A moving train collided with a train stopped ahead of it; the National Transportation Safety Board investigation found that after a June 17 replacement of a track circuit component at what became the site of the June 22 collision, the track circuit had been suffering from parasitic oscillations which left it unable to reliably report when that stretch of track was occupied by a train. The struck train came to a stop because of traffic ahead; because the entire train was within the faulty circuit, it became invisible to the Automatic Train Control system. The train behind it was therefore commanded to proceed at 55 mph; the operator of the striking train applied the emergency brake after the stopped train came into full view, but there was not enough time to prevent the collision, which occurred at 49 mph.

At 4:57 pm EDT on Monday, June 22, 2009, Washington Metro Train 112, bound from Glenmont to Shady Grove, left the Takoma station. Minutes at 5:02 pm, Train 112 rear-ended Train 214, stopped between the Takoma and Fort Totten stations while waiting for another train to leave the Fort Totten station. Nine people were killed, including the operator in the lead car of the moving train, Jeanice McMillan, 42, of Springfield, Virginia; the death toll makes the crash the deadliest in Metro history. The NTSB found that Train 214 had come to a stop within the faulty circuit B2-304, making it invisible to the automatic train control system. Other trains had received speed commands of 0 when traveling through this circuit, but had enough forward momentum to make it to the next circuit and resume detection and receipt of speed commands from the ATC system. Train 214 was going slower than normal because it was being driven in manual mode by its operator, it came to a stop while remaining on circuit B2-304 and was therefore invisible.

Train 112 behind it was given full speed commands by the ATC to proceed on the track. The investigation found that the emergency brakes had been applied by the operator of train 112 when train 214 came into view but it was too late to avert the collision, which occurred with a speed of about 49 mph. A series of almost-collisions in 2005 in similar circumstances in the tunnel between Foggy Bottom and Rosslyn stations led to a new test procedure which would have identified the faulty circuit after installation. However, by 2009, Metro engineers were unaware of this incident or the tests developed to detect the failure condition. Train 112 was made up of 1078, 1071, 1070, 1130 and 1131 -- all from the 1000-Series. Train 214 was made up of cars 3036, 3037, 3257, 3256, 5067 and 5066, from the Breda 3000-Series and the CAF 5000-Series; the cars are equipped with on-board systems called Automatic Train Operation and Automatic Train Control, which allow autonomous operation with little human intervention.

After the collision, WMATA announced a policy of no longer placing the 1000-Series at the end of train consists to prevent a telescoping in a collision, as they were the weakest cars structurally. All 1000-Series cars were put in the middle of trainsets and served for another 8 years until their retirement in June 2017. Cars 5066 and 5067 were never repaired and were used as a source for spare parts until all the 5000-Series railcars were retired in 2018. 1078 was retired following the collision. At 5:20 pm, rescuers first entered car 1079, the lead car of train 112; this car had telescoped over the rear car of the stationary train, trapping many passengers who required rescue by emergency workers using ladders for access. Survivors described the crash as "like... hit a concrete wall," with air clouded by smoke and debris, panic among passengers when car doors did not open. Dennis Oglesby and Martin Griffith, two United States Army soldiers who were in the lead train and were uninjured in the collision, helped passengers evacuate from their train, most of whom appeared to have minor injuries.

Oglesby and Griffith noticed that six to eight people from the other train had been ejected by the force of the collision and were more injured. One person from the overtaking train had been thrown onto the roof of the stationary train and had suffered a severe head wound; the soldiers gave first aid to the more injured victims until help arrived, informed responding emergency personnel that the rails were still powered and needed to be shut down. Following the collision and paramedics from District of Columbia Fire and Emergency Medical Services were dispatched to the Takoma Metro station, arrived at the location of the collision soon after. D. C. Fire Chief Dennis Rubin stated that the initial 9-1-1 emergency calls made the incident seem small, but after firefighters arrived on scene, they dispatched mass casualty incident teams. Within two hours, more than 200 firefighters were on-scene in response to the three-alarm incident. Rescuers worked through the night of June 22, using cranes and heavy rescue equipment to free trapped passengers and search for bodies.

Fire Chief Dennis Rubin confirmed four fatalities and 74 injuries, 14 of which were considered moderate and 6 critical. Five of the dead were discovered in the wreckage and removed from the site of the collision on the morning of Ju

Madara Uchiha

Madara Uchiha is a fictional manga and anime character in the Naruto series created by Masashi Kishimoto. He appears for the first time in "Part II" of the manga and the Shippuden anime adaptation, as a background character. He, along with its first leader Hashirama Senju, is one of the co-founders of Konohagakure village from the ninja world, their power conflict over how to run the village, as well as the long-term feud between clans, leads to Madara's death. However, it is revealed that Madara had used a man named Obito Uchiha as his agent, he transplants his Eye Technique Rinnegan eye power into a child named Nagato, the leader of the terrorist organization Akatsuki, to be preserved for his eventual revival years later. During most of the series, Obito uses Madara's name until a criminal named Kabuto Yakushi reanimates the real Madara who becomes the main antagonist during the Fourth Shinobi World War storyline. Madara has appeared in several pieces including a Boruto feature film. Many Naruto video games have featured him as a playable character.

Kishimoto created the character to provide a strong villain in the series' final arc who would face the series' protagonist, Naruto Uzumaki. This confrontation would be different from that with previous enemies and result in more of a focus on fight scenes. Merchandise has been released in Madara's likeness, including keychains, plush dolls, figurines. In the Japanese anime, Naoya Uchida voices Madara. In the English version, Neil Kaplan provides Madara's voice with Xander Mobus voicing him as a child. Critics have commented on his role as an antagonist in the series. Madara's background was well-received for how integral his story is to the series' scenario. Madara originated from Masashi Kishimoto's desire to elaborate on the ending to the manga series Naruto. Once the series began its second part referred to as "Part II" in the manga and Shippuden in the anime, Kishimoto felt the need to create a story arc that would emphasize the tragedy of wars, leading to the final arc which would include a war.

According to Kishimoto and his group were weak as children in the first part and he wanted to make them stronger in the second part. This is the reason. In an interview, Kishimoto asserted that making the villains "flamboyant" was one of the "guiding principles" he followed, it was his desire, to create villains with "powerful aura". Madara appears late in the manga, as one of the masterminds leading to the fourth shinobi world war, in the final arc, he wanted to make him an engaging antagonist who would clash with Sasuke Uchiha and the creature known as the Ten-Tailed Beast to give the storyline a proper conclusion. According to Kishimoto, Madara is not just like any other character introduced in the series, he created Madara as a character with no weaknesses. As one of the antagonists in the story, Kishimoto designed Madara as the antithesis to the protagonists' values, a perfect anti-hero. Kishimoto said Naruto always defeated his enemies without intending to kill them by settling disputes with words since his battle with Nagato.

Naruto forgave his enemy, instead of having the protagonist kill them, which Kishimoto liked but no shonen manga followed. Kishimoto found the idea of the two characters interacting and settling their differences more interesting and challenging, rather than killing them. So from the Nagato battle onwards, he decided to introduce the resurrected Madara to have someone to beat. According to Kishimoto, this was one of the reasons, but he confessed. It ended up with confusion about the difference between the two characters. Japanese voice actor Naoya Uchida enjoyed voicing Madara's character, most notably in his final recording where he sent regards to fellow actor Kazuhiko Inoue and Wataru Takagi (Obito Uchiha. Neil Kaplan first voiced Madara in the video game Naruto Shippuden: Ultimate Ninja Storm 3; because CyberConnect2's fighting game's storyline preceded the television series' plot, he was asked by their staff to read the series' manga to get an idea of the character. Kaplan said. Madara is mentioned in the second part of the manga Naruto but does not appear in most of the series except the war arc.

During his childhood, he met a child named Hashirama, but amidst the conflicts of their respective clans resulting in war, their friendship ends. After failing at several attempts to kill or defeat his rival, Madara became friends with him and accepted his treaty offering peace between the two clans; the Uchiha and all of their affiliated clans came together to form a village, which Madara named Konohagakure. Madara did not like Hashirama's compassionate methods. After the death of his brother, Madara became cynical and vengeful and held a grudge against the Senju Clan for slaughtering his brothers in the War. After his defection from Konoha, he believed. Losing all of his family and conflicted over his own clanmates, he decided to free the world from its pains, by casting Infinite Tsukiyomi upon the whole world so that there would be no war and no death. Madara is believed to have been killed by Hashirama's hand, he awakens the legendary Eye Technique Rinnegan using Hashirama's DNA. Before dying, Madara takes Obito as hi