Antipodal Points - Numberphile

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Simon Pampena discusses antipodal points on tennis balls.
A little bit extra from this interview: • Antipodal (extra bit) ...
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[@numberphile]

Check out Brilliant: brilliant.org/numberphile (sponsor) More videos with Simon Pampena: bit.ly/Pampena_Videos

[@FlyGirl774]

*Artillery Only*

[@user-ph5wf5ko6x]

*_no._*

[@FlyGirl774]

then do that

[@FlyGirl774]

1111

[@hsterts]

"Whats can you notice between Australia and Australia?" "Well they're the same!" "Y E S."

[@rewrose2838]

*What Euclid: What can you notice between a Right angle and a Right angle? Everyone else: Well they're the same! Euclid: Y E S

[@JorgetePanete]

What*

[@damienjones1487]

*What

[@MrSamwise25]

@@rewrose2838 That's hilarious!!

[@danielroder830]

Everyone: Whats can you notice between the speed of light and the speed of light? Einstein: Well they're the same! Everyone: *What

[@christianburrello811]

"You have to believe the Earth is round" *scrolls to comment section*

[@adrianhdz138]

Christian Burrello Exactly what I was gonna do xddddddd

[@cubethesquid3919]

Same tho XD

[@phoenixstone4208]

:43

[@Игор-ь9щ]

Omg I literally instantly did that lol

[@CiuccioeCorraz]

Did not disappoint

[@SpaceSnaxxx]

4:09 "And he proved it. And he explained to me the proof. And for me, this was like, one of the most beautiful experiences, just someone coming up with math." I love listening to Simon talk about maths. Best way to spend my weekends!

[@Robobrine]

This reminds me of the fact that there will always be at least one pair of antipodal points on earth with the same temperature and air pressure.

[@jojojorisjhjosef]

I too watch Vsauce.

[@carlmmii]

Exactly what I was thinking going into this, and I'm glad this turned out to be much more of an "aha!" moment once it all came together.

[@trucid2]

Shouldn't there be an infinite number of antipodal points with the same temperature or pressure?

[@Robobrine]

trucid2 - Yes, but there will also always be a pair with the same temperature AND the same pressure.

[@andymcl92]

*at least one pair

[@MichaelSteeves]

And here I was hoping to learn the proper pronunciation of Antipodal.

[@lfteri]

antIpodal

[@RDSk0]

Ant-eye-pod-Al

[@Minecraftgnom]

I always say it like "ant I poodle".

[@アヤミ]

Antipode is like that in most dictionaries though, I don’t know what to say...

[@dielfonelletab8711]

an-tip-odal

[@norm58inator]

On the thumbnail he looks like he is in great distress

[@michaelpapadopoulos6054]

holding that tennis ball is real hard you know?

[@GaryDunion]

Please... my tennis ball... he is very sick

[@samuelgladicas3866]

Sorry but the Earth is actually à Klein Bottle.

[@karldavid3127]

Lucas Megaglia Agartha?😂

[@アヤミ]

Lucas Megaglia kek

[@VAFFANFEDE18]

So 4 dimensional?

[@isambo400]

Its actually a Klein bottle inside another Klein bottle holding up that crazy guys house

[@Spiros219]

Hahahahahahah

[@WarmongerGandhi]

If you put two pieces of bread on antipodal points it makes an Earth sandwich.

[@sushzi4937]

How come is your comment an hour ago after 17 minutes of release? Oh yeah your sandwich is soggy

[@Mr4NiceOne]

Have u even watched the video ?

[@Mike-qt4fr]

everything in this universe is either a salad or a sandwich

[@andymcl92]

But only if you get the bread lined up the right way.

[@glennhefel]

Vsauce

[@KazamaShin89]

Trolling the flat-earthers like a boss....

[@AvanaVana]

Another interesting fact about antipodal points intersecting with earth science: 50% of volcanic hotspots are antipodal (some have suggested this is a result of impacts in ocean basins propagating through the earth). Also, notice that the vast majority of continental crust has oceanic crust at antipodal points. It seems as if continents drift towards stable configurations of positive/negative pairs, where continental crust opposes itself at antipodal points. I do not know of an explanation for this fact. It seems to persist throughout deep geological time - consider Pangaea, where one hemisphere was almost completely continental crust and the other almost completely oceanic. Continents seem to drift in a bimodal pattern from so called supercontinental configurations to well distributed (present earth) configurations, both configurations having the property that the continental crust is balanced at antipodal points by oceanic crust.

[@alazrabed]

Regarding your second point, as there is on earth much more oceanic crust than continental crust, couldn't the phenomenon of opposition be just a coïcidence? (Since there is around 40% of continental crust, you'd expect to have 15% percent of it at antipodal points at any given time.)

[@mu_on]

Thank you ! Please do more content with Simon, I just love him

[@superj1e2z6]

Dammit Brady bringing up the tennis ball color question from out of nowhere.

[@BrandonGraham]

Ha. Just drop that bomb and say "just askin" like you didn't just make it sound like a trick question.

[@Varksterable]

Aye. Gave me a big grin that one. Well played, sir!

[@recklessroges]

What colour is it for you? (I see it as yellow.)

[@Varksterable]

Google "tennis ball colour tv itf". Thanks, Tim.

[@PotatoesGottaPotate]

he really did examine the ball and his answer was so unconfident, then he mentioned the fact that he was mildly colour blind and it made me feel bad for him😂

[@JoshuaHillerup]

Everyone knows tennis balls are chartreuse. The issue is if you ask someone if chartreuse is a shade of green or a shade of yellow :P

[@redapplefour6223]

chartreuse is the best color

[@ryaneakins7269]

I sincerely grew up thinking chartreuse was a shade of red. Not colour-blind, just picking things up from non-visual media, i.e.: text.

[@Le_Tchouck]

Here in France, Chartreuse is a monk alcohol, plants liquor. It exists both in green and in yellow!

[@lt3880]

i mean its like asking someone if orange is red or yellow, we should be thinking of it (chartreuse) as its own colour in between yellow and green

[@JoshuaHillerup]

@@lt3880 yup. Just as long as everyone agrees it's not yellow ;)

[@connorwilliams3451]

Wow! Judging from that beard, Simon Pampena is apparently taking a break from his whirlwind tour starring in the live stage production of Castaway, and showing us some of the lesser understudies for the part of Wilson... How nice of him to take the time!

[@ExhaustedPenguin]

Simon has turned into a caveman

[@esdisaysaloha]

He reminds me a bit of Liverpool FC's Mohamed Salah.

[@wmichaelbooth]

I think he's been busy writing a manifesto.

[@keithwilson6060]

John the Baptist.

[@dlens2606]

"I don't even know what's going on here. OK we've got everything we need"

[@encounteringjack5699]

I just love how mathematics takes the simple and makes it complicated.

[@flamencoprof]

The other side of the world (TOSOTW) is a place I have actually been to. In 1996, I and my Ex set off on the so-called trip of a lifetime, an extended driving tour of Western Europe and Britain. Before we left, a yacht-owning friend brought his GPS to my house and I recorded the exact Latitude and Longitude. It is a simple matter to calculate the exact antipode of my house, and whilst driving around Europe, I made a point (ha-ha) of visiting that location (as near as I could get without invading private property). From where I live close to Auckland, New Zealand, the antipode falls in the South of Spain, in the mountains about 50 km almost due North of Gibraltar. I can now sit at my dining table and direct visitors' attention to a modest crystalline rock safely ensconced in the China cabinet, which is an actual piece of the Earth from a point straight down through the Earth from where they are sitting.

[@SpiffyCheese2]

Once I saw that this video was 14 Minutes I thought It wouldn't be worth watching, but I was wrong, kept me interested for a long time. Such a beautiful Proof.

[@markbrown2450]

I waited through the whole video for the big reveal or big "aha" moment, and it never appeared! He took so much time and care to explain something that, for the life of me, I can't understand how someone doesn't know intuitively.

[@JackLe1127]

"You have to believe the earth is round." I like how he even has to say it

[@YouPlague]

He said it because the Earth is a geoid, not a sphere.

[@omikronweapon]

@@YouPlague when people think they're the smartest, because they know more than the absolute dummies, and miss the ACTUAL purpose of the statement.

[@rikwisselink-bijker]

Fun fact: there's an island close to New Zealand that's called Antipodes Island. My teacher told me this was because it's oposite to the Netherlands, but it turns out it was actually named after Londen, but it's actually the French town Tôtes that's the closest. The more you know.

[@hank7281]

That is probably the most satisfying proof that I've ever seen.

[@johnlou844]

Finally Simon is back!!!

[@220Dave220]

80 flat earthers disliked this video

[@EmaDaCuz]

Simon is my favourite Numberphile regular. I love his style and the topics he presents. More videos with him, please.

[@kenhaley4]

As always, I love Simon's presentations. He is so into the subject matter that one can't help being drawn in with him. I won't forget this little proof about antipodal points. One question: Can't you carry this one step further and claim there are at least two pairs of antipodal points on a curve that cuts the sphere into two equal areas? After all, once the curve crosses its mirror image it has to cross it again to close itself. Oops, on second thought that's wrongl there can easily be just one pair of antipodals. The curve crosses its mirror image at one of the points and again at the other; that's all we need. There can be any number of antipodal pairs. (Dunno what I was thinking)

[@ceruchi2084]

I'm curious: What would it look like to have a division with exactly two pairs of antipodal points? I can't think of any shape of loop that would do it. The tennis ball was pretty basic and it netted six pairs! (While the simplest division of all, just a horizontal line, would create infinite antipodes :-O ).

[@kenhaley4]

ceruchi: Easy. Just change that wavy tennis ball curve into a regular sine wave (on the flat projection of the earth). Each of the vertical sets of 3 points would collapse to single point. Four points of inersection (two pairs) total, instead of twelve.

[@puruchaudhary22]

this proof can be probably understood by anyone regardless of age, academic background, etc. and thats just so beautiful.I was giggling and clapping at the end when the "assumed" halves didn't even touch. thank you.

[@CosmiaNebula]

So, this proves that there must be at least 1 pair of antipodal points on any area-bisecting smooth curve (so that it is not weird like some area-filling curve, and so it is possible to unambiguously define the area inside and outside the curve). But it's easy to show that there must be at least two pairs. And further, any number n >= 2, it's possible to make an area-bisecting smooth curve that has exactly n pairs of antipodal points.

[@vegardt3433]

So, I have two questions then. 1) Will it be possible to construct a non-trivial loop containing only antipodal points? A non-trivial antipodal curve? (an "equator like" line, which forms a great circle around the sphere is the trivial one, which obviously contains only antipodal points) 2) Would a similar result be true for higher dimensions? Is it for instance possible to find antipodal curves, or even surfaces, on a four dimensional sphere?

[@andretimpa]

1) Yes. Start from one point in the sphere and draw a curve that end in the antipode of the starting point. Complete the loop with the antipodes of the curve you just drew. 2) You need to make the way you are splitting the hypersphere in 2 more precise, but I think the answer would also be yes. (the main argument would be similar to the one presented, to avoid antipodes you need to break the sphere in 3, not in 2).

[@ITR]

Because for two shapes to be the same, they have to either be mirrored or rotated, or a mix between the two, and since the two shapes have to stick together, rotation isn't possible, or something like that.

[@BattousaiHBr]

"you have to believe the Earth is round [...]" Shots fired

[@bgoggin88]

Dropped what I was doing to watch when I saw Simon in the thumbnail.

[@Azrage]

Yay! Another one with Simon Pampena!

[@delanask]

That's a satisfying proof, I love when the point of a proof clicks and makes sense to me moments before its actually explained.

[@joshyoung1440]

Sure, I always wanted to learn about antipodal points from an actual ancient Greek philosopher

[@kasparsoltero1995]

A map that includes New Zealand Represent

[@ragnkja]

If it didn't include New Zealand, it wouldn't be a map of the whole world, and that would mean it was useless for this video.

[@andymcl92]

Rhys Darby will be most pleased

[@ourboyroy9398]

maps that don't include new zealand has become a bit of a meme

[@HL-iw1du]

Kaspar Soltero what

[@y_fam_goeglyd]

My first thought too lol. Did a quiz on the BBC app the other day "what country is missing?" from different maps. It was kicked off by poor NZ falling off any number of maps :(

[@VibratorDefibrilator]

What a brilliant proof is this! Its simplicity brings together knowledge and understanding, which results in feeling of beauty!

[@austynhughes134]

Glad to see Simon back!

[@750kv8]

A simple method of finding an antipodal point in Google Earth. Press the ruler button, select straight line (that is its default mode), choose an arbitrary starting point, then go to the other side of the Earth, dragging your straight line from the starting point with your cursor (but don't select any ending point). You'll eventually find an area where a small movement of your cursor makes the line jump to a near opposite direction. Keep zooming down until you get just a very small area down to a few meters across, at which point the accuracy of the ruler breaks down, with its end jumping between discrete points, but each connecting to your starting point in a different direction across 360 degrees.

[@willnewman9783]

Best Numberphile video

[@trunksn1]

I hope simon is doing well. His enthusiasm is contagious

[@SamnissArandeen]

I'm the kind of person that stares at a baseball and wonders about antipodal points. Meanwhile, the batter is staring daggers at me in utter confusion.

[@bonecanoe86]

11:32 He has drawn Virginia.

[@masansr]

Yes, the ultimate fly-over state!

[@Jivvi]

New South Wales is Ohio and Queensland is Idaho. GeOgRaPhY cOnFiRmEd!!!!

[@aciloraphael]

0:16 "We have to believe the Earth is round" Flat-Earthers incomming...

[@Ineddiblehulk]

Ha! Thought the exact same thing!!

[@blu3ntv]

Same

[@アヤミ]

in-cumming*

[@grivar]

He said that because the earth isn't actually spherical. Easiest way to remember is that Longitude isn't actually the long one.

[@lars7898]

Just wait, until CoolHardLogic enters the comment section, all Flat Earthers will have gone :D

[@Toobula]

I had observed the antipodal points on a tennis ball. Nice to see the analysis played out. But the coolest thing in the video is the antipodal earth map overlay.

[@songbird6248]

I flew from NZ to JFK via Qatar a couple days ago and used that same map to check when I hit antipodes! I specifically remember crossing Queenstown’s off the west coast of France on the second leg.

[@albert123chern]

This argument can be generalized to a statement in measure theory, right? Suppose we have a measure space X with finite measure, a measure-preserving measurable map f from X to itself, and a measurable subset A in X that has half of the measure of X. Then one expects that the "boundary" of A is not disjoint from its image through f. One can probably play with different notions of "boundary"...

[@chrisriess1298]

Interesting question... I would assume that chosing a different measure will also distort the sphere so that all differences to ordinary Euclidean metric cancel out

[@theoneallinblack]

"We have to believe the earth is round." It physically hurts that he had to say that. . .

[@terdragontra8900]

elegant result, nice

[@Schwuuuuup]

The first 10 Minutes of the video was all well and nice, but the conclusion - although I got it - was not very well laid out... this is a pattern I do recognize quite a bit on this and other channels and with the teachers I had in school: Explaining the obvious and everything that is accessible through practical approach quite lengthly and then rushing the proof and the abstract part where the understanding of the main point becomes quite tricky

[@jaishkhan7442]

"you have to believe earth is round" 😂😂

[@seinfan9]

More from Simon. Most entertaining.

[@kippers12isOG]

I thought that proof was incredible. Delightful even. When it hit me I smiled

[@bruceq8722]

Does the map projection matter when doing this stuff?

[@nomekop777]

No. The horizontal incraments don't change, and the vertical incraments are reflected across the equator, just like the antipode

[@SYSMO00]

No because the distortion caused by the projection is the same for both antípodas points

[@chrisriess1298]

as long as the map preserves the necessary symmetries

[@jeffreyblack666]

To some extent. The key part is the rotation by 180 degrees then flipping it. That only works they way he did it if you have a projection where the equator is a straight line, the north and south are presented symmetrically, and the centre can be any point along the equator where the only manipulation to achieve that different centre is cutting the map and rejoining it. (i.e. you can connect the map to make a cylinder, where at any given height, a section of the ring around the cylinder has the same scale as any other section of the same ring) For other projections, such as dymaxion, azimuthal, butterfly, conical, elliptical and so on; you need to distort the line/area rather than simply translate and mirror.

[@oncedidactic]

I am incredibly moved by the love of maths for maths displayed here.

[@dr.gamjo3931]

Sir, I've found some numbers which have nice property Like 5 5! = 120 And prime factors of 120 are 2*2*2*3*5 and sum of prime factors of 5! is 2+2+2+3+5=14=1+4=5 Some other examples are 1,2,11,13

[@mistafabro]

Simon is the man

[@tommykarrick9130]

The best way I can describe a numberphile binge is “Oops that’s not the video I meant to press I’m gonna go ba- Mm Hmmm I think I’m gonna keep watching...”

[@GuanoLad]

Australians talking about the Antipodes is right and proper. Bonus New Zealand reference also extremely appropriate.

[@AuthenticDarren]

I'd dare say that in the last improvised example that thereis such an enormous gap between the two areas because the first shaded area wasn't nearly half the total area. You should try the idea again on graph paper, carefully making sure that the first area is half the total map area. I'd say the two lines would have to have at least one antipodal pair.

[@VWftw82]

Simon is much better at cutting than I am. On a channel from England, I suspect people would understand my simile that my cuts are about as straight as a British road.

[@maltager5106]

I feel like that doesn't prove anything? All you've done is prove that the arbitrary line you've drawn does not contain half the area while simultaneously not containing antipodal points? You haven't proved that every line which splits the area in 2 must contain antipodal points, you've just given a single case where it is true. Although I understand where to go from this proof, it's as simple as saying in order for the area to be the same, the line and it's "inverse" line must either cross at least twice or touch eachother throughout the sphere. Simply because if they do not, then there will be space on the sphere that neither area contains and there will be no space on the sphere that both areas contain, meaning the total area comprised by the two lines cannot be there total area of the sphere. Thanks for the video though Brady and Simon!

[@tommasoforni]

He said: suppose we have a loop that circles half the area and that doesn't contain antipodal points. Then he showed that such a loop doesn't exist, therefore all loops that contain half the area contain antipodal points. It's a proof by contradiction

[@maltager5106]

He never showed that it cannot exist - he just gave an example of one that doesn't contain antipodal points. That's like saying something like "2x =/= x^2 for any x" by giving an example of "2*3 =/= 3^3". It's not true because 2x=x^2 when x=2.

[@tommasoforni]

the fact that it doesn't contain antipodal points is an hypothesis of the proof by contradiction. All the loops that don't contain antipodal points look like the one in the example because they can't touch their mapped counterpart per hypothesis.

[@maltager5106]

yes but it still didnt prove anything, he didnt make the final conclusion, he just said "okay thats a proof by contradiction" and done.

[@tommasoforni]

@@maltager5106 he proved that the two initial hypothesis lead to a contradiction, that's enough for the proof

[@pierreabbat6157]

The Antipathies! (Alice) What's the difference between Java and Colombia? All the difference in the world.

[@zeldajerk]

Is there a theorem that says that there must be X or more pairs, therefore implying there can't be less? (clearly I mean where x >= 1, and the areas are equal) In other words, do we have a range that tells us how many specific pairs there can be? (We already know 6 is a possible amount of pairs)

[@108titan]

Wow. Just wow. Amazing episode!

[@westronic]

Been digging these topology videos

[@legendhero-eu1lc]

Thank you for the video! All of you friends are super awesome!

[@nilswitt6314]

Is there some kind of formal proof of this for an arbitrary closed loop, cutting the surface into two equally big pieces? I did indeed understand the proof presented in the video however I am not sure whether and how it's possible to generalize it, so that it shows the validity of the statement for any given sphere (it is rather obvious however that it works for any r>0 if it works for a specific r) and closed loop cutting it into two pieces of equal area.

[@Sporkabyte]

I think this might be something you could prove using linear transformations. I guess you would have to find the linear transformation that maps the points on the curve to their antipodal points and show that the kernel of the transformation is never the empty set, but I’m not sure what the connection with the equal area bit would be exactly.

[@nikanj]

Am I right is saying it's also impossible to have exactly one antipodal point? I can picture seams which generate 2, 3 or infinite points but never only one.

[@hydra314]

i wonder if the "antipodal water for land mass" observation has anything to do with why land masses are distributed the way they are. This could also apply to the distribution of lipid rafts the the surface of cell membranes.

[@vs-cw1wc]

Seems like the assertion that the area between two non-intersecting curves is strictly greater than zero requires further assumptions about the curves themselves that are not explicitly stated in this video.

[@rowanprice3291]

Could you make a similar statement where, under the same conditions, there exists pairs of antipodal points where one point exists in each surface? What about proving that there are antipodal points that do or don't exist such that both points exist on the same surface?

[@TrakDah]

Question. If you end up at the end (without antipodes), would you prove that the areas need to be the same by sliding the overlaid map upwards until such a point that the distance from the bottom to the tangent of the largest curve is the same as the distance for the same tangent on the original map to the border? What I mean is this: If you slide the overlaid map up to that point, wouldn't the two areas be the same at that point?

[@v.e.theorist8838]

I watched the first 50 seconds of this video right after I watched the "five-sided square" vid with Cliff, and that's when I said something mathematical for once in my life: "YOU CAN'T MAP OUT A SPHERE!"

[@Zerotan]

I found out a few days ago that I live in the only populated North American point that has an antipode on land - Medicine Hat AB, Canada to the Kerguelen Islands. Most of North America maps to the Indian Ocean

[@OrangeTheMarker]

Does this proof really hold? Since you need to perfectly project the sphere onto a 2D (the sheet of paper) and we know that can't be done? I feel like when you do the projection, the points would get warped and it wouldn't make a valid comparison back to the sphere...

[@manla8397]

Beautiful proof. Love it. Thank you.

[@martinconrad9260]

If I'm thinking correctly, only *exactly* at the north and south poles can antipodal points both be in daylight at the same time-twice a year at the equinoxes....?

[@ffggddss]

Well, no, *any* two points on the terminator (the light/darkness boundary), that are at opposite points on that circle. Fred

[@maxravenwood3877]

I loved this video, because it reminds me how important proofs are in maths but doesn't go really deep into technical details.

[@ThePs3isgay]

Is there any requirement for the 2 sections to be congruent? Or is congruency happen because of equal area?

[@jpe1]

Does the “cut the transparent map in half, swap sides, flip over” trick work for any cylindrical projection, or just Mercator? That map didn’t quite look like a Mercator projection but clearly was cylindric (also, now that I think about it, that map didn’t go all the way to the poles, did it? Does that matter? At the very least I suspect the cutoff must be the same for both hemispheres.) would something like a Peirce quincuncial also work? Could a similar trick be formulated for conic projections?

[@ffggddss]

"That map didn’t quite look like a Mercator" - That's right, it's a cylindrical [ -equidistant- - no, CORRECTION: equal area] projection. The method shown for antipodal inversion, will work with any cylindrical projection that's equidistant in longitude, and has reflectional symmetry in latitude. In general, I'm not sure that there aren't other projections in which it would work; I sort of doubt that there are, though. Fred

[@angharadhafod]

That was very interesting. Right from the start of the video, I was thinking: "but it's obvious - because of equal areas". I was thinking in 3D, and I don't know how I would have verbalised it. So I suppose it was nice to see it demonstrated in a 2D reduction. There is also an assumption that the object in question is a perfect sphere (or, intuition tells me - without too much thinking I hasten to add - an ellipsoid). So for example, with an egg, it would be quite possible to draw a line (indeed, a perfect circle) around it, dividing the surface into two equal areas, which did not have any antipodal points. The earth is not a perfect sphere, although it does approximate to an ellipsoid.

[@shruggzdastr8-facedclown]

(@11:33): What you have drawn here is a crewd version of the outline of the US state of Virginia!

[@TheThe-lg5hq]

Do you guys know this: Sam's Number is a "number" described in an article written on Googology Wiki in Feb. 2014 by a user under the alias "SammySpore":[1][2] Sam's number is so gigantically huge it cannot be described. It boggles the mind. Actually, it would boggle a megafugafzgargoogolplex minds. If you want a small glimpse of how big it is, here. Sam's Number is enormously larger than Rayo's Number. It can fill a greagol multiverses. Actually it can fill so much more than that, it is undescribable. The article was quickly deleted due to the site's restrictions on unsourced original content, but it grew into an inside joke in the googology community as an example of lazy and badly defined googology. Sam's Number was mentioned in a joke article by Sbiis Saibian about Easter bunnies.[3]

[@kevin_b]

That is a really nice proof.

[@kevwang0712]

Fun fact: The northern part of Taiwan (old name Formosa) is the exact antipode of Argentina's Formosa province.

[@AaronHollander314]

Could one draw the loop so that the two accurate half areas meet in the middle? In the false example he gave, just make the original area bigger. I'd guess that that would count as ALL the points on the loop having antipodal points.

[@christosvoskresye]

Your formulation was a bit vague. Through more than half the presentation I was thinking of examples of simple cuts that would divide a sphere into equal areas but in which some pairs of points are not antipodal. You were claiming that AT LEAST SOME pairs of points are antipodal, not that ALL points are antipodal.

[@ceruchi2084]

I think the claim was that ALL seams of this type will have some antipodal points.

[@thehyperfinestructure6550]

What about the fact that-even if a curve doesn't divide a sphere into two different areas,still it can contain antipodal points?

[@sushitime9496]

Now I am intrigued because if you do a map like that: (let the ones be the curve) | 1 | | 1 | and you do the cut in half and flip, you have four halves of ones, two on each side like that plus they are reverted: |1 1| |1 1| So they don't "stack" with the previous ones. They obviously are antipodal but can you explain why it doesn't work ? (Sorry for the bad engrish)

[@frederf3227]

Your initial curve doesn't bisect the surface into two regions. It's just a line from the north to south pole along one meridian. And they would share points with the second curve: the north and south pole. We must remember that the whole top edge of the map is the same point.

[@DArtagnonW]

In other words, your original map is more like: [ 1 1] [ 1 1]

[@sushitime9496]

@@DArtagnonW yeah and since the extreme left is the extreme right you could also do [1 1 1] [1 1 1]

[@samtibbitts]

Are there ways to divide the surface of a sphere into 2 even parts where the dividing line is entirely composed of antipodal points other than the cricket ball? That would be practical for making a ball?

[@Veptis]

So platonic shapes inside a sphere. All points have the same distance minimum distance, they are evenly spaced. But what about 3 points or 5?

[@Pfhorrest]

"we have to believe the Earth is round" is a weirdly circumlocutious way of saying "the Earth is round".

[@211222222332]

the Lusternik-Schnirelmann theorem, aka Lusternik-Schnirelmann-Borsuk theorem or LSB theorem, says as follows. If the sphere Sn is covered by n + 1 open sets, then one of these sets contains a pair (x, −x) of antipodal points.

[@xCorvus7x]

Do you mean by Sn the n-dimensional sphere S?

[@211222222332]

@@xCorvus7x yes

[@xCorvus7x]

@@211222222332 Well, you might want ro clarify that, perhaps, for easier understanding without looking the theorem up. For my part, I have not immediately understood n as the sphere's number of dimensions.

[@anweshanghosh8753]

Can this proof be extended to all platonic solids?

[@thephoenix3155]

What’s the website you used called?

[@jaxblonk5127]

That's kind of what defines the diameter of a circle, since that is a fixed line segment, so any shapes made into a sphere would indeed have to have such line segments ends tangent and not intersecting the surface. The diameter is a fixed length in 3d space relative the midpoint, where the surface is just the maximum length of a line segment where it reaches this equidistant, spherical surface, and no more. So to fit on the surface of a sphere-map in a sense, of course it would be as such. This just seemed obvious, really. Apologies if anyone was more enthused, however.

[@cinvhetin8054]

And it's intermediate value theorem all over again. When I first heard that in University, I was like "Well, duh. Would is that such a big deal" but it just keeps popping up EVERYWHERE ;)