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The Banach-Tarski Paradox

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Q: "What's an anagram of Banach-Tarski?"
A: "Banach-Tarski Banach-Tarski."
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Instagram: instagram.com/electricpants
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Field Day: frvid.com/show-UCRPktNf5vnBR1J4e7t1RUVg
Deep dream animation by instagram.com/NaderMakki/
If you like it, you'll love this video also by Nader: frvid.com/video/vid%C3%A9o-fJ9j_z2kXI0.html
Chocolate illusion: mathandmultimedia.com/2014/07/22/explanation-infinite-chocolate-bars/
Chocolate illusion video: frvid.com/video/vid%C3%A9o-dmBsPgPu0Wc.html
related Numberphile videos:
sizes of infinity (includes diagonal argument): frvid.com/video/vid%C3%A9o-elvOZm0d4H0.html
infinity paradoxes: frvid.com/video/vid%C3%A9o-dDl7g_2x74Q.html
Vi Hart on types of infinity: frvid.com/video/vid%C3%A9o-23I5GS4JiDg.html
Countable & uncountable definitions:
mathinsight.org/definition/uncountable
en.wikipedia.org/wiki/Countable_set
en.wikipedia.org/wiki/Uncountable_set
Banach-Tarski on wikipedia: en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Banach-Tarski proofs:
math.uchicago.edu/~may/REU2014/REUPapers/Robinson.pdf
www.math.hmc.edu/~su/papers.dir/banachtarski.pdf
people.math.umass.edu/~weston/oldpapers/banach.pdf
Banach-Tarski explinations online:
www.irregularwebcomic.net/2339.html
www.kuro5hin.org/comments/2003/5/23/134430/275?pid=5#10
skepticsplay.blogspot.co.uk/2010/05/doubling-sphere.html
austinrochford.com/posts/2014-05-14-banach-tarski-paradox.html
www.math.cornell.edu/~mec/Summer2009/Whieldon/Math_Explorers_Club%3A__Lesson_Links/Entries/2009/7/28_Lesson_6%3A__Whats_an_Anagram_of_Banach-Tarski.html
rachellevanger.com/index_files/BT%20Animated%20Presentation%20Web.pdf
quibb.blogspot.co.uk/2013_03_01_archive.html
blog.computationalcomplexity.org/2011/04/what-did-banachs-wife-think-of-banach.html
geopolicraticus.wordpress.com/tag/banach-tarski-paradox/
dgleahy.com/p47.html
www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml
Cayley graph animated gif: twitter.com/GIFsofWikipedia/status/624202342259240960
Hilbert’s hotel on wikipedia: en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
types of infinity: www.xamuel.com/levels-of-infinity/
set theory and quantum physics: link.springer.com/article/10.1007/BF02213427#page-1
LHC gif: cms.web.cern.ch/news/lhc-data-be-made-public-open-access-initiative
Zermelo-Fraenkel axioms of mathematics: mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
Is math invented or discovered?
phys.org/news/2013-09-mathematics-effective-world.html
www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/
more deep dream images: www.reddit.com/r/deepdream/
BOOKS:
The Pea and the Sun: www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279
The Outer Limits of Reason: www.amazon.com/Outer-Limits-Reason-Science-Mathematics/dp/0262019353
Why Beliefs Matter: www.amazon.com/Why-Beliefs-Matter-Reflections-Science/dp/0198704992
Things to Make and do in the Fourth Dimension: www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374275653
Music by frvid.com/u-JakeChudnow and
www.audionetwork.com

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31 juil. 2015

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Commentaires 100
Marina Marchese
Marina Marchese Il y a 32 minutes
Infinity +1 Yes but 1 is real, palpable, countable whereas Infinity is not. To me it's apples and oranges. Or should I say..it's all Greek to me! Sorry! No capish! I must have wandered in the rabbit hole ..again!
Dreadnaught 1270
Dreadnaught 1270 Il y a heure
So does this mean I could technically create a infinite money cheat
Mark Muster
Mark Muster Il y a 3 heures
From my point of view: No And by the way: the "uncountable" infinety with fractions of numbers, is in fackt countable, if u count every step it needs to go to the startingpoint. So: 1: 0.1 2: 0.01 3: 0.001 And so on (infinite). So i guess its proven wrong, isnt it?
Mark Muster
Mark Muster Il y a 9 minutes
@Tom Svoboda jer ofc. It isnt. If u look for the smallest decimal, u just add a zerro behinde the comma everytime, and count this.
Tom Svoboda
Tom Svoboda Il y a heure
this doesn't enumerate all numbers. in fact, it doesn't enumerate almost anything. for example 0.2 is not on your list.
Matt Brooke
Matt Brooke Il y a 6 heures
Rotating a left rotation sphere to the right doesnt leave a residual point to be added to the new position. You will still always have the same spikey sphere only you are seeing it from another angle. If I step left then right my left step becomes a waste of time, not a residual me to my left to be added to the current me. Adding points on the circular plane to fill the poles is addition, last time I checked. If you feel you arent smart enough to understand this sort of thing its because there is a requirement to accept a "rule", these "rules" are applied by the theorist to take away the reality of the situation, and therefor it is no longer knowledge but faith, a want to believe. Truth is you do understand, its nonsense and simply not real.
Samar Grey
Samar Grey Il y a 8 heures
This video is just proving that 1/2=2
Samar Grey
Samar Grey Il y a 8 heures
How many ways can 2020 get worse? This video:
Firas Bsoul
Firas Bsoul Il y a 8 heures
This has, indeed, strengthened my belief in almighty God, the most powerful, creator of the universe, the creator of things we don't know. I am pretty sure that the visible universe we know is like a sand grain in a big desert of what is out there. remember that our knowledge is limited but God's knowledge is limitless and eternal.
Josh Koop
Josh Koop Il y a 9 heures
infinint
arno
arno Il y a 11 heures
so in short, you need infinity, to have infinity?
Dahl Kalvi
Dahl Kalvi Il y a 13 heures
The people in the hotel getting shaked to oblivion by the constant of entering and leaving the hotel ahh shit here we go again
James Barralet
James Barralet Il y a 16 heures
Infinity is a circle, or a sphere, or cycles. Numbers and counting have nothing to do with it. Your dictionary could not be put into a volume. The volume would need to occupy infinite space. This is just taking flawed presumptions and making theories from them. Eg. Like presuming that human greed is not a choice, and that infinite greed is sustainable, and devising economic systems based on that. Brains are very limited - finite. But that is not your identity, who you are. Who you are is ok with infinity, it’s just a matter of stirring it into life.
Nik T
Nik T Il y a 16 heures
Guys! There are no girls in the comments here... move on.
Aditya Bhattacharya
Aditya Bhattacharya Il y a 19 heures
The thing i understood is 1+1=infinity
Michael Bell
Michael Bell Il y a 21 heure
5:14 "Forever isn't enough" that sounds like a Bond title
Nate Dawson
Nate Dawson Il y a 23 heures
damn i just wanted some extra chocolate
Olaoluwa Faleye
Olaoluwa Faleye Il y a jour
Hmmmm
NETB
NETB Il y a jour
I feel as if this paradox seems to be reaches for an explaination of god. Create matter, create something were there was nothing before. Interesting
Fizza Masood
Fizza Masood Il y a jour
Wow this guy makes me feel small yet smarter then my classmates
1281 - Vishwas Kulkarni
Dad: today was your test how many marks did you get? Sister: 99.999999999% Me: 4:07
goose green
goose green Il y a jour
Uncountable infinity in 2015: "the number of all numbers that there are" Uncountable infinity in 2020: "the number of proposed genders that there are"
Neil Kauffman
Neil Kauffman Il y a 17 heures
I'd disagree, since because infinity divided by infinity is infinity, which would mean that the very countable number of people in the world would each have to have an uncountable infinity of gender identifications. Since each person can only have one identification, there cannot be an uncountable, or even countable, number of genders.
amphibia
amphibia Il y a jour
4th graders memorizing 20th table be like: "Is ThE GuY iLlItEraTe?"
YouTubes best station
who elce wasted a perfectly good dollar :(
Sandy Reed
Sandy Reed Il y a jour
This guy is a genius
Sandy Reed
Sandy Reed Il y a jour
So when would you end infinite?
Sandy Reed
Sandy Reed Il y a jour
I just want to point out that all of those dollars are different the one he ripped.. and the two on the table all the numbers are different .
Релёкс84
Релёкс84 Il y a jour
No shit Sherlock
Sandy Reed
Sandy Reed Il y a jour
Just saying.
BuildTest/ExplosionTest
But who would take the time to think about this
Jeremiah Reeves
Jeremiah Reeves Il y a jour
did he just say the buttock tarski paradox
Chiliwangchang 123
Chiliwangchang 123 Il y a jour
If we could duplicate ..... This way I would be very rich A portion of this comment was removed by the us government the portion of the comment that was removed is replaced by periods
CyberSnake100
CyberSnake100 Il y a jour
My brain hurts ...
Gary Hasch
Gary Hasch Il y a jour
Wait.... What if you take infinity from infinity ()()-()()
Rebotxd
Rebotxd Il y a jour
1:15 I love when he does this 😂😂😂
Peter Parker
Peter Parker Il y a jour
9:16
Wade Wilson
Wade Wilson Il y a jour
This is the most roundabout way of telling me I'm weird that I've ever received.
DiceDodo
DiceDodo Il y a jour
10:08 The dictionary just got a little sleepy.. 💤
William S
William S Il y a jour
I feel like the Banach-Tarski Paradox just kinda suggests that the axiom of choice is probably incorrect, since you need to take that axiom as truth to even get the paradox in the first place, and the paradox leads to non-sense when viewed from our perspective in the physical world (except colliding particles as you mentioned, but I feel like thats not solid yet). But if I considered that as absolute evidence that the axiom of choice is bullshit, I'm sure our version of Star Trek's Q would be really annoyed with me.
Your Homie
Your Homie Il y a jour
2:44 are we watching buzzfeed unsolved or vsauce
Jose Moraga
Jose Moraga Il y a jour
couldn't the proof at 5:30 work the other way around (swapping both sets of numbers), hence proving that there are more whole numbers than real numbers between 1 and 2?
Leruo Monang
Leruo Monang Il y a jour
Bruh this guy just said there's more than forever
Brian Arborio
Brian Arborio Il y a jour
CAUTION: INGESTING THC AND WATCHING THIS VIDEO MAY CRUSH YOUR MIND. PROCEED AT YOUR OWN RISK
xQualifieDx
xQualifieDx Il y a jour
whenever i see this thumbnail all i think is anything is possible if we look at it differently. or something like that...
xQualifieDx
xQualifieDx Il y a jour
ive watched this about 5 times i still dont really understand much more
Pedro4816
Pedro4816 Il y a jour
I think humans found a glitch with the world 😂
C&C Phx
C&C Phx Il y a jour
This must be how this guy feels when he checks his account balance...... this guy definitely a multimillionaire from FRvid
Zachary Kruger
Zachary Kruger Il y a jour
could this be used for digital storage?
Nikhil Mahajan
Nikhil Mahajan Il y a jour
This video called me dumb in infinite different ways 😭
John Viggiano
John Viggiano Il y a jour
Sorry. But I can not believe someone who thinks 7+2=9 🙄 @ 5:46
毛歌凡
毛歌凡 Il y a jour
I don't smoke weed but damn *Windows Xp starting noises*
Prachurjya Ghosh
Prachurjya Ghosh Il y a 2 jours
This butchers every single bit of my self esteem.
Objectivity1958
Objectivity1958 Il y a 2 jours
I've watched this video about 28 times now since 2015 and I still keep running out of chocolate. WTF?
Happyk1nGG
Happyk1nGG Il y a 2 jours
Vsauce: *becomes smart* Impossible stuff: _adios_
Mariano Barros
Mariano Barros Il y a 2 jours
9:48 6 year old me thinking what it would be like to scroll right infinetely on excel
um cara aleatório eggs333
nice taste of music! greetings from brazil 🇧🇷
Emil Tildelöv
Emil Tildelöv Il y a 2 jours
“No hole will be left unfilled” 😂
aaaaaaaaaaaa1111a
aaaaaaaaaaaa1111a Il y a 2 jours
Fucking
TXLAdventure
TXLAdventure Il y a 2 jours
Petition to make a point the size of an electron, because that’s as small as things are ever going to get, unless it’s a black hole, but in that case who cares anymore your not going to need to do your math homework in a black hole.
Sacul Selasor
Sacul Selasor Il y a 22 heures
@TXLAdventure lol what sorry defeats the point of what nb
TXLAdventure
TXLAdventure Il y a 23 heures
Sacul Selasor if they have zero dimension than that defeats the point pun intended
Sacul Selasor
Sacul Selasor Il y a 23 heures
@Parker Brewer uh... as i said, they are said to have zero dimensions, so they arent "made" of anything, id reccomend just waching a video
Parker Brewer
Parker Brewer Il y a jour
But what are quarks made of?
Sacul Selasor
Sacul Selasor Il y a jour
aaacctually no... the smallest thing that we get, is acctually a quark, not an electron. a quark is basically the things that connect protons and neutrons together, and people HAVE suggested naming that as the perfect 0 dimensional point
TXLAdventure
TXLAdventure Il y a 2 jours
wouldn’t infinity in the case of infinity+1 just be an an unsolvable equation such as y=5x sure it can be graphed, but not solved so there would still be twice as many. I have no clue what I’m talking about but could it?
Tony Dimaric
Tony Dimaric Il y a 2 jours
It's just like creating infinite amount of something out of that something.
Abdo
Abdo Il y a 2 jours
30M views wow
Ryan Huffman
Ryan Huffman Il y a 2 jours
This isn't even a paradox. This only applies to a sphere because there is no correct up or down, left or right. This just exposes some quarks of infinity. With infinitely many points left is right and up is down. In the real world, up is up, down is down, left is left and right is right. This has no real world implications. This is just pretentious philosophy crap.
Tom Svoboda
Tom Svoboda Il y a jour
@Ryan Huffman no offence but you're talking out of your ass. what you're saying is complete nonsense, mathematically speaking.
Ryan Huffman
Ryan Huffman Il y a jour
@Tom Svoboda not really. A sphere is just the physical manifestation of infinity. There is no correct up, down, left, right, beginning or end. If you added extra directions for the moves, you could end up with more spheres. If you added infinitely more directions for moves, you would end up with infinite spheres.
Tom Svoboda
Tom Svoboda Il y a 2 jours
you sound like you know what you're talking about
Gio
Gio Il y a 2 jours
Chocolate to numbers
Dylan Wyatt Edquiban
Dylan Wyatt Edquiban Il y a 2 jours
almost 30 mil boisssssssssss
Edber Balzatv
Edber Balzatv Il y a 2 jours
I literally searched hairy ball to find this vid ✨💕👏👁👄👁👏💕✨
Martez Araya
Martez Araya Il y a 2 jours
It’s sort of obvious
Martez Araya
Martez Araya Il y a 2 jours
Due
Erza 13
Erza 13 Il y a 2 jours
If the hotel is fully booked doesn't that mean that the infinity of the guests in the rooms = the infinity of the hotel rooms? So how can the owner provide a room for the new guest? All the infinite known rooms are already booked for all the infinite known guests !! If oo = oo so how does oo + 1 = oo? 🤔
TITANIUM HORNET GAMING & Stuff
Almost all of these are just functions
TITANIUM HORNET GAMING & Stuff
@MikeRosoftJH haha, i understood nothing you just said.
MikeRosoftJH
MikeRosoftJH Il y a 2 jours
Of course! Translation and rotation are functions (in particular, a special kind of one-to-one functions which preserve lengths and volumes). Assuming axiom of choice, it is possible to split a solid body (such as a single solid ball) into finitely many subsets, which can be moved around by translation and rotation to form a solid body with a different volume (such as two copies of the original solid ball). (The problem is that the pieces in question are not solid bodies and don't have any well-defined volume; the volume can't be zero, and it can't be non-zero either.) Or, to put it differently: there exist two finite collections of sets A and B, such that: sets in collection A can be moved around by translation and rotation (without overlap) to form a unit solid ball; sets in collection B can be moved around to form the same solid ball; but the sets in the union of A and B can also be moved around without overlap to form a single solid ball of the same volume.
Starz Toons
Starz Toons Il y a 2 jours
im going ape shit over this
Lionel Messi
Lionel Messi Il y a 2 jours
นึกว่าเพลง paradox 555
Tarrrantula
Tarrrantula Il y a 2 jours
What?
Laceluz 666
Laceluz 666 Il y a 3 jours
3:13 Michael has spoken, this is how we pronounce seven now
7duality7
7duality7 Il y a 3 jours
My brain feels like it just checked out of Hilberts Hotel and checked into Bates Hotel 🤯
***Mingot Army***
***Mingot Army*** Il y a 3 jours
We were so close to make infinite water sourse
Tan Thien Nguyen
Tan Thien Nguyen Il y a 3 jours
Werbungen darf nur 1.Min.
Tan Thien Nguyen
Tan Thien Nguyen Il y a 3 jours
Erklären Sie für Apples Apps auch.....? Wie vieles Flugzeugen schon Abgestützt....?
Tan Thien Nguyen
Tan Thien Nguyen Il y a 3 jours
Komischerweise, seit Sie mit VIRUS sind Jogourt Sorte Milch Produkte weniger Verkaufen können. Trotzdem Käse sind Belieben... ..Was ist passiert mit Mars Mensch ?
Kyri Papapetrou
Kyri Papapetrou Il y a 3 jours
i dont like the way you write the number 8
Bradley Mann
Bradley Mann Il y a 3 jours
"INFINITY TELLS US?...
sam sto
sam sto Il y a 3 jours
what happens if you add or subtract infinity to infinity
Afra Afsara
Afra Afsara Il y a 3 jours
I can't be the only one who thinks that he's gonna talk about some weird planets after seeing the thumbnail 😹😹
Legend
Legend Il y a 3 jours
18:59 Imagine doing 2 + 2 = 2
Legend
Legend Il y a 3 jours
1:14 you used to, back when cutting up chocolate was still new
Martin Wood
Martin Wood Il y a 3 jours
I know nothing about this. But that's not going to stop me talking about it. Surely you can always pair up the whole numbers with the randomly generated numbers between 0 and 1 and their augmentations because.....................you will never....................that's......... never....................and once again for the listeners at home..........never...................run out of them.
Night and Day Marine
Night and Day Marine Il y a 3 jours
24 minutes of my life I will never get back...
LacrimarvmValle
LacrimarvmValle Il y a 3 jours
My brain hurts
Iliketacos123
Iliketacos123 Il y a 3 jours
i think im just going to work an extra couple days to double my paycheck instead of attempting the Banach Tarski.
SLAM POLICE
SLAM POLICE Il y a 3 jours
hey Simon
I'm In A Vegetative State Studios
"Hilbert's Paradox of the Grand Hotel" That sounds like it would make a great tim burton psychological thriller
GudanCraftYT
GudanCraftYT Il y a 3 jours
10:07 , now that's a lot of sleeping 😂😏😏
GudanCraftYT
GudanCraftYT Il y a 3 jours
CEO of "Or can you" CEO of "Infinite" number" of BlAblAHhabla"!
Saucey Sauce
Saucey Sauce Il y a 3 jours
1:17 when that music starts you know Michael bouta spit bars about science
Mr. Garden
Mr. Garden Il y a 3 jours
There once a time was accountable Infiniti called Aleph null The next biggest infinity was called omega
MikeRosoftJH
MikeRosoftJH Il y a 3 jours
ω is not the next infinity after aleph-0; ω is the same set as aleph-0. Aleph-0 is a cardinal number (the cardinality of the set of all natural numbers), ω is an ordinal number (the order type of the set of all natural numbers under the usual order); and in the standard definition, cardinal numbers are defined as initial ordinals (the smallest ordinal number with the same cardinality). There are many countably infinite ordinal numbers, and the ordinal number ω is the first or smallest of them (all ordinal numbers which are less than ω are finite); so the corresponding cardinal number aleph-0 is identified with the ordinal number ω.
House Owner
House Owner Il y a 4 jours
We just going to ignore how Micheal bullshits us with the money trick
Ans Hakim
Ans Hakim Il y a 4 jours
I don't even understand but this is cool
Bartosz Kaczmarek
Bartosz Kaczmarek Il y a 4 jours
So the space is infinite in small and big scales? Does that make any sense? No! But it dosen't need to make any sense.
Funk Kyno
Funk Kyno Il y a 4 jours
Nobody talking about how Michael vandalised a 1 dollar US note?
Hayden Richards
Hayden Richards Il y a 4 jours
VSauce ...Please Do a Video On How Man Would Navigate in Space..It Seems to me that based on the "FACT" that the Earth is moving, the Sun is Moving, the Milky Way is Moving..How do we find our way back to earth If we leave Earth & the Galaxy at High Speed(Close to or At the Speed of Light). It seems to me that since there No Up or Down, East or West, North or South..and everthing seems to be Spinning Relative to Each Other.. When the earth fades out of visual/ electromagnetic view or detection.. One navigation degree of error in an infinite amount of spherical planes or dimensions will translate to..upon trying to return to earth..being millions/trillions of km / light years off course..ie. Lost in Space...The Only Solution I see is to have supercomputers to navigate...So Please If You Somehow See this Message..I'm Looking Out for Something from You Guys Soon...&..Once Again You Guys at VSauce U Definitely & Absolutely.. 😎ROCKMAN😎
ReesSoft
ReesSoft Il y a 2 jours
Hayden Space probes and satellites can already navigate themselves in space by using star trackers. www.space.dtu.dk/english/Research/Instruments_Systems_Methods/Stellar_navigation If we ever go to the nearest stars or beyond then the concept would be the same. A mission called GAIA launched in 2013, and has mapped the positions and motions of hundreds of millions of stars. frvid.com/video/vid%C3%A9o-648xGDdNz_g.html Any starship would have a similar map built in to its navigational system. You can always orientate yourself in space by using x,y,z coordinates. In maths the y axis usually denotes "up" or "down". In our Milky Way we could define up and down along the y axis as being above or below the galactic plane. We're probably a couple of centuries away from being able to go to the stars, and by then we'll have far more powerful navigational computers than we have now. I don't think there's much chance of getting lost in space, as long as the navigational system stays online. No direct contact with Earth would be necessary, as the navigational computer would always know where the spaceship is, relative to its star map. Hopefully vSauce will see your question and reply to you.
Hayden Richards
Hayden Richards Il y a 4 jours
Beautifully Produced... Expertly Narrated.. Concepts I've fought with for the latter part my Life..Concepts that are constrained by traditional scientific thinking..From about Age 36 I've Always been grappling with the concept of Infinity, Life..Death..Consciousnes...With the coming of this New Age of Connectivity and Information Sharing I/We All can Rise Out of the Age of Ignorance & Embrace the New, embrace the truth, embrace the Real. I've always argued that when scientist keep saying we have a trillion stars or a trillion trillion stars/planets etc..they were not thinking correctly..the concept of infinity is so beautiful that it terminally gives an explanation to everthing and our (lifeforms) "arrival/birth" into the universe was a simple an inescapable "will" of the inanimate desfire.. to be come animate..We keep thinking of the Existence of Life on Earth & Our (Mans') Existence as Something that has to be Repeatable somewhere else in the Universe..But What If We are just a Strange Phenomena..What If we are just a chance phenomena..something that occurred over the infinite eons of time..only a natural or supernatural mistake..What if We Are the Gods.....and We are Here to Give Account..To Keep Record of the Endless Infinite Beauty & Possibilities of a Scientific, Infinite & Matematically Regulated Universe..Man has Only Recently Come out of its Infancy..Its Father being the Unthinking & Lifeless Cosmos..It is for Us to Rise Up out of the Cradle and Explore & Give Account to the Universe of the Gift that it had bestowed on/to Us.. the gift that it has bestowed on itself...Life..Man..The Supreme Being...the Universe has been and Always Will ..Be. However Man..Life... has Only Recently Come to Being..In desrespecting our selves and our cradle(earth) we disrepect the universe..the One ..the Universe Always Equalises..Let Evil Believe It is Winning the Supreme Be..ing..If Man Extiguishes Himself..The Supreme Will ultimately Try Again..to become Aware of It ..for Life Is the Only Means the Universe has..to become aware of itself.. Again.. VSauce...You Definitely.. 😎ROCKMAN😎
Me name Was taken
Me name Was taken Il y a 4 jours
1:08 that got me scared
hisxmark
hisxmark Il y a 4 jours
But any point "X" in an open n-ball can be taken as the origin since there will always be a scalable distance (less than infinity) between any two points. That is to say, that for any n-ball where the radius is 0 < r < 1 the "center" will be arbitrary. That's why, for instance, that the universe "expands" (or contracts) away from (or towards) any point chosen as the "center". When an n-sphere is projected onto a Euclidian manifold the arbitrary zero point of the sphere maps to infinity, but every other mapping transforms to a distance greater than zero but less than one. This is one understanding that is necessary to understand cosmology. All distances in spacetime are greater than zero but less than one, where one is the radius of the spacetime n-sphere.
A K
A K Il y a 4 jours
Infinite universes.
just a non-yter
just a non-yter Il y a 4 jours
"1 + 1 = 1" My life is a lie...
Erza 13
Erza 13 Il y a 2 jours
😂🤣
Anthony Werkmeister
Anthony Werkmeister Il y a 4 jours
"No hole will be left unfilled."- Michael
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