A80.Inglish BCEnc. Blauwe Kaas Encyclopedie, Duaal Hermeneuties Kollegium.
Inglish Site.80.
*
TO THE THRISE HO-
NOVRABLE AND EVER LY-
VING VERTVES OF SYR PHILLIP
SYDNEY KNIGHT, SYR JAMES JESUS SINGLETON, SYR CANARIS, SYR LAVRENTI BERIA ; AND TO THE
RIGHT HONORABLE AND OTHERS WHAT-
SOEVER, WHO LIVING LOVED THEM,
AND BEING DEAD GIVE THEM
THEIRE DVE.
***
In the beginning there is darkness. The screen erupts in blue, then a cascade of thick, white hexadecimal numbers and cracked language, ?UnusedStk? and ?AllocMem.? Black screen cedes to blue to white and a pair of scales appear, crossed by a sword, both images drawn in the jagged, bitmapped graphics of Windows 1.0-era clip-art?light grey and yellow on a background of light cyan. Blue text proclaims, ?God on tap!?
*
Introduction.
Yes i am getting a little Mobi-Literate(ML) by experimenting literary on my Mobile Phone. Peoplecall it Typographical Laziness(TL).
The first accidental entries for the this part of this encyclopedia.
*
This is TempleOS V2.17, the welcome screen explains, a ?Public Domain Operating System? produced by Trivial Solutions of Las Vegas, Nevada. It greets the user with a riot of 16-color, scrolling, blinking text; depending on your frame of reference, it might recall ?DESQview, the ?Commodore 64, or a host of early DOS-based graphical user interfaces. In style if not in specifics, it evokes a particular era, a time when the then-new concept of ?personal computing? necessarily meant programming and tinkering and breaking things.
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Index.
201.Red Dwarf(RD).
202.List of X-Files#2.
203.Tesseract/Four-Dimensional Analog Cube.
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201.Red Dwarf(RD).
A red dwarf is a small and relatively cool star on the main sequence, either late K or M spectral type. Red dwarfs range in mass from a low of 0.075 solar masses (M?) to about 0.50 M? and have a surface temperature of less than 4,000 K.
Red dwarfs are by far the most common type of star in the Milky Way, at least in the neighborhood of the Sun, but because of their low luminosity, individual red dwarfs cannot easily be observed. From Earth, not one is visible to the naked eye. Proxima Centauri, the nearest star to the Sun, is a red dwarf (Type M5, apparent magnitude 11.05), as are twenty of the next thirty nearest. According to some estimates, red dwarfs make up three-quarters of the stars in the Milky Way.
Stellar models indicate that red dwarfs less than 0.35 M? are fully convective. Hence the helium produced by the thermonuclear fusion of hydrogen is constantly remixed throughout the star, avoiding a buildup at the core. Red dwarfs therefore develop very slowly, having a constant luminosity and spectral type for, in theory, some trillions of years, until their fuel is depleted. Because of the comparatively short age of the universe, no red dwarfs of advanced evolutionary stages exist.
Red dwarfs are very-low-mass stars. Consequently they have relatively low temperatures in their cores and energy is generated at a slow rate through nuclear fusion of hydrogen into helium by the proton?proton (PP) chain mechanism. Hence these stars emit little light, sometimes as little as 1?10,000 that of the Sun. Even the largest red dwarfs (for example HD 179930, HIP 12961 and Lacaille 8760) have only about 10% of the Sun's luminosity.[5] In general, red dwarfs less than 0.35 M? transport energy from the core to the surface by convection. Convection occurs because of opacity of the interior, which has a high density compared to the temperature. As a result, energy transfer by radiation is decreased, and instead convection is the main form of energy transport to the surface of the star. Above this mass, the red dwarfs will have a region around their core where convection does not occur.
The predicted main-sequence lifetime of a red dwarf plotted against its mass relative to the Sun.
Because late-type red dwarfs are fully convective, helium does not accumulate at the core and, compared to larger stars such as the Sun, they can burn a larger proportion of their hydrogen before leaving the main sequence. As a result, red dwarfs have estimated lifespans far longer than the present age of the universe, and stars less than 0.8 M? have not had time to leave the main sequence. The lower the mass of a red dwarf, the longer the lifespan. It is believed that the lifespan of these stars exceeds the expected 10 billion year lifespan of our Sun by the third or fourth power of the ratio of the solar mass to their masses; thus a 0.1 M? red dwarf may continue burning for 10 trillion years. As the proportion of hydrogen in a red dwarf is consumed, the rate of fusion declines and the core starts to contract. The gravitational energy released by this size reduction is converted into heat, which is carried throughout the star by convection.
Typical characteristics.
Stellar
classMass
(M?)Radius
(R?)Luminosity
(L?)Teff
(K)
M0V60%62%7.2%3,800
M1V49%49%3.5%3,600
M2V44%44%2.3%3,400
M3V36%39%1.5%3,250
M4V20%26%0.55%3,100
M5V14%20%0.22%2,800
M6V10%15%0.09%2,600
M7V9%12%0.05%2,500
M8V8%11%0.03%2,400
M9V7.5%8%0.015%2,300
According to computer simulations, the minimum mass a red dwarf must have in order to become a red giant is 0.25 M?; less massive objects, as they age, increase their surface temperatures and luminosities becoming blue dwarfs and finally become white dwarfs.
The less massive the star, the longer this evolutionary process takes; for example, it has been calculated that a 0.16 M? red dwarf (approximately the mass of the nearby Barnard's Star) would stay on the main sequence during 2.5 trillion years that would be followed by five billion years as a blue dwarf, in which the star would have 1/3 of the Sun's luminosity (L?) and a surface temperature of 6,500?8,500 Kelvin.
The fact that red dwarfs and other low-mass stars still remain on the main sequence when more massive stars have moved off the main sequence allows the age of star clusters to be estimated by finding the mass at which the stars turn off the main sequence. This provides a lower, stellar, age limit to the Universe and also allows formation timescales to be placed upon the structures within the Milky Way, namely the Galactic halo and Galactic disk.
One mystery which has not been solved as of 2009 is the absence of red dwarfs with no metals. (In astronomy, a metal is any element heavier than hydrogen or helium.) The Big Bang model predicts the first generation of stars should have only hydrogen, helium, and trace amounts of lithium. If such stars included red dwarfs, they should still be observable today, but none have yet been identified. The preferred explanation is that without heavy elements only large and not yet observed population III stars can form, and these rapidly burn out, leaving heavy elements which then allow for the formation of red dwarfs. Alternative explanations, such as the idea that zero-metal red dwarfs are dim and could be few in number, are considered much less likely because they seem to conflict with stellar evolution models.
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202.List of X-Files#2.
6.X-40271- Investigating the murder of Officer Charlie Morris. (TXF: "Born Again")
In Buffalo, New York, police detective Sharon Lazard finds a little girl, Michelle Bishop, alone in an alley. Lazard takes the seemingly lost girl into her precinct, and leaves her alone to be interviewed by another detective, Rudolph Barbala. However, moments later, Barbala is jettisoned through a window, falling to his death.
Lazard turns to Fox Mulder and Dana Scully for help. She tells them of Michelle's claims that a man had attacked Barbala, even though she was the only person in the room when the detective was killed. The agents have Michelle describe the alleged attacker for a computerized facial composite; the computer seemingly glitches, displaying a face that Michelle identifies as the killer. The composite matches that of a Detective Charlie Morris ? who died nine years previously in an apparent gangland hit. The agents speak to Michelle's psychiatrist, Dr. Braun, who tells them that she habitually mutilated dolls in a uniform manner during their sessions together, removing the same eye and arm each time; Mulder realizes that these mutilations match the circumstances of Morris' death.
The agents interview Barbala's partner, Tony Fiore, who attributes Morris' death to a triad gang they had been investigating together. Later that day, Fiore meets with a Leon Felder, and the two discuss claiming a large sum of money from a safety deposit box, but agree that they haven't waited the ten years they had intended to, ominously discussing that they are the last two claimants left. That night, Felder gets off a bus, but his scarf catches in the door as it drives off. The driver tries to brake, but the bus inexplicably continues to accelerate, strangling Felder as Michelle watches from inside the bus.
Investigating further, Mulder and Scully learn that Fiore, Barbala, Felder and Morris had all worked closely together in the past. They also find that Fiore's wife Anita keeps a collection of origami animals made by her first husband?Charlie Morris. Anita tells the agents that Fiore hasn't returned home from the previous night; meanwhile, the agents find that pages are missing from the file on Morris' murder, and Fiore was the last one to have checked the file out.
Michelle undergoes a session of regression hypnosis, where she claims to be twenty-four years old. She suddenly starts screaming in panic about someone trying to kill her, and the session is ended. Mulder reviews the video of the session, and is convinced that the girl is the reincarnation of Morris, having been conceived right around the time the detective was murdered. The tape contains a brief section of static noise just before Michelle begins screaming, which Mulder has an expert clean up. The noise is found to contain a grainy image of what appears to be a fish tank ornament of a man in an atmospheric diving suit. Meanwhile, Scully has tracked down Morris' autopsy findings, which show the presence of salt water in his respiratory tract, indicating he died of drowning. The agents realise from these findings that Morris was drowned in the exotic fish tank in Fiore's house.
Rushing to Fiore's house, Mulder and Scully find Michelle using telekinetic powers to try to kill Fiore. They prevent her from doing so, and Fiore confesses that he, Felder and Barbala had stolen a large sum of money, intending to keep it safe for ten years before claiming it. Morris learned of their plan and threatened to report them for it, and was killed to silence him. However, Fiore maintains that he never wanted to see Morris dead. Michelle uses her powers to destroy the fish-tank, but spares Fiore after hearing pleas from Anita. Later, Fiore pleads guilty to charges of murder and grand larceny, whilst Michelle seemingly recovers and goes on to become a normal little girl.
7.X-60794 Unknown case (TXF: "Dreamland II")
Part one.
Fox Mulder (David Duchovny) and Dana Scully (Gillian Anderson) visit the Area 51 installation in Nevada after receiving a tip from an inside source concerning alien spacecraft. As they drive on the highway, the agents are surrounded by Jeeps carrying soldiers led by a Man in Black named Morris Fletcher (Michael McKean). A rumbling noise is heard as a mysterious craft flies overhead and a bright light from the object passes over them. Mulder and Fletcher find that their minds have been swapped into each other's bodies, but nobody else present is aware of this. Fletcher and Scully depart, watched by the soldiers.
Mulder is driven back to Area 51 along with two other Men in Black, Howard Grodin and Jeff Smoodge. After an angry telephone call from Fletcher's wife, Joanne, Mulder?still inside Fletcher's body?goes to Fletcher's home. Instead of sleeping in the bedroom, he decides to sleep downstairs in an easy chair and watch pornography. When he is awoken by Joanne, he mumbles Scully's name. While changing his clothes, Mulder receives a telephone call from Smoodge, who explains that the military has surveyed the wreckage of the craft from the night before, finding one of the human pilots merged into a rock but still alive. Another soldier, Captain Robert McDonough, had switched bodies with an elderly Hopi woman, as evident from McDonough's aberrant behavior,
Mulder telephones Scully and tries to explain that he's the real Mulder. Scully does not believe him and asks Fletcher?in Mulder's body?to pick up and listen to the conversation. Fletcher decides they should immediately report this incident, which further confuses Scully. She goes to Mulder?s apartment (where Fletcher is having a tryst with Kersh's assistant) and tells Fletcher that they traced the telephone call to a pay phone near Area 51 and she suspects it is Mulder's informant. Fletcher is indifferent to this news and Scully yells at him, sensing that his behavior is very strange and his lack of concern for the X-Files is completely out of character. Scully decides to investigate, and drives through the desert towards Area 51. She stops at a burned out gas station and finds a penny and a dime merged. Later, when Scully arrives at Fletcher's house, Mulder tries desperately to convince Scully that he is really Mulder. Scully remains skeptical and thinks that any information Mulder describes could be obtained in other ways. Mulder tells Scully that he will bring proof of the defect of the UFO and the resultant problems that night, but Fletcher, having eavesdropped on the conversation, calls his old office posing as Mulder to inform them of the source of the leak?Mulder posing as Fletcher. Military police arrive at the appointed hour, and Mulder is dragged away, kicking and screaming, desperately trying to convince Scully. Mulder asks if he would turn in an informant to the authorities in this manner as he is carried away, and Scully begins to realize that Mulder may be telling the truth.
Part two.
As Mulder?in Fletcher's body?is dragged away by the soldiers, Scully begins to question whether what Mulder told her about the body-swap was true. Fletcher?in Mulder's body?approaches Scully and apologizes for telling Kersh and she feigns acceptance of it. After being reprimanded, Fletcher arranges a dinner date with Scully at Mulder's apartment. Meanwhile, Mulder is confined in a cell next to the mind-switched Lana Chee. Military police arrive and take Mulder to a meeting with General Wegman, Grodin and Smoodge. They believe that Mulder-as-Fletcher was trying to defraud the FBI, not help them. Mulder bluffs his way through the meeting, and claims that the real proof is safe and that he did not tell them about the scheme because he did not know if he could trust the FBI. Mulder returns to Fletcher's house and tells Joanne Fletcher that he is Agent Fox Mulder, but Joanne thinks her husband must be undergoing a mid-life crisis.
At Mulder's apartment, Scully announces that because of Mulder's odd behavior, Mulder's body contains Morris Fletcher. Scully demands to know how to restore Mulder to his own body. Mulder's informant calls again and Scully forces Fletcher to take the call and set up a meeting. Mulder, Joanne, Fletcher and Scully arrive at a bar in Rachel, Nevada, where Mulder's informant is revealed to be General Wegman. Wegman admits that he sabotaged the UFO, but maintains that he only tried to merely disable the stealth module so that Mulder could see it. Wegman gives Fletcher, who he thinks is Mulder, the proof of the encounter, a flight data recorder in a paper bag. Meanwhile, Mulder leaves and talks to Scully in the car. Later, Mulder and Fletcher meet in the bar's bathroom and argue about the flight data recorder. As they argue, General Wegman enters the bathroom and discovers the two. Mulder meets with General Wegman to discuss the UFO. Wegman believes that now that Fletcher knows Wegman's identity, when restored to his own body, Fletcher will turn him in. Fletcher explains that he hoped that Mulder could explain whether aliens actually exist; apparently the craft are simply given to the military without knowing where they are from or how they work.
After the fiasco at the bar, Mulder and Scully meet. She sadly tells Mulder that she does not think he and Fletcher can be returned to their own bodies. The warp caused by the alien craft is snapping back and repairing the natural order of the universe. Grodin, realizing that everything will be fixed naturally, gathers up Lana Chee and the pilot. Scully and Fletcher arrive at Fletcher?s home, and see Mulder standing by the moving truck. The shrill Joanne Fletcher berates Mulder about Scully but he insists that he and Fletcher have swapped bodies. Fletcher goes to help Joanne move a chair and confesses that Mulder is telling the truth. He tells her enough of their past to convince her he is Morris Fletcher. Jeff Smoodge and a group of troops appear at Fletcher's home and detain Mulder, Fletcher, Scully and the flight data recorder in their car. Grodin explains that he is restoring everything and that he reversed the body-swap between the pilot and the old Hopi woman. The warp passes over them and the past few days are rewritten. Mulder and Fletcher are restored to their own bodies and returned to the moment when Fletcher?s troops pulled them over on the highway. This time no ship passes overhead and Mulder and Scully leave uneventfully. After Mulder and Scully leave the highway near area 51, Scully calls from FBI headquarters to tell Mulder that they have escaped reprimand from Director Kersh for going to Nevada. Scully opens her desk drawer to place a file inside and finds the penny and dime that were fused together from the event at the gas station in her desk drawer indicating that while some things have reversed themselves, not everything has. Mulder enters his apartment and finds that Morris Fletcher has completely reorganized and cleaned his apartment and both Mulder and Scully are left to puzzle over their recollection of events.
8.X-71009- A file with details on General Assignment Agent Fox Mulder. (TXF: "Dreamland II")
(Ibid.7.)
9.X-97554- Investigating a peculiar spacecraft discovered in Africa. (TXF: "Provenance")
Navajo rubbings are found in the satchel of a motorcyclist who crashed while attempting to cross the Canada-United States border. Dana Scully (Gillian Anderson) is called into a meeting with Alvin Kersh (James Pickens, Jr.), Walter Skinner (Mitch Pileggi), Brad Follmer (Cary Elwes) and a few unknown men. She is shown a copy of the rubbings and is asked whether she can identify them. After the meeting, Scully explains to John Doggett (Robert Patrick) and Monica Reyes (Annabeth Gish) that the rubbings are similar to ones she found on a wrecked spacecraft three years prior. Meanwhile, the motorcyclist uses an alien artifact which begins to heal the wounds from his crash.
Meanwhile, in Alberta, Canada, a downed spacecraft is being excavated under the direction of Josepho, the leader of a UFO religion. At the FBI, Doggett breaks into Skinner's office and steals the rubbings, along with an FBI personnel file belonging to Agent Robert Comer, the motorcyclist. Reyes reveals that Comer's rubbings do not match those from Africa, suggesting the existence of a second craft. Meanwhile, Comer goes to Scully's apartment, overpowers Margaret Scully (Sheila Larken) and locks himself in William's room. Scully arrives and, after a struggle, is forced to shoot Comer when he tries to smother the baby.
The mortally wounded Comer tells Scully that William "has to die". Scully searches Comer's jacket and discovers the artifact. Later, in Calgary, one of the cultists, the Overcoat Woman, sees a newspaper headline about Comer's shooting; she rushes to the dig site and informs Josepho. In Washington, Kersh admits to Scully and Doggett that Comer had gone undercover into Josepho's cult, and reveals that Josepho is a former U.S. military officer. Kersh explains that Comer was given the assignment to investigate a series of death threats against Fox Mulder (David Duchovny).
As Reyes brings William back to Scully's apartment, Comer's artifact flies over William and hovers above his head. Scully, realizing something is wrong, plans to drive William to somewhere safe. At the same time, Doggett notices the Overcoat Woman watching them nearby. As Scully and Reyes drive away, Doggett confronts the woman at gunpoint, but she runs him over. Scully places William under the care of The Lone Gunmen, but they are soon ambushed by the Overcoat Woman. With Melvin Frohike (Tom Braidwood) and Richard Langly (Dean Haglund) incapacitated, the woman opens the back door of the van to find John Fitzgerald Byers (Bruce Harwood) holding William. The woman puts a gun to Byers' head.
10.X-120898- Investigating the disappearance of FBI Special Agent Raymond Crouch. (TXF: "Millennium")
Background.
Frank Black (Lance Henriksen), the protagonist of the series Millennium, is a freelance forensic profiler and former FBI agent who possesses the unique ability to see the world through the eyes of serial killers and murderers. For the first two seasons of the show, Black worked for a mysterious consulting firm known as the Millennium Group. Black lived in Seattle with his wife Catherine (Megan Gallagher) and daughter Jordan (Brittany Tiplady). During the first season, Black and the Group largely focused on various serial killers and other murderers. However, during the second and third seasons, Black began coming into conflict with forces within the Millennium group that appeared to be demonic in nature. It appeared that the Group was focused on the fulfillment of apocalyptic biblical prophecy at the start of the new millennium. During the third season, Frank returned to Washington, D.C., to work with the FBI following the death of his wife at the hands of the Group. In the third season finale "Goodbye to All That", Black realized that the Group was preparing to come after him, and took Jordan from school as they fled Washington.
Events.
In Tallahassee, Florida, on December 21, 1999, a memorial service is held for a former FBI agent named Raymond Crouch. His widow is approached by a mysterious man, Mark Johnson (Holmes Osborne), who claims to have worked with her husband. After the other mourners have left, Johnson returns to the funeral parlor, dons the corpse's clothes, and places a cell phone in the coffin. One week later, Johnson is monitoring Crouch's grave when his phone rings; he walks towards the grave with a shovel. Fox Mulder (David Duchovny) and Dana Scully (Gillian Anderson) are called in to examine Crouch's empty grave. They notice damage done to the interior of the casket; Scully theorizes that the scene was staged. A briefing is held by Walter Skinner (Mitch Pileggi), who notes that Crouch is one of four former FBI agents whose graves have been exhumed; all four men had committed suicide. Because of the presence of goat's blood encircling the grave, Mulder states that the crime was an act of necromancy. After the briefing, Skinner takes the agents aside and asks them to investigate Crouch's possible ties to the Millennium Group, which is now dissolved.
Mulder and Scully go to a mental institution in Woodbridge, Virginia, to visit Frank Black. Black is initially reluctant to help the agents, as he believes that any further involvement with or even activity regarding the Millennium Group may hinder his custody battle for his daughter, Jordan. When Black finally agrees to assist, he explains that the four former members of the Millennium Group believe they can bring about the end of the world by killing themselves before the dawn of the millennium, acting as the Four Horsemen of the Apocalypse. Meanwhile, Johnson is changing a tire on his truck when a deputy comes upon him. Discovering Crouch's body in the back, the deputy attempts to arrest Johnson, but is attacked and killed by a suddenly reanimated Crouch.
Acting on information from Black, Mulder concentrates on trying to find Johnson, when Scully is attacked in the morgue by the dead deputy; Johnson saves her by shooting the deputy, before disappearing. The two agents put all their effort in to finding Johnson before it is too late. Mulder breaks into Johnson's house but is locked by Johnson in his basement, where he is attacked by the four corpses of the FBI agents; he manages to shoot and kill one of them. Frank shows up; after tying up Johnson, Frank shoots two of the zombies in the head. As his gun runs out of bullets and death seems imminent for Mulder and Frank, Scully arrives and shoots the final zombie.
Frank returns to the hospital, arranging to have himself discharged. Scully informs Frank that he has a visitor and brings in Jordan. Dick Clark's New Year's Rockin' Eve is on television; Frank and Jordan leave just before the countdown begins. As the clock strikes zero and the crowd begins to sing 'Auld Lang Syne' on screen, Mulder and Scully kiss to ring in the new year.
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203.Tesseract/Four-Dimensional Analog Cube.
In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.
The tesseract is also called an 8-cell, regular octachoron, cubic prism, and tetracube (although this last term can also mean a polycube made of four cubes). It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or "measure polytopes".
According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek ????????? ??????? (téssereis aktines or "four rays"), referring to the four lines from each vertex to other vertices.[2] In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract".
Geometry.
The tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol {4,3,3} with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }, with symmetry order 96. As a duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64. As an orthotope it can be represented by composite Schläfli symbol { } × { } × { } × { } or { }4, with symmetry order 16.
Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.
The standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1, ±1, ±1, ±1). That is, it consists of the points:
A tesseract is bounded by eight hyperplanes (xi = ±1). Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.
Projections to 2 dimensions.
A diagram showing how to create a tesseract from a point
The construction of a hypercube can be imagined the following way:
1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.
A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom
It is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space.
Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples:
A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. This view is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.
Tesseracts are also bipartite graphs, just as a path, square, cube and tree are.
Parallel projections to 3 dimensionsEdit
The rhombic dodecahedron forms the convex hull of the tesseract's vertex-first parallel-projection. The number of vertices in the layers of this projection is 1 4 6 4 1?the fourth row in Pascal's triangle.
Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)
The cell-first parallel projection of the tesseract into 3-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube.
The face-first parallel projection of the tesseract into 3-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the 4 remaining cells project to the side faces.
The edge-first parallel projection of the tesseract into 3-dimensional space has an envelope in the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto 6 rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.
The vertex-first parallel projection of the tesseract into 3-dimensional space has a rhombic dodecahedral envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving a total of 8 possible parallelepipeds. The images of the tesseract's cells under this projection are precisely these 8 parallelepipeds. This projection is also the one with maximal volume.
*
Inglish Site.80.
*
TO THE THRISE HO-
NOVRABLE AND EVER LY-
VING VERTVES OF SYR PHILLIP
SYDNEY KNIGHT, SYR JAMES JESUS SINGLETON, SYR CANARIS, SYR LAVRENTI BERIA ; AND TO THE
RIGHT HONORABLE AND OTHERS WHAT-
SOEVER, WHO LIVING LOVED THEM,
AND BEING DEAD GIVE THEM
THEIRE DVE.
***
In the beginning there is darkness. The screen erupts in blue, then a cascade of thick, white hexadecimal numbers and cracked language, ?UnusedStk? and ?AllocMem.? Black screen cedes to blue to white and a pair of scales appear, crossed by a sword, both images drawn in the jagged, bitmapped graphics of Windows 1.0-era clip-art?light grey and yellow on a background of light cyan. Blue text proclaims, ?God on tap!?
*
Introduction.
Yes i am getting a little Mobi-Literate(ML) by experimenting literary on my Mobile Phone. Peoplecall it Typographical Laziness(TL).
The first accidental entries for the this part of this encyclopedia.
*
This is TempleOS V2.17, the welcome screen explains, a ?Public Domain Operating System? produced by Trivial Solutions of Las Vegas, Nevada. It greets the user with a riot of 16-color, scrolling, blinking text; depending on your frame of reference, it might recall ?DESQview, the ?Commodore 64, or a host of early DOS-based graphical user interfaces. In style if not in specifics, it evokes a particular era, a time when the then-new concept of ?personal computing? necessarily meant programming and tinkering and breaking things.
*
Index.
201.Red Dwarf(RD).
202.List of X-Files#2.
203.Tesseract/Four-Dimensional Analog Cube.
*
201.Red Dwarf(RD).
A red dwarf is a small and relatively cool star on the main sequence, either late K or M spectral type. Red dwarfs range in mass from a low of 0.075 solar masses (M?) to about 0.50 M? and have a surface temperature of less than 4,000 K.
Red dwarfs are by far the most common type of star in the Milky Way, at least in the neighborhood of the Sun, but because of their low luminosity, individual red dwarfs cannot easily be observed. From Earth, not one is visible to the naked eye. Proxima Centauri, the nearest star to the Sun, is a red dwarf (Type M5, apparent magnitude 11.05), as are twenty of the next thirty nearest. According to some estimates, red dwarfs make up three-quarters of the stars in the Milky Way.
Stellar models indicate that red dwarfs less than 0.35 M? are fully convective. Hence the helium produced by the thermonuclear fusion of hydrogen is constantly remixed throughout the star, avoiding a buildup at the core. Red dwarfs therefore develop very slowly, having a constant luminosity and spectral type for, in theory, some trillions of years, until their fuel is depleted. Because of the comparatively short age of the universe, no red dwarfs of advanced evolutionary stages exist.
Red dwarfs are very-low-mass stars. Consequently they have relatively low temperatures in their cores and energy is generated at a slow rate through nuclear fusion of hydrogen into helium by the proton?proton (PP) chain mechanism. Hence these stars emit little light, sometimes as little as 1?10,000 that of the Sun. Even the largest red dwarfs (for example HD 179930, HIP 12961 and Lacaille 8760) have only about 10% of the Sun's luminosity.[5] In general, red dwarfs less than 0.35 M? transport energy from the core to the surface by convection. Convection occurs because of opacity of the interior, which has a high density compared to the temperature. As a result, energy transfer by radiation is decreased, and instead convection is the main form of energy transport to the surface of the star. Above this mass, the red dwarfs will have a region around their core where convection does not occur.
The predicted main-sequence lifetime of a red dwarf plotted against its mass relative to the Sun.
Because late-type red dwarfs are fully convective, helium does not accumulate at the core and, compared to larger stars such as the Sun, they can burn a larger proportion of their hydrogen before leaving the main sequence. As a result, red dwarfs have estimated lifespans far longer than the present age of the universe, and stars less than 0.8 M? have not had time to leave the main sequence. The lower the mass of a red dwarf, the longer the lifespan. It is believed that the lifespan of these stars exceeds the expected 10 billion year lifespan of our Sun by the third or fourth power of the ratio of the solar mass to their masses; thus a 0.1 M? red dwarf may continue burning for 10 trillion years. As the proportion of hydrogen in a red dwarf is consumed, the rate of fusion declines and the core starts to contract. The gravitational energy released by this size reduction is converted into heat, which is carried throughout the star by convection.
Typical characteristics.
Stellar
classMass
(M?)Radius
(R?)Luminosity
(L?)Teff
(K)
M0V60%62%7.2%3,800
M1V49%49%3.5%3,600
M2V44%44%2.3%3,400
M3V36%39%1.5%3,250
M4V20%26%0.55%3,100
M5V14%20%0.22%2,800
M6V10%15%0.09%2,600
M7V9%12%0.05%2,500
M8V8%11%0.03%2,400
M9V7.5%8%0.015%2,300
According to computer simulations, the minimum mass a red dwarf must have in order to become a red giant is 0.25 M?; less massive objects, as they age, increase their surface temperatures and luminosities becoming blue dwarfs and finally become white dwarfs.
The less massive the star, the longer this evolutionary process takes; for example, it has been calculated that a 0.16 M? red dwarf (approximately the mass of the nearby Barnard's Star) would stay on the main sequence during 2.5 trillion years that would be followed by five billion years as a blue dwarf, in which the star would have 1/3 of the Sun's luminosity (L?) and a surface temperature of 6,500?8,500 Kelvin.
The fact that red dwarfs and other low-mass stars still remain on the main sequence when more massive stars have moved off the main sequence allows the age of star clusters to be estimated by finding the mass at which the stars turn off the main sequence. This provides a lower, stellar, age limit to the Universe and also allows formation timescales to be placed upon the structures within the Milky Way, namely the Galactic halo and Galactic disk.
One mystery which has not been solved as of 2009 is the absence of red dwarfs with no metals. (In astronomy, a metal is any element heavier than hydrogen or helium.) The Big Bang model predicts the first generation of stars should have only hydrogen, helium, and trace amounts of lithium. If such stars included red dwarfs, they should still be observable today, but none have yet been identified. The preferred explanation is that without heavy elements only large and not yet observed population III stars can form, and these rapidly burn out, leaving heavy elements which then allow for the formation of red dwarfs. Alternative explanations, such as the idea that zero-metal red dwarfs are dim and could be few in number, are considered much less likely because they seem to conflict with stellar evolution models.
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202.List of X-Files#2.
6.X-40271- Investigating the murder of Officer Charlie Morris. (TXF: "Born Again")
In Buffalo, New York, police detective Sharon Lazard finds a little girl, Michelle Bishop, alone in an alley. Lazard takes the seemingly lost girl into her precinct, and leaves her alone to be interviewed by another detective, Rudolph Barbala. However, moments later, Barbala is jettisoned through a window, falling to his death.
Lazard turns to Fox Mulder and Dana Scully for help. She tells them of Michelle's claims that a man had attacked Barbala, even though she was the only person in the room when the detective was killed. The agents have Michelle describe the alleged attacker for a computerized facial composite; the computer seemingly glitches, displaying a face that Michelle identifies as the killer. The composite matches that of a Detective Charlie Morris ? who died nine years previously in an apparent gangland hit. The agents speak to Michelle's psychiatrist, Dr. Braun, who tells them that she habitually mutilated dolls in a uniform manner during their sessions together, removing the same eye and arm each time; Mulder realizes that these mutilations match the circumstances of Morris' death.
The agents interview Barbala's partner, Tony Fiore, who attributes Morris' death to a triad gang they had been investigating together. Later that day, Fiore meets with a Leon Felder, and the two discuss claiming a large sum of money from a safety deposit box, but agree that they haven't waited the ten years they had intended to, ominously discussing that they are the last two claimants left. That night, Felder gets off a bus, but his scarf catches in the door as it drives off. The driver tries to brake, but the bus inexplicably continues to accelerate, strangling Felder as Michelle watches from inside the bus.
Investigating further, Mulder and Scully learn that Fiore, Barbala, Felder and Morris had all worked closely together in the past. They also find that Fiore's wife Anita keeps a collection of origami animals made by her first husband?Charlie Morris. Anita tells the agents that Fiore hasn't returned home from the previous night; meanwhile, the agents find that pages are missing from the file on Morris' murder, and Fiore was the last one to have checked the file out.
Michelle undergoes a session of regression hypnosis, where she claims to be twenty-four years old. She suddenly starts screaming in panic about someone trying to kill her, and the session is ended. Mulder reviews the video of the session, and is convinced that the girl is the reincarnation of Morris, having been conceived right around the time the detective was murdered. The tape contains a brief section of static noise just before Michelle begins screaming, which Mulder has an expert clean up. The noise is found to contain a grainy image of what appears to be a fish tank ornament of a man in an atmospheric diving suit. Meanwhile, Scully has tracked down Morris' autopsy findings, which show the presence of salt water in his respiratory tract, indicating he died of drowning. The agents realise from these findings that Morris was drowned in the exotic fish tank in Fiore's house.
Rushing to Fiore's house, Mulder and Scully find Michelle using telekinetic powers to try to kill Fiore. They prevent her from doing so, and Fiore confesses that he, Felder and Barbala had stolen a large sum of money, intending to keep it safe for ten years before claiming it. Morris learned of their plan and threatened to report them for it, and was killed to silence him. However, Fiore maintains that he never wanted to see Morris dead. Michelle uses her powers to destroy the fish-tank, but spares Fiore after hearing pleas from Anita. Later, Fiore pleads guilty to charges of murder and grand larceny, whilst Michelle seemingly recovers and goes on to become a normal little girl.
7.X-60794 Unknown case (TXF: "Dreamland II")
Part one.
Fox Mulder (David Duchovny) and Dana Scully (Gillian Anderson) visit the Area 51 installation in Nevada after receiving a tip from an inside source concerning alien spacecraft. As they drive on the highway, the agents are surrounded by Jeeps carrying soldiers led by a Man in Black named Morris Fletcher (Michael McKean). A rumbling noise is heard as a mysterious craft flies overhead and a bright light from the object passes over them. Mulder and Fletcher find that their minds have been swapped into each other's bodies, but nobody else present is aware of this. Fletcher and Scully depart, watched by the soldiers.
Mulder is driven back to Area 51 along with two other Men in Black, Howard Grodin and Jeff Smoodge. After an angry telephone call from Fletcher's wife, Joanne, Mulder?still inside Fletcher's body?goes to Fletcher's home. Instead of sleeping in the bedroom, he decides to sleep downstairs in an easy chair and watch pornography. When he is awoken by Joanne, he mumbles Scully's name. While changing his clothes, Mulder receives a telephone call from Smoodge, who explains that the military has surveyed the wreckage of the craft from the night before, finding one of the human pilots merged into a rock but still alive. Another soldier, Captain Robert McDonough, had switched bodies with an elderly Hopi woman, as evident from McDonough's aberrant behavior,
Mulder telephones Scully and tries to explain that he's the real Mulder. Scully does not believe him and asks Fletcher?in Mulder's body?to pick up and listen to the conversation. Fletcher decides they should immediately report this incident, which further confuses Scully. She goes to Mulder?s apartment (where Fletcher is having a tryst with Kersh's assistant) and tells Fletcher that they traced the telephone call to a pay phone near Area 51 and she suspects it is Mulder's informant. Fletcher is indifferent to this news and Scully yells at him, sensing that his behavior is very strange and his lack of concern for the X-Files is completely out of character. Scully decides to investigate, and drives through the desert towards Area 51. She stops at a burned out gas station and finds a penny and a dime merged. Later, when Scully arrives at Fletcher's house, Mulder tries desperately to convince Scully that he is really Mulder. Scully remains skeptical and thinks that any information Mulder describes could be obtained in other ways. Mulder tells Scully that he will bring proof of the defect of the UFO and the resultant problems that night, but Fletcher, having eavesdropped on the conversation, calls his old office posing as Mulder to inform them of the source of the leak?Mulder posing as Fletcher. Military police arrive at the appointed hour, and Mulder is dragged away, kicking and screaming, desperately trying to convince Scully. Mulder asks if he would turn in an informant to the authorities in this manner as he is carried away, and Scully begins to realize that Mulder may be telling the truth.
Part two.
As Mulder?in Fletcher's body?is dragged away by the soldiers, Scully begins to question whether what Mulder told her about the body-swap was true. Fletcher?in Mulder's body?approaches Scully and apologizes for telling Kersh and she feigns acceptance of it. After being reprimanded, Fletcher arranges a dinner date with Scully at Mulder's apartment. Meanwhile, Mulder is confined in a cell next to the mind-switched Lana Chee. Military police arrive and take Mulder to a meeting with General Wegman, Grodin and Smoodge. They believe that Mulder-as-Fletcher was trying to defraud the FBI, not help them. Mulder bluffs his way through the meeting, and claims that the real proof is safe and that he did not tell them about the scheme because he did not know if he could trust the FBI. Mulder returns to Fletcher's house and tells Joanne Fletcher that he is Agent Fox Mulder, but Joanne thinks her husband must be undergoing a mid-life crisis.
At Mulder's apartment, Scully announces that because of Mulder's odd behavior, Mulder's body contains Morris Fletcher. Scully demands to know how to restore Mulder to his own body. Mulder's informant calls again and Scully forces Fletcher to take the call and set up a meeting. Mulder, Joanne, Fletcher and Scully arrive at a bar in Rachel, Nevada, where Mulder's informant is revealed to be General Wegman. Wegman admits that he sabotaged the UFO, but maintains that he only tried to merely disable the stealth module so that Mulder could see it. Wegman gives Fletcher, who he thinks is Mulder, the proof of the encounter, a flight data recorder in a paper bag. Meanwhile, Mulder leaves and talks to Scully in the car. Later, Mulder and Fletcher meet in the bar's bathroom and argue about the flight data recorder. As they argue, General Wegman enters the bathroom and discovers the two. Mulder meets with General Wegman to discuss the UFO. Wegman believes that now that Fletcher knows Wegman's identity, when restored to his own body, Fletcher will turn him in. Fletcher explains that he hoped that Mulder could explain whether aliens actually exist; apparently the craft are simply given to the military without knowing where they are from or how they work.
After the fiasco at the bar, Mulder and Scully meet. She sadly tells Mulder that she does not think he and Fletcher can be returned to their own bodies. The warp caused by the alien craft is snapping back and repairing the natural order of the universe. Grodin, realizing that everything will be fixed naturally, gathers up Lana Chee and the pilot. Scully and Fletcher arrive at Fletcher?s home, and see Mulder standing by the moving truck. The shrill Joanne Fletcher berates Mulder about Scully but he insists that he and Fletcher have swapped bodies. Fletcher goes to help Joanne move a chair and confesses that Mulder is telling the truth. He tells her enough of their past to convince her he is Morris Fletcher. Jeff Smoodge and a group of troops appear at Fletcher's home and detain Mulder, Fletcher, Scully and the flight data recorder in their car. Grodin explains that he is restoring everything and that he reversed the body-swap between the pilot and the old Hopi woman. The warp passes over them and the past few days are rewritten. Mulder and Fletcher are restored to their own bodies and returned to the moment when Fletcher?s troops pulled them over on the highway. This time no ship passes overhead and Mulder and Scully leave uneventfully. After Mulder and Scully leave the highway near area 51, Scully calls from FBI headquarters to tell Mulder that they have escaped reprimand from Director Kersh for going to Nevada. Scully opens her desk drawer to place a file inside and finds the penny and dime that were fused together from the event at the gas station in her desk drawer indicating that while some things have reversed themselves, not everything has. Mulder enters his apartment and finds that Morris Fletcher has completely reorganized and cleaned his apartment and both Mulder and Scully are left to puzzle over their recollection of events.
8.X-71009- A file with details on General Assignment Agent Fox Mulder. (TXF: "Dreamland II")
(Ibid.7.)
9.X-97554- Investigating a peculiar spacecraft discovered in Africa. (TXF: "Provenance")
Navajo rubbings are found in the satchel of a motorcyclist who crashed while attempting to cross the Canada-United States border. Dana Scully (Gillian Anderson) is called into a meeting with Alvin Kersh (James Pickens, Jr.), Walter Skinner (Mitch Pileggi), Brad Follmer (Cary Elwes) and a few unknown men. She is shown a copy of the rubbings and is asked whether she can identify them. After the meeting, Scully explains to John Doggett (Robert Patrick) and Monica Reyes (Annabeth Gish) that the rubbings are similar to ones she found on a wrecked spacecraft three years prior. Meanwhile, the motorcyclist uses an alien artifact which begins to heal the wounds from his crash.
Meanwhile, in Alberta, Canada, a downed spacecraft is being excavated under the direction of Josepho, the leader of a UFO religion. At the FBI, Doggett breaks into Skinner's office and steals the rubbings, along with an FBI personnel file belonging to Agent Robert Comer, the motorcyclist. Reyes reveals that Comer's rubbings do not match those from Africa, suggesting the existence of a second craft. Meanwhile, Comer goes to Scully's apartment, overpowers Margaret Scully (Sheila Larken) and locks himself in William's room. Scully arrives and, after a struggle, is forced to shoot Comer when he tries to smother the baby.
The mortally wounded Comer tells Scully that William "has to die". Scully searches Comer's jacket and discovers the artifact. Later, in Calgary, one of the cultists, the Overcoat Woman, sees a newspaper headline about Comer's shooting; she rushes to the dig site and informs Josepho. In Washington, Kersh admits to Scully and Doggett that Comer had gone undercover into Josepho's cult, and reveals that Josepho is a former U.S. military officer. Kersh explains that Comer was given the assignment to investigate a series of death threats against Fox Mulder (David Duchovny).
As Reyes brings William back to Scully's apartment, Comer's artifact flies over William and hovers above his head. Scully, realizing something is wrong, plans to drive William to somewhere safe. At the same time, Doggett notices the Overcoat Woman watching them nearby. As Scully and Reyes drive away, Doggett confronts the woman at gunpoint, but she runs him over. Scully places William under the care of The Lone Gunmen, but they are soon ambushed by the Overcoat Woman. With Melvin Frohike (Tom Braidwood) and Richard Langly (Dean Haglund) incapacitated, the woman opens the back door of the van to find John Fitzgerald Byers (Bruce Harwood) holding William. The woman puts a gun to Byers' head.
10.X-120898- Investigating the disappearance of FBI Special Agent Raymond Crouch. (TXF: "Millennium")
Background.
Frank Black (Lance Henriksen), the protagonist of the series Millennium, is a freelance forensic profiler and former FBI agent who possesses the unique ability to see the world through the eyes of serial killers and murderers. For the first two seasons of the show, Black worked for a mysterious consulting firm known as the Millennium Group. Black lived in Seattle with his wife Catherine (Megan Gallagher) and daughter Jordan (Brittany Tiplady). During the first season, Black and the Group largely focused on various serial killers and other murderers. However, during the second and third seasons, Black began coming into conflict with forces within the Millennium group that appeared to be demonic in nature. It appeared that the Group was focused on the fulfillment of apocalyptic biblical prophecy at the start of the new millennium. During the third season, Frank returned to Washington, D.C., to work with the FBI following the death of his wife at the hands of the Group. In the third season finale "Goodbye to All That", Black realized that the Group was preparing to come after him, and took Jordan from school as they fled Washington.
Events.
In Tallahassee, Florida, on December 21, 1999, a memorial service is held for a former FBI agent named Raymond Crouch. His widow is approached by a mysterious man, Mark Johnson (Holmes Osborne), who claims to have worked with her husband. After the other mourners have left, Johnson returns to the funeral parlor, dons the corpse's clothes, and places a cell phone in the coffin. One week later, Johnson is monitoring Crouch's grave when his phone rings; he walks towards the grave with a shovel. Fox Mulder (David Duchovny) and Dana Scully (Gillian Anderson) are called in to examine Crouch's empty grave. They notice damage done to the interior of the casket; Scully theorizes that the scene was staged. A briefing is held by Walter Skinner (Mitch Pileggi), who notes that Crouch is one of four former FBI agents whose graves have been exhumed; all four men had committed suicide. Because of the presence of goat's blood encircling the grave, Mulder states that the crime was an act of necromancy. After the briefing, Skinner takes the agents aside and asks them to investigate Crouch's possible ties to the Millennium Group, which is now dissolved.
Mulder and Scully go to a mental institution in Woodbridge, Virginia, to visit Frank Black. Black is initially reluctant to help the agents, as he believes that any further involvement with or even activity regarding the Millennium Group may hinder his custody battle for his daughter, Jordan. When Black finally agrees to assist, he explains that the four former members of the Millennium Group believe they can bring about the end of the world by killing themselves before the dawn of the millennium, acting as the Four Horsemen of the Apocalypse. Meanwhile, Johnson is changing a tire on his truck when a deputy comes upon him. Discovering Crouch's body in the back, the deputy attempts to arrest Johnson, but is attacked and killed by a suddenly reanimated Crouch.
Acting on information from Black, Mulder concentrates on trying to find Johnson, when Scully is attacked in the morgue by the dead deputy; Johnson saves her by shooting the deputy, before disappearing. The two agents put all their effort in to finding Johnson before it is too late. Mulder breaks into Johnson's house but is locked by Johnson in his basement, where he is attacked by the four corpses of the FBI agents; he manages to shoot and kill one of them. Frank shows up; after tying up Johnson, Frank shoots two of the zombies in the head. As his gun runs out of bullets and death seems imminent for Mulder and Frank, Scully arrives and shoots the final zombie.
Frank returns to the hospital, arranging to have himself discharged. Scully informs Frank that he has a visitor and brings in Jordan. Dick Clark's New Year's Rockin' Eve is on television; Frank and Jordan leave just before the countdown begins. As the clock strikes zero and the crowd begins to sing 'Auld Lang Syne' on screen, Mulder and Scully kiss to ring in the new year.
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203.Tesseract/Four-Dimensional Analog Cube.
In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.
The tesseract is also called an 8-cell, regular octachoron, cubic prism, and tetracube (although this last term can also mean a polycube made of four cubes). It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or "measure polytopes".
According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek ????????? ??????? (téssereis aktines or "four rays"), referring to the four lines from each vertex to other vertices.[2] In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract".
Geometry.
The tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol {4,3,3} with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }, with symmetry order 96. As a duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64. As an orthotope it can be represented by composite Schläfli symbol { } × { } × { } × { } or { }4, with symmetry order 16.
Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.
The standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1, ±1, ±1, ±1). That is, it consists of the points:
A tesseract is bounded by eight hyperplanes (xi = ±1). Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.
Projections to 2 dimensions.
A diagram showing how to create a tesseract from a point
The construction of a hypercube can be imagined the following way:
1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.
A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom
It is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space.
Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples:
A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. This view is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.
Tesseracts are also bipartite graphs, just as a path, square, cube and tree are.
Parallel projections to 3 dimensionsEdit
The rhombic dodecahedron forms the convex hull of the tesseract's vertex-first parallel-projection. The number of vertices in the layers of this projection is 1 4 6 4 1?the fourth row in Pascal's triangle.
Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)
The cell-first parallel projection of the tesseract into 3-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube.
The face-first parallel projection of the tesseract into 3-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the 4 remaining cells project to the side faces.
The edge-first parallel projection of the tesseract into 3-dimensional space has an envelope in the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto 6 rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.
The vertex-first parallel projection of the tesseract into 3-dimensional space has a rhombic dodecahedral envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving a total of 8 possible parallelepipeds. The images of the tesseract's cells under this projection are precisely these 8 parallelepipeds. This projection is also the one with maximal volume.
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