# Dimensions: A Walk Through Mathematics

A film for a wide audience! Nine chapters, two hours of maths, that take you gradually up to the fourth dimension. Mathematical vertigo guaranteed!

**Dimension Two** - Hipparchus shows us how to describe the position of any point on Earth with two numbers... and explains the stereographic projection: how to draw a map of the world.

**Dimension Three** - M.C. Escher talks about the adventures of two-dimensional creatures trying to imagine what three-dimensional objects look like.

**The Fourth Dimension** - Mathematician Ludwig Schläfli talks about objects that live in the fourth dimension... and shows a parade of four-dimensional polytopes, strange objects with 24, 120 and even 600 faces!

**Complex Numbers** - Mathematician Adrien Douady explains complex numbers. The square root of negative numbers made easy! Transforming the plane, deforming images, creating fractal images...

**Fibration** - Mathematician Heinz Hopf explains his "fibration". Using complex numbers he constructs pretty patterns of circles in space. Circles, tori... everything rotating in four-dimensional space.

**Proof **- Mathematician Bernhard Riemann explains the importance of proofs in mathematics. He proves a theorem concerning the stereographic projection. Support the authors and buy the DVD here.

Uhh I love this kind of documentaries ! They are soooo inspiring !

loved it! watch, dont fish and drive.

Highly recommended to dose yourself with LSD to fully comprehend this documentary. Great documentary that requires an open mind and a good amount of attention. Something most people don't have without some influence. <('.''.')><('.''.')>

Way over my head. I am great with algebra, trig, even a little calculus, but geometry I only get in three demensions. I have watched this several times on another site and I just don't get it. Tried again tonight, I really want to understand, but I don't. It makes me feel bad, I love mathematics and feel like I am missing out on a whole new level of understanding. If they represented it in stereo graphic projection I could count the faces and all and tell you what shape it is (if I memorized the amount of faces on each 4th demensional shape that is), but I would only be working from memory. I want to be able to see it in my head. Next quarter I am taking geometry maybe I will finally get the help I need to get it.

@Waldo, you know it always stroke me as odd when people seem to say that 2 dimensions are easier to understand then 4! So they try to explain leap from 2 dimensions to 3, as if 2 dimensions are innately intuitive. The peculiar thing is, no one ever encountered 2 dimensions, because in 2 dimensional plane anything beyond x,y is infinite.

It's almost impossible to imagine a one dimensional plane as it simply doesn't exist in any possible way of measure.

As for 4 dimensions, something they didn't explain is basically in 3 dimensions you have x,y,z which are 90? to each other, and in 4 dimensions you have a new direction t which is 90? to x,y and z.

This doc is rather hard to understand, though very mind boggling! It's good when you already know the subject, and occasionally thought and imagined it, but it's impossible to grasp it point blank, I think.

Was going to watch this doc as it looked interesting but after reading the posts about it and finding them confusing lol, there is no way my limited knowledge will allow me to understand any of it.

@ iraoksman

Well I can understand two demensions just fine, I can see it in my head. It is just a flat plane like a piece of paper. Of course three demensions are also intuitive to me, I live in three demensional space. But four demension just doesn't make sense. You can not draw a perpendicular axis for the fourth demension, it is physically impossible. To me the four demensional shapes they drew simply looked as if they included a face inside the three demnsional object. But they said that was just the projection of the object not the actual shape as they could not represent the actual shape in our three demensional world. I'll get to it in geometry eventually, I am a physics major. Maybe the professor I get can make my hick brain understand. The idea is interesting to me, I am sure it would open my mind to all kinds of possibilities. Thanks for trying to help me though, I am glad one of us got it.

@Waldo

Lol I dunno if I go it per say either, we are not really wired for it..

it's very hard to _imagine_ the forth dimension, which kinda goes inside of the object but at the same time it's all on the outside ! It's... crazy ! I luv it !

I am physics major too btw :D

*I dunno if I understand it per say

^^

I didn't understand either but it was pretty to look at :D

What makes it both difficult and easy to see is that the projection changes the size of the face and side of the object you are trying to look at just as the projection changed the size of the continents when projecting the three dimensional earth onto a two dimensional plane.

Realize that ninety degrees from x, y, and z is t or time. Put a pen on the table in front of you. Now, what is the difference between the pen you just placed on the table and the one you are looking at now. The one you see now is older than the one you placed there at first. Its' three dimensions didn't change at all but it still aged while is was sitting there and it continues to age.

If you could travel at the speed of light, according to Einstein, then time would stop progressing forward and the pen wouldn't age but you would also stop progressing forward so you wouldn't be able to tell anyway. Now if you went a little faster than the speed of light the pen would be getting younger as would you. Now a good question would be 'If you could go back in time to before the pen was made would it still be in front of you or because you have it with you while you are traveling through time it would still be there?'

If time stops at the speed of light then time for light doesn't exist. While it takes eight minutes for light to get to earth from the sun, that light isn't eight minutes older than when it left.

It is hypothosized that there is an absolute zero in temperature but not necessarily an absolute maximum temperature. I will put it to you that there is an absolute zero in speed but no absolute maximum even past the speed of light.

Assume there is a galactic center where all matter orriginated from ie. the big bang starting point and you were to stop moving away from that stationary point. You would be at absolute zero speed. As you speed up time slows down so that as you slow down time speeds up and you would age at the fastest rate possible. Now, even if you were to move towards that point as apposed to away from it, as we are now, you would be speeding up from absolute zero and therefore slowing down your time progression.

If the universe stops expanding and begins to contract, we would reach absolute zero time, which would seem to last a very long time as we would age as rapidly as possible with no seeming movement of the earth or stars. We may age a lifetime in a single year or month or week or day. Since we only understand time relative to our own speed/time relationship (which, by the way, is constantly varying) this absolute zero value hasn't been calculated although it could be.

When the contraction begins, our speed would begin to increase and time would begin slowing down and we would be ageing slower and slower the faster we go but time wouldn't go backwards as some physicists believe. Time only goes backwards when moving faster than the speed of light (if Einsteins' calculations are correct).

To be honest I didnt know much about the universe and all the amazing maths associated with it until I came to this website..but now I am truly hooked and watched hours of it every night.

I've always been puzzled and stumped trying to visualize and understand the concept of four dimensions.

Tonight I smoked a joint and was blown away with finally having it visually explained and gaining some real understanding of what it is.

Tie that into the concept of things appearing a certain way based on HOW we look at them amazes me (2 slit experiment correct?)

It's still very confusing but I grasp the concept and am loving it!

Any further insights would be much appreciated.

Thank you

Thank you for this guys. When I first noticed it my heart leaped with joy like a Salmon making its way through the treacherous Canadian rapids on it's seasonal migration; Like Santa soaring through the winter skies fuelled on Rum and Cola.

This series makes me want to get out my compass and straight edge and have some fun, Euclid style. If you love mathematics or want to get into the beautiful subject then this documentary introduces some nice concepts and some pretty pictures too.

It is not possible to visualise higher spatial dimensions, but by analogy with flat-world we can see a 3-D part of a 4-D object in our reality of 3 spatial dimensions.

For anyone taking undergraduate Geometry in the future there are a few things to be aware of: firstly that modern geometry courses are the hardest part of a pure math degree, I certainly found that to be true. Secondly, perverse as it may sound, its easier to do modern Geometry analytically (using equations) than Geometrically (using pictures) lest your brain be stretched into a noodle of spaghetti. You may have noticed this feeling while watching this documentary and trying to have a meaningful thought abut it. Lol.

My favourite part of this was the final segment; the proofs hosted by the great German Riemann, although he is dead, and has been for a few hundred years he still managed to roll back time and deliver a rousing proof of stereographic projections. What a man.

Now where did I put that astrolabe?

Ohh carp. This will blow my mind. Gonna go watch now since it is a slow day at work.

Ok, great doc. Question to those who know; The recurring spiral patterns within the fractals..Is this a representation of the orrigional equation or a resultant ratio whic occurs? I ask because it reminds me of a documentary about the golden ratio I once watched. And makes me wonder what the base equation would be if one applied that ratio to a fractal and extrapolated backward.

On a less mind boggleing note. The use of Echer in this made me quite happy. He even could have been used when discussinge 4th dementionality as much of his art incorporated it. I got one of those pieces tattooed on me much to the chagrin of the tattoo artest. In fact I think he almost kicked me for screwing with his head, and sued me for giving him carpal tunnel.

The fact that multidimensional math can be seen in advanced artworks makes me believe that somewhere in the depths of the mind is an intuitive knowledge of these things. That implication is very intense if you let it soak in a bit.

@ RV but here are some thoughts on your questions.

Mathematics has been inspired by art on many occasions as has art with math.

(Its just that one will never admit it to the other)

The big black bug is the Mandelbrot set and the never-ending recurring spirals exist at the boundary of the M-set. The M-set itself lives in the complex plane between the coordinates (1,i), (-1,i),(-1,-i) and (1,-i) i.e. a square of area 1 unit. The iteration formula for generating the M-set (arguably the most complex known mathematical structure) is, Z goes to Z^2 +C. Without going into it too much each complex numbers in the complex plane is inputted as a starting value of Z and iterated! If the result becomes very large the point is not in the M-set and if the iteration tends to a fixed number it does. This criterion is what determines the colouring process and hence the recurring spiral patterns that we observe.

In contrast the Golden ratio (1.61803399...)is formed is formed from real (as opposed to complex) numbers and the easiest way to form this ratio is by dividing two consecutive (neighbouring) numbers in the Fibonacci sequence, 1,1,2,3,5,8,13,21,…

So they are both spirals as you correctly noticed but are generated by fundamentally different math. M-set generating uses iteration and Golden section uses dividing neighbouring numbers. M-set requires the use of complex numbers and Golden ratio uses Integers.

The spiral shapes are intrinsically built into the equations as you guessed although it is not clear how!

Again, it is not clear how one would link these two concepts which is what you ask in your final question.

'I believe that all students should be given cannabis upon entering the examination hall, that way they can re-create the feelings that they had while they were studying' - Dr Robert Leverer

Chapter 3 which deals with 4th dimension..

Is it fair to say that every 3d object i see in real life is in fact a 4 dimensional object on the molecular/sub atomic level?

Or is it confirmed that TIME IS the 4th dimension.

If the latter is the case. Does that mean if i sit in a chair my brother just stood up out of, that because moments before HE was there...that it is in fact a 4 dimensional situation we are dealing with?

I'm sorry if my question/explanation is confusing... i am having a tough time comprehending how time fits in.

From my amateur mind I get the feeling if anything time would be the 5th dimension.

Any insight is greatly appreciated.

@Epic; Thanks for that great explaination. I was wondering about it due to the similarity as I mentioned. The golden ration occurs so frequently in nature it is spooky. I wonder if that number corresponds to some atomic bonding structure since it has such widespread appearance. Meaning perhapse the atomic binds themselves form in a way that frequently bears out into a spiral formation. I knew this doc was gonna get my head humming.

i really enjoyed this, i might buy the dvd from their site.

are there any other great math documentaries ?

Excerpt from a 1970's comic book: The evil scientist says, ' Wohaha, I have discovered the secrets of the 6th dimension' the hero is thinking to himself, ' Hmmm, what ever happened to the 4th and 5th dimensions'.

@emanuel.

A brief history of 4-Dimensional maths: In the mid 1800's the great German mathematician Riemann discovered that by not getting bogged down in the question of, 'Can I picture a 4-D object?' he could generalise the methods of 2-D and 3-D math to also work in 4-D! Four spatial dimensions makes no sense to us but the maths works just as well as with two and three spatial dimensions.

The previous paragraph is concerned with the mind stretching, impossible-to-visualise fourth dimension of space. i.e. the mathematically beloved, abstract notion of higher spatial dimensions. It's not based on observations in the outside world but mathematicians love it because it is an abstract generalisation of known facts; one of the mathematicians main hobbies.

The next part of the story happens in the late 1800's-early 1900's with the physicist Einstein. He took Riemann's work, jiggled the math to suit his needs and made a creative leap of faith in using the 4th dimension as time. Thus space-time was born. The mathematical fusion of space and time.

Three spatial and one temporal dimension is very easy to understand, not at all like it's four spatial dimensional cousin:

Think of a Bee buzzing around a room. At time 0 it is on the window, at time 1 it is on the ceiling. At ANY time in between 0 and 1 we can write down the location of the Bee by recording 4 numbers - 3 to record its position in the room (spatial) and one to record the time between 0 and 1 (temporal). That’s it! The physical concept of 4 dimensions.

It is a fascinating postulate that the observable world is a lower dimensional manifestation of a higher dimensional reality!

This was... Ok. I feel like some things were drawn out too much, while other ideas weren't explained enough. For example, the idea of a sphere as a "complex projective line" was interesting but it seems like they just glossed over it, and should have elaborated on it more. This stuff is all really cool but it's almost too abstract. It would've been more enjoyable and enlightening if they'd tried to make it more applicable to real life.

For all you Tool fans out there the song Lateralis is based on the Fibonacci sequence. Break the syllables down, ((1) Black,: (1) then,: (2) white are,: (3) all I see,: (5) in my infancy,: (8) red and yellow then came to be,) Also at the end of the song, (ride the spiral to the end we may just go where no ones been) He is referring to the spiral patterns contained in the sequence, as referenced above. Tool is an amazing band, there is much more than meets the ears, so to speak, when listening to them.

They also have a song called 46 and 2, ( forty-six and two just ahead of me) He is referring to the 46 chromosomes in the human DNA sequence and the symbology of evolving into what he calls the christ conscience, by adding two more. He also talks about listening to his muscle memory, a Carl Jung concept of genetic memories. In reality though apes have 48 chromosomes so you can't take this literally. Many Tool fans do and fail to realize that adding two chromosomes to humans causes a congenital gentic disorder called Klinefelter Syndrome, found only in males. It makes the sufferer slightly hemaphroditic, and can be associated with mental retardation. Maynard often says he thinks we will all evolve into a unisex creature of sometype one day, so maybe thats why he used these lyrics.

I know I will get some flack for being a Tool fan from someone, it never fails. Everyone seems to think we are conceded or think Maynard is some god or something. Tool is just a really good band and Maynard is just a guy that sings well and knows how to write interesting lyrics. We are not anything special and Maynard is no god, so please don't start with me. I can't help the fact that Tool seems to draw the desperate to appear smart crowd or social rejects, but not all of their fans fit this catagorey.

Common on man, what in the world could have been in that comment to warrant it being caught in moderation?

Wow I love this doc my 2nd time watching it :D

Still don't understand it but the images make me feel like I almost can.

Anyone can see 4 dimensions with an infrared camera =P

What in the world would a complicated description of a doughnut be good for?

@Cool E Beans

What I don't understand is if you are moving past the speed of light and started to move back into time, I believe that would mean you would end up in the same place/time at which you began moving past the speed of light.

In other words... You are at point A at 12:00pm moving at 299,000 km/s then one hour later (1:00PM) at point B you started moving past the speed of light for an amount of time equal to -1 hour, wouldn't that leave you at the same place/time as point B, therefore proving that traveling back into time further that when you started moving by breaking the speed of light is impossible since you will begin to move backwards and therefore no longer moving forward at the speed of light.

I know all the scientists in the world couldn't have missed something so trivial so can someone please explain to me how my above calculations are wrong?

and I thought, I am in touch with some basic concepts of universe. I read a book on 4D space, and am aware of these concepts, but either it was not in the book, or I just forgot, but the connection of time to fourth dimension... someone told, that we should not think of it as a cousin... so is just "next" number of description... cause my inner feeling (dreaded common sense :), is just telling me, that by adding dimensions you never get concept time... althou I remeber reading something about gravity and other dimension... well in mathematical sense I have no problem with whatever dimension, but a connection to a physical world... ok I have a mess in head. any can help me, I am aware I did not pose any question? I am lost, does anyone know where? :D:D

Great video, especially, chapter 5 onwards.

I done this face watching:

enjoyed it still very much!

@henry

lol.. no i think you actually got it quite right and I never thought about it that way.. although along with opinin I think there is something wrong with making time a physical dimension.. leads to a mess in ur head.

@opinin

I think you might enjoy the series of videos on youtube by user "bgaede" although I disagree with him on the whole "we are the last generation of humans," i believe you will find the rest of his stuff fascinating to say the least!..

The example about the 4th dimension being like 3D to the 2D is misleading.

there is no such thing as 2D in physical reality... it appears that way, but that is only because we chose to ignore the 3rd dimension to simplify representations.. in truth all instances of 2D do have a 3rd dimension to them, may be flat or tiny, but nevertheless it is there!

like the conceptualization, but I think i problematic to extrapolate to physical reality which can only possibly have 3 dimensions..

Nothing exists without the fourth dimension which is time. Think of the amoeba. It lives in only two dimensions of left and right and while it has thickness (the third actual dimension) it probably doesn't have a concept of up or down like we do. But without time passing, there is no living as movement requires time to make the movement. True two dimensional existance is the movie screen. The images have no depth to them at all but again without time passing there is no movement or existance.

@Henry Moving forwards or backwards in time has nothing to do with your position in space. Think of climbing a ladder with rungs on both sides. If you are climbing up one side of the ladder while going slower than the speed of light, reaching the speed of light at the top of the ladder and then going faster than the speed of light while climbing down the other side, you would be able to see yourself climbing up the ladder at the same time that you are climbing down. Now if you got to the top of the ladder and backed back down after crossing the light speed barrior, you would bump into yourself while you are both climbing up and climbing down the same ladder at the same time. I recommend always moving forward when you go faster than light speed unless you really know what you are doing.

@cool e beans

i agree that nothing exists without time.. . but i am not so sure it is the fourth physical dimension.. I think that is like mixing apples and toyotas... your right in your analisys of the ameoba.. even in 2D there still has to be a concept of time for existence to be possible, which kinda precludes time from being another physical dimension.. but hey, just my amateur take on things

so from your description of things, when you go faster than the speed of light and accordingly are traveling back in time, then there are two "you's", one moving faster than the speed of light and the other you going backwards as you travel back in time?

youtube / watch?v=N7e39D-tVag

@princeton

There is only one of you. If you are traveling faster than the speed of light and see yourself as you move backwards then you are looking at what the real you has already done to get you to the place you are now to see you do it. Think of the ghosts of Christmas past. They are only shadows of what has been. You couldn't interact with them while you are traveling backwards in time as you would be interacting retroactively. If you stopped moving backwards and resumed progressing forward at Earth time then you could interact with yourself but only if your past self had already met your future self in the past which means you are going to meet yourself. This would create a time loop but only one since past you is going to become future you and future you is not going to replace past you. A self-correcting paradox could occur if your future self eliminated your past self and resumed his/her life from that point on but at each occurence your future self would be getting older and older until death which would be the correction.

There is a good book providing one possible view of time travel called "The Man Who Folded Himself" by David Gerrold. Though I don't think it actually represents true time travel it is an interesting read.

There is some great information in this doc, but to be honest, as a doc, it's horrible. I had a hard time focusing because the animation is very very crude, the script is very unnatural sounding and the music is more distracting than not. I wish these people would have just made a video with people talking about the points without trying to incorporate a bunch of effects and aids they didn't know how to properly execute. - Would have been much more effective that way.

Picture all of these comments coming from those 2D lizards.

I think Its a good perspective of how we should urge ourselves to understand science.

i havent read all the posts, but hes talking about edges and vertices, that is discreet math and sounds like counting factorially to me

This is one of the most mind boggling things I have ever came across. If anyone knows of any other docs on the subject that may give you a broader explanation.

wWOW very cool, 4d became much more conprehencable

had an experiance , a year ago, of turning my conciencensnes into a 4D structure and only now i finaly get it

I note a few comments here from people who seem to be struggling to "picture" or "imagine" multiple dimensions.

Here's a litte help: you can't picture them - we live in a 3 dimensional space, so "picturing" higher timeframes is nigh on impossible with this as our only frame of reference. What you need to do is just understand their characteristics.

For example, picture a person or an object (i.e. a wooden block). It has height, length & width. Now imagine it moving through time - i.e. the block exists on Monday, Tuesday, Wednesday etc. You've just visualized a 4 dimensional object.

Higher dimensions work the same way - by adding an additional layer of complexity for each dimension. Imagine 3 blocks existing in 3 separate universes. That's 9 dimensions. Now imagine them all moving through the same Monday, Tuesday, Wednesday etc. That's 10 dimensions. Now imagine a string tied around all three universes containing the blocks as they move through time, with the string touching all three universes. That's 11 dimensions. Got it? Congratualations - you've just grasped the rudiments of modern string theory which predicts 11 dimensions (or 9 depending on the theoretical approach used).

Now you have two choices:

1) Go and do a degree in Maths so you can use & understand the tools that predict the theory or...

2) Sit back, smoke a huge joint and just think about it's implications (^_^).

Good luck!

One other thing. There seem to be alot of people struggling with the implications of E's TOR and it's implications for moving through time at or beyond the speed of light (Will I bump into myself climbing up the ladder in the past as I climb down in the present? etc.)

You people are needlessly torturing yourself. 10 years ago I was fascinated with the implcations of time. I read 3 books in a row purely on this subject to try and understand it better. All you need to know is this:

The 2nd Law of Thermodynamics states that entropy always increases (i.e it NEVER decreases). You don't even need to understand what this means (although it certainly would help you if you can be bothered doing a little reading on it). All you really need to understand is what this means - which is that time is and can ONLY ever be uni-directional. Time flows forward, always has and always will. It's impossible for it to do otherwise else it contradicts the 2nd Law. Therefore, entertaining notions about moving backwards in time is a pointless excercise - you can't resolve it in your mind because it was never possible in the first place, no matter how many Hollywood movies speculate about it being so (Back to the Future anyone? Lol).

If you really want to f@*k you head up though, it has been experimentally proven that the sub-atomic world is not subject to the limitation imposed by the speed of light. Information can (and does) move faster that the speed of light, and so do virual particles (VPs). VPs exist in pairs, but only exist 1 at a time (one exist now while the other half of the pair does not). Then one dissapears and the other simutaneously comes into existence, and each time this happens they exist at totally random points in the universe. They move around the universe like this in a way which is, again, not subject to the limit of the speed of light predicted by E's TOR. This is the essence of "Quantum Fluctuation". If there is a physical, rational explanation for the mechanics of consciousness and intuition, this is it. So the bottom line is, we cannot move backwards in time (or faster that we do), but maybe our thoughts can.

That's a mindf@*k (^-^).

Now where'd I put my weed...?

I hesitate to post because until I press the button, there exist 42 posts, which I find amusing given the topic.

@ren

Here's a wacky thought. All spaces have subspaces, which I'll call borders and nodes. In 2D space, lines become borders and points become nodes. In 3D space a 2D space is a border between two 3D spaces and a line is a node or point of intersection between 2 or more borders. By analogy in 5D space, a 3D space becomes a node, analogous to a mere point.

All matter is really energy. Solid objects are a mere illusion. Energy is vibrating in all its possible dimensions. If for an instant a particle's vibrations in 3D are zero and increase in the fifth dimension, it could reappear anywhere in the 3D universe. Who's to say exactly how many dimensions our universe consists of or whether or not there are lower and higher dimensional universes? It's all an endlessly fascinating mindf--k.

@princeton

That guy bgaede just plays word games. His arguments have more holes in them than Swiss cheese. They may 'sound' logical but are really full of illogical leaps.

@ Ren,

There is no maybe about it, our thoughts can and do move backwards in time and it's a skill that improves with age. Our thoughts can only move forward in time to speculate over very short periods of time by considering causality: as the fist comes swinging towards your face you know that in the short term future you will feel a stinging pain. Does that count as thoughts moving forward in time? I guess that it is still a speculation because one could dodge the blow or not, but as the time interval becomes smaller and smaller the speculation tends to fact. Lol.

By moving at FTL speeds it has been shown that a persons body can move forward in time. And yet as @Ren said it is physically impossible for us to move backwards in time. Are you still with me?

Thoughts can only move backwards and body can only move forwards in time.

When it comes to temporal quantum movement… Well, anything goes... It's like a zoo and the animals have all escaped from their cages... Bloody statistics, never give a straight answer to anything.

@ Ren, So in 5-D space all of the 3-D universe is just a point… Nice…

A good documentary film. Though, it requires full attention and a good mind for imagination. I really enjoy watching it just as much reading the comments given by all the smart guys.

@hugs

that is an interesting take and i've heard it before.. I wonder what exactly you mean though.

I don't agree with all his arguments, but i believe his take on physics is dead on and i don't think he plays word games, he is just consistent with definitions and points out where people tend to misuse words or rather redefine them to fit their ideas with no regard to what the words actually mean.

i think he gets accused of that same as stefan molineux unfairly. there is no knowledge or communication without concrete definitions that don't get bent and twisted for convenience.

I think this 'projection' method of explanation is not compatible with my brain. I don't understand why we need to add in the extra layer of the sphere in order to project something onto a 2d surface and then do the same thing for 4d onto 3d, why not just project the actual shape instead of project the shape onto a sphere and then onto a something else. And where is Carl Sagan when you need him!

brilliant documentary.

This has interrested me for a very long time. And I've never been able to really get my head around the subject.

I must say this documentary has given me renewed enthousiasm to try and 'see' in 4D :-)

What would really interrest me is seeing a 'shadow' of a 4D object built in 3D space. And then, using vanishing points (or planes ?) - similar to those we use to create the illusion of 3D on a 2D paper or screen - to create the illusion of 4D in the 3D space.

If Escher were still alive, he would surely be of great help in this area...

What we see in books and on the screen, is really a 2D object giving us the illusion of a 3D object using perspective. So we're always using the perspective trick to show 3D objects on a flat surface. Maybe losing the perspective and seeing in real 3D, and then using the perspective trick for the 4D to 3D downscaling would work.

Anyone know if this has been attempted/done yet ?

Are there any theories about 4D/3D vanishing points/planes ?

I see your point in the 4d/3d, but when you think about it, it gets kind of freaky and mindblowing. I dont think it is possible to trick a persons mind to see another perpendicular axis no matter what technology you use, because that is what you would have to do...

Also when you think about if we were living in a 2-dimensional world, how should we then see the third dimension when the way we would look at the world was as if it had only 1 dimension?

When you think about it, we only see a plane of which we interpret in our minds in such way, that it is converted in to another, 3rd, dimension.

As i see it, it could only work, if we could see in 3 dimensions instead of only 2. This would require that we could watch all sides of an object simultaneously / what we are looking at.

Maybe if we had enough mind capacity, and the right configuration of our neural network, it would be possible to grasp the concept of actually having a model of an object in 3d, which might could lead to the individual persons understanding of a 4d object, but i consider that totally indigestible to the human mind. Can you see my point?

I do not really know anything about 3d technology, but i know for sure, that no technology alone could make us see in 4d afterall

(email censored)

At introduction I think: "of course another Euro-centric history of science." Why is it Westerners are so obsessed with 1. attributing all science to the Greek, while there is so much literature on science earlier and everywhere else on the globe 2. why they keep thinking they are the Greek's successor, while the Greeks themselves associated with Asia much more and abhorred Western "barbarians"?

I see your point in the 4d/3d, but when you think about it, it gets kind of freaky and mindblowing. I dont think it is possible to trick a persons mind to see another perpendicular axis no matter what technology you use, because that is what you would have to do...

Also when you think about if we were living in a 2-dimensional world, how should we then see the third dimension when the way we would look at the world was as if it had only 1 dimension?

When you think about it, we only see a plane of which we interpret in our minds in such way, that it is converted in to another, 3rd, dimension.

As i see it, it could only work, if we could see in 3 dimensions instead of only 2. This would require that we could watch all sides of an object simultaneously / what we are looking at.

Maybe if we had enough mind capacity, and the right configuration of our neural network, it would be possible to grasp the concept of actually having a model of an object in 3d, which might could lead to the individual persons understanding of a 4d object, but i consider that totally indigestible to the human mind. Can you see my point?

I do not really know anything about 3d technology, but i know for sure, that no technology alone could make us see in 4d afterall

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Best math lesson I ever got! I have to rewatch this to understand what the 4th dimension is. All the examples of 4d objects looked 3d... which I think is the point...since we can't see in 4d... but then what's 4d and how can anyone explain it if humans only see in 2 dimensions? So many things to follow through on... great documentary.

If you live in flatland, how do you observe an object of any SHAPE, if your line of sight is limited to, left/right, forwards/backwards? Is it possible you only see a line representing the width of the object at the time it passes through flatland. I assume from representations of people like Carl Sagan & Lois De Broy that there is no observational up and down in flatland, suggesting the only way to see shapes in flatland is by observing them from from above or below. If you live in flatland, there is NO above OR below, because there is no up or down.

i see 3d in different shapes, or 1thats without imagination but i still cant see or imagine 4d? what differentiates 3d between 4d?

Too much space between informormation...speed it up x1.5 and its proper

Maybe I'm doing my usual, over complicating, but if you live in flat land surely you have no horizon, how would you see the stereographic projections. Also silly with the cross sections, you could slice things any number of ways and get a different shape. The music is irritating and the man is too slow :)

Lets do this: We observe three dimensional objects using our two eyes (stereoscopic vision, it's amazing.) we use two points of vision to observe a third point, which we will call the focus. (keep focused!) Here's where geometry is important: Euclid Book 1, Definition 3: Every plane contains at least three points not all on the same line. Guess what? It's guaranteed not to be on a straight line; the focus always has an angle. From our eyes & focus we can construct a plane! from this we can visualize a 2d plane which constitutes our horizontal and depth components of vision. Toss in our vertical component; put it together: Two points of "origin" (our eyes), a focus create a plane that we can rotate around a line which goes thru both of the origins we have certain limitations on the angles we see in exchange for incredibly accurate depth perception; this is necessary for tool use, like typing (which I am doing to much of).imagine now that we remove that ability to rotate or focus; so all we can do is perceive the plane directly in front of us it becomes a line of various colors, right? This is where you are having a creative mind-block (or mathematician's way of saying I must have been on drugs to think this way)- We have two eyes for three dimensions, shouldn't the two dimensional creatures have only one eye? think of ultrasound- it emits from one point and creates a two dimensional image. Therefor the single 'eye' of the two dimensional creatures is able to visualize everything in front of it in a similar way to a certain depth of field (much like our limitations) Lizards in this dimension can see deeper than three meters in front of themselves, right? Imagine a three meter tall ultrasound screen, it would have to be two and a quarter meters wide (if it saw in 4:3 aspect ratio.) Suddenly seeing the projections is a whole lot easier, since the two dimensional single eyed critter sees everything within its field of vision limitations which are a certain depth and rotation and able to see "thru" other two dimensional objects, as seen in ultrasounds.

Redundant Double Post :(

Math Rules.

i got lost in flatland, but it was so so cool that i'm bound and determined to understand it eventually.

The reason you can tell that a ball is round and not just a flat circle plane, is light.

The reason you can tell that a box is not just a square plane, is light AND size (the edge furthest away seems the smallest)

That's basically how we perceive 3D, off course stereo-vision enhances our our perception, you could argue that sound also enhances our 3D perception

3D exists..

dimention cero is a point

1st dimention is a point in movement, a line

2nd dimention is a line in movement, a surface

3rd dimention is a surface in movement, a volume

4th dimention is a computericed movement of 3rd dimention

so what

4d is time, 3d in movement. It's 5d and beyond thats hard to imagine.

Very very good. Excellent job. The graphics, the classical music, the soft slow deliberate tone, and meter of the explanations excellent work.

I can see some lucky children learning this essential knowledge, and thinking fondly of this video 30-40 years from now. Hopefully in a new world of peace, and tranquility.

just as space is visualized by the 1st-3rd dimensions, imagine the 3rd dimension travelling through time in a line as the 4th dimension. The 3rd dimension without time is just a volume of no movement, everything frozen.