n-skeleton

This hypercube graph is the 1-skeleton of the tesseract.

In mathematics, particularly in algebraic topology, the $n$ -skeleton of a topological space $X$ presented as a simplicial complex (resp. CW complex) refers to the subspace $X n$ that is the union of the simplices of $X$ (resp. cells of $X$ ) of dimensions $m \leq n .$ In other words, given an inductive definition of a complex, the $n$ -skeleton is obtained by stopping at the $n$ -th step.

These subspaces increase with $n$ . The 0-skeleton is a discrete space, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when $X$ has infinite dimension, in the sense that the $X n$ do not become constant as $n \to \infty.$

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Hi Vsauce. Michael here you can practice speaking backwards so when your words are reversed their intelligible but here's something else that is weird the digits in the speed of light are exactly the same as the latitude of the Great Pyramid of Giza and as the anagram genius has revealed all the world's a stage but if you rearrange the letters in the meaning of life it becomes be engine of a film or more pessimistically the fine game of nil what does all this mean? are these just coincidence? Or are greater powers at work? Why is it so easy for us to find hidden messages like in a mere coincidence give us chills and why is it so fine. When you reverse Neil Armstrong saying 'small step for man' you can hear what sounds like man will space walk Lee Harvey Oswald assassinated President John F Kennedy and this interview he defends the fair play for Cuba committee of which he was a member. now listen to what it sounds like when we reverse him saying 'and the fair play for cuba' is that a coincidence or a subconscious confession hidden in his own words its a coincidence for crying out loud if anybody says and the fair play for Cuba and then reverses it it sounds the same this app by the way is called virtual recorder it's really easy way to quickly reverse your own speech Matthew Hudson in the seven laws of magical thinking points out that if you record yourself saying and then reverse it it sounds a bit like happy birthday to you kind of. If a word can be spelt the same forward and backward it's a palindrome but if a word or phrase sounds the same whether spoken forward or rewound it is a phonetic palindrome for example say yes. Reversed pretty cool. But check out this poem by Karsten Johansson by the way some people can speak in reverse on the fly it is really cool to see them in action watch Guys lean back after this video its linked down in the description and its full of pretty cool coincidence videos. Apophenia is the perception of connections or patterns in information. One type of apophenia is pareidolia the scene or hearing of things that weren't meant to be there for instance hearing your name being called or your phone ringing in the sound of running water or hearing english words in a non-English song or seeing faces that weren't purposely placed there. Our brains are good at this kind of work probably because being hyper attentive to patterns and faces can save your life. If there's ambiguity as to whether that thing hiding in the shadows is a threat or just a shadow it's advantageous to air on the side of threat. Organisms with a healthy sense of Apophenia live longer, long enough to have kids and raise them and naturally become the norm. We connect with faces so well Hudson relates a story of a friend who draws faces on things she doesn't wanna lose like her bags she says the faces make her less likely to forget about them if you like it you should have put a ring on it if you like not losing it you should have drawn a face on it we are so good at teasing out patterns and faces from random noise actual random sequences don't always feel random to us originally Apple's iTunes shuffle feature generated complaints from users they said the similar songs or songs from the same artist appeared in a string which of course is to be expected from randomness but it didn't feel random enough so Apple introduced a smart shuffle that avoided totally random sequences that nonetheless didn't seem random to our pattern loving brains as Steve Jobs explained we're making it less random to make it feel more random our impressive ability to imagine patterns also expresses itself when it comes to connect songs and moving images this dancing spider-man animation will famously sync up with any music you play try it. What kind of black magic is going on here? well as it turns out most of it is in our heads Radiolab reported that Michigan State University explains that the major movements of dancing animations like this one or this one move at typical song tempo's but also contain like most dance various other different rhythms of movement allowing them to seemingly fit many different tempos helps a lot to we fall prey to this when we reject all the times the animation doesn't really sync up focusing instead on the more surprising times when it does. The bizarre pyramid coincidence mentioned earlier is a lot less bizarre when you consider the fact that we got to control where we placed the decimal point and that a number of degrees this precise isn't necessary to locate the pyramid by the fourth decimal we're only talking about a matter of a few meters so it's easy to make the rest fit the speed of light exactly and have still picked a point on the pyramid confirmation bias also comes into play here if you really want two things to sync up they will we often look for evidence that supports what we already believe while marginalizing things against it as Marshall McLuhan said 'I wouldn't have seen it if I hadn't of believed it' these biases also help explain the seemingly mind-blowing coincidence that famous movies and famous albums can line up one the most popular states that if you start playing Pink Floyd's Dark Side of the Moon at the same time as the wizard of oz they will eerily line up. Entire communities have sprouted around the syncing of movies and albums. Some of my favorites are the Yellow Submarine sountrack and the Little Mermaid, Lordes pure heroin and Twilight Saga Breaking Dawn 2 and the end of 2001 a Space Odyssey with Pink Floyd's echoes there are conspiracy that these were somehow secretly planned though in reality they're just accidental music videos the product of selection bias confirmation bias and... a behavior of valid pattern sensitive minds two things don't have to line up exactly or literally for us to see a connection this is why vague predictions are a great way to look psychic these are also actually unsurprising when you consider the fact that the number of narrative paces and rhythms we enjoy and typically use are much smaller than the number possible in the 'Improbability Principal' David J Hand calls this what may be rare on average or when considering all possible scenarios can be less rare for specific scenarios even if they are only marginally different. Getting struck by lightning is a proverbially unlikely event, but Walter Summerford wasn't just struck by lightning once during his life he was struck three times. It never killed him but four years after his death his gravestone was also struck by lightning what are the chances? I mean clearly Summerford was some sort of robot built out of lightning rods or had somehow angered Zeus right probably not. You see while for the average person the chance have been struck by lightning is quite low for an avid outdoor sportsman like Somerford it's not as low... also comes into play here with lightning striking Earth forty to fifty times a second billions of people for it to strike and thousand of years of recorded history it's actually not surprising at all that at least once a story like Summerfords would have happened. Given the truly large number of people who visit Disneyworld every day and the fact that they take photos and lots of them it's actually not surprising at all that at least one so far a story like Alex and Donna Voutsinas' has happened while sorting through old photos before their wedding Alexa and Donna found a photo of Donna at disney world fourteen years before the couple met but then Alex noticed something. He too had visited disneyworld as a child and there in the background was his father pushing him in a stroller sometimes coincidences can be tragic In 1864 Abraham Lincoln's son Robert Lincoln was saved from serious injury or possibly even death when a stranger grabbed him by the shirt collar moments before he plunged on the train tracks below that stranger turned out to be Edwin Booth one of the most famous Shakespearean actors of the time so famous in fact Robert recognized him and had a letter sent thanking him for saving his life. Less than a year later Edwin Booth's brother John Wilkes Booth undid the favor by assassinating Abraham Lincoln. Strange as they seem at first math says that given enough time and psychology says that given enough interest in finding them coincidences and connections will be found even unlikely ones. The coincidences between Abraham Lincoln and John F Kennedy are famous both were elected to the presidency in the year ending with sixty. Lincoln was shot at Fords Theatre Kennedy was shot in a 1961 Lincoln Continental four-door convertible made by Ford both presidents last names have seven letters and both assassins had 15 letters in their names the list goes on as it should if you look long enough you can find coincidences between any two people or things or events they may seem strange at first but tend to wind up being in the end pretty expected. For just one example name length isn't that wildly variable seven-letter names are pretty common. Lincoln Kennedy. Michael. Stevens In the famous spooky presidential coincidences contest held by the Skeptical Inquirer in 1992 one contestant alone found similar lists of crazy coincidences between 21 pairs a former presidents given the vast amount of details in any one of our lives its pretty easy. This court can be exploited to almost comedic Heights when it comes to over-analyzing. of course hidden messages and signs are often intentionally included in media for fun or to reward attentive viewers but unintentional extraordinary things happen all the time its not really that extraordinary there's a famous calculation that is known as Littlewoods law given the number of hours we are awake every day and assuming an event only takes about a second to occur If you calculate the odds of something happening to you are only one in a million well you should expect that thing to happen to you about once every 35 days David J Hand took this even further with seven billion people on Earth the chance that an event with a one in a million probability of happening to each of us won't happen today is one In ten to the three thousand and fourty. As Persi Diaconis put it the truly unusual day would be a day where nothing unusual happens and as always thanks for watching you may have noticed a lot of YouTube channels making videos about learning this week well that is not a coincidence it is school of YouTube week many people are going back to school or college right now but across the world millions a children won't be either because they work to support their families or live without a home. Or in areas where there is conflict they may experience overcrowding at school or a lack of teaching and school supplies. But luckily we can help donations to comic relief's school of YouTube campaign can help disadvantaged young people all around the world get an education it doesn't take much to change a life you can learn more in the description below or donate right now and as always, thanks for helping and thanks for learning

In geometry

In geometry, a k-skeleton of n-polytope P (functionally represented as skel_k(P)) consists of all i-polytope elements of dimension up to k.^[1]

For example:

skel₀(cube) = 8 vertices

skel₁(cube) = 8 vertices, 12 edges

skel₂(cube) = 8 vertices, 12 edges, 6 square faces

For simplicial sets

The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set $K_{*}$ can be described by a collection of sets $K_{i},\ i\geq 0$ , together with face and degeneracy maps between them satisfying a number of equations. The idea of the n-skeleton $sk_{n}(K_{*})$ is to first discard the sets $K_{i}$ with $i>n$ and then to complete the collection of the $K_{i}$ with $i\leq n$ to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees $i>n$ .

More precisely, the restriction functor

i_{*}:\Delta ^{op}Sets\rightarrow \Delta _{\leq n}^{op}Sets

has a left adjoint, denoted $i^{*}$ .^[2] (The notations $i^{*},i_{*}$ are comparable with the one of image functors for sheaves.) The n-skeleton of some simplicial set $K_{*}$ is defined as

sk_{n}(K):=i^{*}i_{*}K.

Coskeleton

Moreover, $i_{*}$ has a right adjoint $i^{!}$ . The n-coskeleton is defined as

cosk_{n}(K):=i^{!}i_{*}K.

For example, the 0-skeleton of K is the constant simplicial set defined by $K_{0}$ . The 0-coskeleton is given by the Cech nerve

\dots \rightarrow K_{0}\times K_{0}\rightarrow K_{0}.

(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)

The above constructions work for more general categories (instead of sets) as well, provided that the category has fiber products. The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.^[3]

References

^ Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)
^ Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1, section IV.3.2
^ Artin, Michael; Mazur, Barry (1969), Etale homotopy, Lecture Notes in Mathematics, No. 100, Berlin, New York: Springer-Verlag

External links

Weisstein, Eric W. "Skeleton". MathWorld.

This page was last edited on 30 May 2022, at 15:18

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