[PROF SEQUIN]: What do you make of this? 5, 6, 3, 3. What do you think comes next? [BRADY]: I’m gonna go… 7! [PROF SEQUIN]: 7. No, actually, it is another 3. Now you have a guess at what’s coming next. [BRADY]: 3. [PROF SEQUIN]: Good! Yes. As a matter of fact it continues to be 3, 3, 3. And you wonder, this is really a strange sequence, you know, what are we going to do with this? Well, this is the number of regular polytopes that exist in 3, 4, 5, 6, 7, 8, 9, and all higher dimensions. And that’s what we are going to talk about today. Regular polytopes in N dimensions. A polytope is just a more general term that generalizes for 2-dimensional polygons, 3-dimensional polyhedra, and everything higher we call a regular polytope. That sounds a little scary, so let’s start with something much simpler; the platonic solids. So here is a tetrahedron, that is the first platonic solid. Next we have a simple cube. Third one — octahedron, made from eight equilateral triangles. Number four, twelve faces — the dodecahedron.

This one has twelve pentagons. And finally, the real jewel… — Ahhh…. the icosahedron, made out of twenty equilateral triangles, and this is probably a Swiss crystal. [BRADY]: Of course. [PROF SEQUIN]: And here is another view of an icosahedron — this is a pattern by M.C. Escher. Could you show to a non-mathematician, in an intuitive way, why there are exactly five platonic solids — not more, and not less…? Well, one way to do it is to look at the surface of these objects. So this dodecahedron, for instance,

is made out of regular pentagons. And so all of these platonic solids

are made out of regular polygons. So probably we should step down a dimension

and look at what regular polygons are possible. Simplest one would be a regular equilateral triangle.

Then we can do a square. Then we can do a pentagon. Then we can do an hexagon, and so on, and if you go all the way to infinity, we end up with something that looks like a circle. So there are infinitely many of these regular

polygons in two dimensions. So really, this sequence up here, it has another character before the five, if you go to 2 dimensions. And what’s that? It’s not an eight—it’s infinity. And if you really wanna go to the extreme,

what about 1 dimension? Well, it’s just a line segment — nothing else. What about 0 dimension? Well, it’s just a point;

nothing else. But now, 1, 1, ∞, 5, 6, 3, 3, 3, 3, 3, 3 – that’s a really weird sequence, you know, and, um,

I guess that would really confuse people. Okay, all of the platonic solids are made

from one of these regular polygons. And so, how many different ways can we use these regular polygons to make one of these solids? Start with the equilateral triangle, and ask ‘how can we make a platonic solid out of that one?’ Well, we’ll need a minimum of three of those

to be put around the corner in order to make a valid corner in three dimensions. And if we put three of these triangles around a shared vertex, then the bottom is also an equilateral triangle. That would give us a tetrahedron. Okay, so next we can try to put four equilateral

triangles around the shared vertex. And we can bring them together and make a four-sided pyramid. And, remember, to make it regular all of the dihedral angles here have to be exactly the same. So that means that the bottom now is a perfect square. We can take this square pyramid

and match it with an identical square pyramid. So that makes the second platonic solid;

the octahedron. Now we can try to take five of those

and put them around a vertex. And this makes this five-sided pyramid —

so it’s completely symmetrical. If we continue forming these kinds of corners,

and try to sort of wrap it around, will this actually get a closed surface? Will this actually end up in something,

something useful? Well, we know we have this icosahedron, okay?

But if you have to really figure out whether it works, it takes a little… thinking. You can see on top here — we will have

this five-sided pyramid. So we can take one of these five-sided pyramids — we cannot put this one just up on a mirror — we can

take another one, but the two have to be rotated against one another by 36°. And if we do that, then in between we can see there’s a triangle strip,

with triangles going up–down, up–down, up–down… essentially forming a five-sided anti-prism.

Adding ten more triangles, it’s kind of beautiful that indeed this surface does close and result in number three platonic solid. [BRADY]: And what about six triangles? [PROF SEQUIN]: Oh, good question! Glad you asked that. So, let’s try to take six triangles, and put them together, and we get a wonderful nice hexagon, here. So, unfortunately, it’s totally flat. And so this doesn’t really curve and it doesn’t

help you to make some kind of closed object. You can make a kitchen floor with that one. Now if we go to seven triangles,

they don’t even fit into the plane – it’s kind of get the warped chip. And that is no good.

So we’re done with triangles. For now we have to go to squares. Fold them up, so basically three squares around the shared vertex leaves off an opening of 90 degrees. And we forcefully try to close this opening. And now we have a valid 3D corner. If you take a contraption like that, and match it on the backside with exactly the same contraption, then you get a cube made out of six squares. Now I know what you’re gonna ask next… ‘What about four squares?’ Well… glad you asked. —Same problem… —Same problem here. So we can put four squares together — and they make either a complete flat. Or if we were to somehow bend it in weird ways, then (it’ll) unknot all the dihedral angles

and all the edges would be the same; and so it not would be a regular corner — that’s unuseful. So we’re done with squares. We can put three pentagons together,

and form a nice 3D corner. And again the question arises: does that really close if we continue the process? And the easiest way to see it is to take one pentagon and put five pentagons around it, completing five of these corners. And this makes this nice kind of salad bowl. And now if we put another salad bowl like that

on top of it, — again rotate by 36 degrees so it fits nicely together — then, indeed, we can convince ourselves very quickly that that leads to yet another regular platonic solid. Trying to put four of these pentagons around the corner exceeds a total of 360 degrees, so you already get something that’s

warped like a potato chip — and it’s no good. We’re done with pentagons! Hexagons!

—Hexagons! —You know what happens with three hexagons,

of course: three flat tiling. That’s no good. What about heptagons? Well, they don’t even verily fit into a plane. It’s too much. So, from there on up, we’re sort of done. Nothing new can happen. Now you know exactly why there are five platonic solids — and exactly five platonic solids. —Professor, is a sphere a platonic solid? —No, it isn’t. Because even if you think about a very fine tessellation, you could not find a tessellation on a sphere where all the faces would be exactly the same, and all the edges are all the same,

and all the vertices are the same… So we’re basically done with three dimensions… And so, I must admit that for many of us

there is indeed a sixth platonic solid. And it is … the Utah teapot! For the people in the computer-graphics community,

this is such a famous object which has been used again and again to test our rendering software and solid-modeling software. And it just shows up… it’s just in about every single graphic paper in some form or another. So we like to think of it as the sixth platonic solid. Now these regular polytopes are made out

of regular polytopes one dimension lower. In the same way that we made platonic solids

out of 2D regular polygons, we can now try to make four-dimensional regular polytopes out of three-dimensional platonic solids. Each one of these four-dimensional regular polytopes is of an object that has, as a surface, a thick crust of three-dimensional platonic solids. And just like we made a cube out of squares, we’re now trying to make a hypercube out of cubes in its surface. We used to take three squares, and then forcefully

try to close this gap of ninety degrees. And by closing it, we force this corner

to pop out of two dimension — become a three-dimensional valid polyhedral corner. And that’s the corner of a cube. Now, we’re doing exactly the same thing, except — think that behind each one of these squares, there is a complete cube. And these three cubes now are sharing a joint edge, perpendicular to this plane in that particular corner. So there’s a ninety degree wedge of open space and we’re trying to forcefully close that. And we cannot close that in three dimensions if there are cubes behind each one of these four visible squares. But trust me, if we really do that and we can pop out

in the fourth dimension, we can close up, you know, these cubes — and we get a hypercube. [BRADY]: Those cubes behind these that I was being asked to imagine… they’ve kind of overlapped with each other now… [PROF SEQUIN]: No, because they pop out into four dimensional space, in the same way that when we took these two dimensional squares — and we force them to close. They didn’t overlap; they just popped out

of two dimensions into three dimensions. And four-dimensional space is really much

bigger than three dimensional space, so there’s ample room for these cubes to simply

pop out and form a true four dimensional corner. [BRADY]: But to my three-dimensional eyes,

he is sitting in 3D space… I’m like… how did that happen? But if I had four-dimensional eyes, I’d be thinking… [PROF SEQUIN]: That’s correct. And now you’re hitting on the real problem. So we can figure out that this must be possible,

but then… can we visualize that? I mean, how do we know what these thing looks like…? Well… for that, we have to use some kind of a shadow,

or a projection of a four-dimensional object down into three-dimensional space. And I’m sort of torn between, you know, should we use a shadow or should we use what I would call a ‘wireframe’? And you see… the trade-off between those two options. Just like if I showed you the wireframe

of the dodecahedron — you can really see through it, and it tells you much more than if you just had a faceless shadow of this object. You can eventually start to get a feeling

by looking at three-dimensional wireframes or projection of such wireframes — what the four-dimensional object might indeed look like. And — doing this projection — we still have an option. We can project in various different ways. We’re going from a two-dimensional square

to a three-dimensional object. And in an oblique projection, I start with the red square in front and then the blue square is the ones off in the back, and the green lines essentially show me the depth. We can do the same thing by starting with a complete cube — the red cube is in front the blue cube is in the back — and then the green lines show essentially the extrusion of this cube in this oblique direction. Alternatively, we can use a perspective projection

— everything in the back would appear smaller. And we can look straight at the face. And by doing that, you know, the back face will be a smaller square (that’s off) really behind the red square; and the green lines show kind of the depth

going from the front to the back. And the same we can do with the complete cube. And we get, you know, in the back

a smaller cube shown in blue, and then the edges that go from

the front to the back are shown in green. So here is the oblique projection. You can see the fatter cubes being the ones in front,

and then the thinner cube in the back… and then the slightly conical edges

leading from the front to the back. So that’s a valid depiction of

a wireframe of a hypercube. And here is the alternative model —

a perspective projection. So the bluish cube is the front cube, and the yellow cube is the one in the back — which is here much smaller. The red edges that go from the front to the back. And this hypercube has a total of eight cubes: the front cube in blue, the back cube in yellow, and then some squashed cubes

here showing one on each of six faces. So there’s a total of eight cubes making the surface

— or the thick crust — of the hypercube. [BRADY]: So that yellow cube isn’t inside the blue cube… [PROF SEQUIN]: No. In four dimensions, it would be up

in the fourth dimension (at) a certain distance. Because of perspective projection, what’s

further up there gets projected into the back… and appears smaller in this projection. Now, we’re going to systematically look at all

the platonic solids; see how many we can group around the shared edge, how much empty space there is, and then we push that one out and push that into a valid 4D corner. The simplest object to start with is the tetrahedron. The tetrahedron has a dihedral angle of 70 and a half degrees. We need at least three of these tetrahedrons

around the edge to make a valid corner. Three, of course, fits very easily. Four fits. Even five fits, but just barely:

we just have a few degrees left. And so, when we try to fit five tetrahedrons around this, we just get a little bit bending in four-dimensional space. And we will need — as we will see — a whole lot

of these tetrahedra (to) make it actually work out. So we can start out with three tetrahedra and ask what happens if we forcefully, you know,

bend that into a corner. Can we repeat it? And the answer is we will get the ‘simplex’,

or the ‘5 Cell’. So, because of the projection, you know,

it becomes just a three-dimensional object; and it loses some of its symmetry. So we have five vertices, but the fifth one, — to me, to make it asymmetric as possible — I put it here right in the middle. So we get the outer tetrahedron, and then we get the temples here on each side. They all represent additional four tetrahedra. And they don’t look very regular,

and that’s because of the projection. But, in four dimensional space, all five of these tetrahedra are completely regular. And so that’s the 5 Cell or the simplex. That’s what happens when you take

three tetrahedra around a shared edge and essentially force out the empty space

and bend it into a true four-dimensional corner. Four tetrahedra around the corner,

you know, we get this object here, okay? It’s called the ‘cross-polytope’. And it’s actually the dual of the hypercube

that we’ve seen before, okay?… So that’s the cross-polytope —

the second regular polytope in four dimensions. Five tetrahedra put around a shared edge —

you get something that has a lot of tetrahedra. As a matter of fact, this thing has 600 tetrahedra. Most of them are so crunched up in

the middle here because of the projection… and you just see one giant tetrahedra on the outside. And then, adjacent to each faces, you see smaller…

fairly much flattened-out tetrahedra… And you have to just believe me that, in there, there are essentially — in addition to this one out here — another 599 tetrahedra. This is called the ‘600 Cell’. Six tetrahedra would exceed the dihedral

angle of three-hundred sixty degrees… And that’s like, you know, something warped

that, uh… is not regular. So that’s of no use… We have seen cubes; they have dihedral angles of 90°. Three of them will fit around an axis till it’ll leave an open gap … and then we fold this one up — we’ve seen we get the hypercube. If we try to put four cubes around the joint axis

— you can readily visualize — that would readily fill the space without any bending. And so we can tile all of three-dimensional space

with that, but it will never bend and make a regular polytope in a higher dimension. So we’re done with the cubes. The next platonic solid you may want to try is the octahedron. It has a dihedral angle of 109 and a half degrees. So that’s less than a hundred and twenty degrees here. And that means we can indeed

fit three around a joint edge, and still leave a little bit open space, which we can squeeze out and make this then pop into the fourth dimension. And the result is what I believe to be the most

beautiful four-dimensional regular polytope: the ’24 Cell’, made out of twenty-four octahedra. You can see the outermost octahedron, because

I chose that to be, you know, preserved in the projection. But, then, you see these rather flattened octahedra:

this one triangle here… another triangle, this small one here…, and sort of a… distorted antiprism in between… and then there are more on the inside. I really like this particular object the most,

because it’s not too complex — you can still see what’s going on,

you can still look in the inside, see the innermost, very tiny little octahedron at the center. Now, it turns out this object has 1152 symmetries in four-dimensional space. And then you simply get that by multiplying the symmetries of an octahedron — which are 48 — with the number of octahedra — twenty-four. You multiply this out — you get to 1152. Because you can take — in four-dimensional space — any one of the octahedron and put it in the place — by suitable rotation — of any one of the other 24 octahedra, in any one of the 48 possible positions. I live at the street address 11-52. And then I found out that this object has 1152 symmetries. I thought that somehow fate really meant

me to be, you know, into geometry. The next one on the list is the dodecahedron, with a 116 and a half degrees of a dihedral angle. That is still less than 120 degrees,

so we can still force three of them around an edge and then join it into a valid four-dimensional corner. It’s not bending all that much, And so we will need a hundred and twenty

of those objects in the crust to form a valid four-dimensional regular polytope. So this is a model of the 120 Cell. Of course you can see the outermost dodecahedron. You can see kind of these flattened pancake-like,

you know, dodecahedra on top here. And each one of the faces has one of those. And then stacked on the inside there are a few more, adding up to a total of a hundred and twenty of those. We still have the icosahedron. Unfortunately it has a very shallow dihedral angle. It is more than 120 degrees; it is actually 138 degrees, roughly. And so even three of them would not fit around this edge without overlapping. And so we cannot form a valid corner. And, unfortunately, this is useless, as far as making a four-dimensional regular polytope. You basically can figure it out yourself, right? You know how to make five-dimensional regular polytopes. You look at the four-dimensional regular polytopes, look at their dihedral angles, and figure out, you know, can I fit at least three of them around

an edge so I can make a valid corner? Most of these regular polytopes in four dimensions are very round, and they’re not very useful There are really only two of them that actually have some hope of generating a new regular polytope in higher dimension. One of them is the simplex, or the 5 Cell; and the other one is the hypercube. [Brady]: The other ones have got too shallow … ? [PROF SEQUIN]: Yeah, the other ones have too shallow of dihedral angles, and they just don’t make valid corners in the next higher dimension. And so, from five dimensions onwards, there are really only just three regular polytopes in each one of these dimensions. One is the simplex series — you can always make a simplex. By taking a simplex in a particular dimension

— I’m starting with the tetrahedron — and then I put another vertex at the center of gravity. And now I use the fourth dimension, and essentially raise this vertex up in the fourth dimension until it has exactly the same distance from all the other four vertices. That makes a new simplex. And then I can take that particular simplex, put a new vertex at its center, raise it up in the next higher dimension until it is exactly the same distance from all the previously existing vertices and make the next simplex. That always works. So I can work my way up to infinity by always

making an additional simplex in exactly that manner. The problem now is how do we depict those things. ‘Cause clearly projecting it down — you could do anything. So, it’s better to figure out how do we make a model

that has the right connectivity, even though the geometry really sort of bogus at this point because it’s so much distorted from what we originally have. And I have make a few more simplices in higher dimensions. You just need to always add one more vertex,

and then figure out how to make a nice little graph that will get the right connectivity. So, this one has six vertices.

I claim this is a projection of the 5D simplex. ‘Cause the simplex also is the complete graph; every vertex is connected directly to every other vertex. And you can see in this case — you start at any one vertex, there’re five edges going off, one to each of the other vertices. So we want to make a 3D model, that has six corners, is reasonably symmetric, but not too symmetric. So we could start with an octahedron. But in the octahedron then, when you connect opposite vertices, those three edges will all intersect in the middle. That would be so nice. You wouldn’t see what’s going on. So you’d have to deliberately distort this octahedron, warp it a little bit, so that those three spaced diagonals that go through the middle of the regular octahedron — they have separated out. And now they

do not intersect with each other anymore. So the trick was really just finding six vertices

in a relatively symmetrical arrangement, so we can connect every one with

every other one without any intersections. And that’s the game that we have to play for every one of the dimensions that we want to build a simplex model. We already had this object that had

six vertices completely connected — the warped octahedron — and that actually

has an extra free space in the center. It’s so easy therefore to put a seventh

vertex right here in the center. This one can again be connected to all the other vertices. And so now we have a complete graph of seven vertices. And that would be a nice model of a six-dimensional simplex. There is a second sequence that works for all dimensions. And it’s called the ‘measure polytope’ or ‘hypercube sequence’. The cube is the object with which we measure

the volume of three-dimensional space. Just like the square is the geometrical element

that we use to measure two-dimensional space. Similarly, the hypercube would

measure four-dimensional space. So the measure polytopes are always the equivalent

of a cube in the higher dimensions. And the best way of visualizing what’s going on

is essentially going in stepwise extrusions. So, we start with a line. And then we take this line,

and extrude it basically into a square. And then we sort of take that square and extrude it perpendicular to itself into a cube. And then, we’re doing one more extrusion

of that cube into a hypercube, to an extrusion of the hypercube into the 5D measure polytope. And you could continue that way. Or you can change the scale. So, what I prefer — maybe in six dimensions — would be to take a regular cube and essentially sweep that in three different dimensions to make yourself a thickened cube frames which is showing every edge as the

‘sweep path’ of one of those cubes. So this particular diagram would be

a depiction of a sixth-dimensional hypercube, which you get by extrusion. Another projection of the sixth-dimensional hypercube results in this ‘Rhombic Triacontahedron’, which has thirty rhombic faces on the outside. But, on the inside, there’s a whole lot of intersecting edges going on. And you may not like that. So it all depends on what you really

want to get out of your model. In some sense, this makes a very nice

sort of climbing structure for children. [Brady]: That’s not very cuby though, is it? It’s hard to see the cube. [PROF SEQUIN]: Yeah, for some angles, if you look at… here it looks like an oblique kind of squashed cube that starts with one of the cube face. Or a very flattened cube, you know,

we have just like three faces here. And then on the inside there will be another three flat faces, but they of course already intersect with

edges from some of the other cubes. —We can go cubes all the way to infinity.

—That is correct. So, you know, we started out with five platonic solids. We went to the six regular polytopes in four dimensions. And after that I said 3, 3, 3, 3, 3… I have shown you two of those three series; the simplex series, and then

the measure polytopes series. So there must be another one. We’ve seen that we can make a series of measure polytopes through all the dimensions. In each one of these dimensions we can also form the ‘dual’. Just like in three dimensions we have the cube,

and then we have each dual which is the octahedron. By holding it that way, you can see the top and bottom vertex in my right hand correspond to the top and bottom face

in this particular cube. For each of the six faces in the cube,

I now have a vertex. And for each of the vertices in the cube,

I now have a face. [BRADY]: Like its evil twin…

[PROF SEQUIN]: Yeah. Oh, nice twins actually… [BRADY]: Yeah. [PROF SEQUIN]: So they’re duals of one another. We also have the models that show the same relationship — not quite as obvious — in 4 dimensions. So here we have the measure polytope — the hypercube. And, if we take the eight cubes in the crust of

the hypercube, and replace them by eight vertices, we get the corresponding measure polytope, which is the dual to this hypercube. [BRADY]: Evil twin! [PROF SEQUIN]: And the same principle goes on. It gets, you know, near impossible to kind of visualize in higher dimensions, but again, every measure polytope in d-dimension is made out of measure polytopes of d-minus-one-dimension. You replace each one of those cells with a vertex at its center, and connect them properly, and you get the ‘cross polytope’, which is the dual to the measure polytope in any one of these dimensions. And that’s the third series that we can form. Um, here is again our sequence. You know, we have infinitely many polygons in two dimensions. We have five platonic solids in three dimensions. Six polytopes in four dimensions. And from there on, it’s just always

only three regular polytopes. So, in conclusion, I think I can say we’re sort of lucky that we live in one of the two dimensions where it’s really interesting — and we have a variety of different regular polytopes. [BRADY]: We don’t live in the most interesting dimension though. We live in the second most interesting… [PROF SEQUIN]: Well, if you think that we live in space-time continuum, which is four-dimensional, so we live in that one too. So, what, I’m inside, a four-dimensional polytope right now? Yes, you’re inside the 120 cell space is entirely filled with dodecahedra because in this spherical space, dodecahedra tiled space. It’s non-Euclidean…

Maybe I missed it, but where was the 16-cell, the sixth 4D-polytope?

This is the best explanation of a dmt experience ever.

š

Wait, arenāt those still all 3D shapes. Really want to understand other dimensions but havenāt yet understood where this other dimension comes from or how itās measured.

With the triangles, each of those steps changes the process of closure to become the given solid. š¤

Are there 11 other pentagons? yikes town.

Will artificially intelligence be bound by our 3d brains? Will Brain interfacing allow us to experience higher dimensions?

Is there a way to purchase these models or do I need to find a 3D printer?

What

You lost me at 5

want to see in real life have some dmt case closed hahaahahaa

Niki Lauda meets Menotti

icosahedron has no polytope š so sad

Thank you so much. very interesting

OMG I SAID 3!!! YAYYYYYY

That other guy sounds like maxmoefoe.

Glad you made the video.

2 + 2 = 3 (extra 1 popped out in 4d)

Just kidding.. š

All of this is as useless as it is man made.

im from a diamension so high that scientists are still trying to figure out what you would consider what diamension we live in would be

Actually, as we, 3D creatures, see infinite polygons in 2 Dimension, a 4D creature actually sees infinite platonic solids in 3 Dimension, and we can only see 5 of them. Now think about 6D creatures and so on.

Me encantĆ³ la traducciĆ³n de youtube xd

This is too much! I came home drunk, ate leftover pizza and thought dimensions were pretty cool… Now I have a polytope on order from Amazon and I need a new mind cause mines blown

One of my greatest hopes for future technology is the ability to truly visualize 4D space and shapes

amazing Brazilian following up

Feel Better

Minecraft Beacon is a HyperCube

I have self destructive personality…. This video burns my nervs but i watched at least 8 times as of now

4d people are drawing in 3d right now

So cool. many questions arise.

If you want to visualise the 4th dimension, inhale vaporized N,N DMT crystals

č·é»ē²åē ²ęć”ćć

3d boring. We were 5th dimensional beings, but due to the powers that be, through indoctrination, foods, liquids, frequencies most are stuck in the 3rd. Alignment of the chakras, raising our kundalini to awaken the 3rd š puts you in the 5th and you will see the truth of everything that is really here. The earth is rising so most peopleās vibration will get higher. Thereās 11 dimensions, but most wont get anywhere near it. Donāt put restrictions on yourself.

1:35 these are dices for D&D

Can someone answer this question for me?

Do objects that exist within the presumed 11 dimensions have to exist in all 11? Like, does the human body extent into higher dimensions without us realising OR can objects exist up to 3D/6D/7D etc.

Also, are the shapes he is holding in the video bending into 4D space or are they trapped in 3D?

24:30 A perfectly square monitor. What else would he use?

7:05 Go Utah!

Is this the thing from neon Genesis evangelion?

The 3D Platonic solids are all the dnd dice except for the d10

24:00 hellraiser is real…

You didnt say how many symmetries the 120 cell had!! WHYYY!! T>T I say it has 28800 symmetries. Thought? o>O

I clicked because the thumbnail reminds me of the monstrous geometric angels in Evangelion lol. [screams geometrically]

yikes

Came to learn about hyperdimensional geometry, stayed for the Swiss accent.

What happens at 10 dimensions?

Waiting for when we can get Dungeons and Dragons-style dice sets modeled after each of those wireframesā

7:24 Here begins the best explanation to apprehend tesseract.

so swiss

@numberphile Could a sphere be thought as a polytope at a infinite-Dimension? Since the solids on the faces of the polytopes are getting smaller and flatter at higher dimensions.

3D space just ripped off D&D dice

i'm still confused about this dimension stuffs.. is this the same "dimension" as in string theory?

So we are looking to 2d projection of 3d projection of 4d projection of 5d thing

Whatās the point to all these shapes?

Thou all those geometrical objects are still in 3 dimension. What should we really see in here?

It would have been more fun if he'd started the dimension count with 0, 1, 2

giving an even more strange result sequence of : 0, 0, infinity, 5, 6, 3, 3, 3…

Geometry is sacred, and is a part of everyone's lives .And if anyone on our planet truly understood, how, what and why 3, 6, 9 is unsolvable .. is the key ….hurry up !

I would love to see and hear you explain with your models, how to square a circle or cube a sphere please and thank you, outstanding videos by the way….

the thumbnail looks like a playstation 2 cover

Does this mean the 4th dimension is essentially "wherever you look"?

If you were doing these higher demons extrusion and projections in their respective dimensions, they would match perfectly without the need to force the shape…

Bollocks its all 3d

Close your eyes and listen closely and he sounds like a robot.

You shall not pass,… into the 4th dimension.

It's important to know that these 3D printed objects are not in fact 4 dimensional objects. They are ony depictions of them in 3D space, like drawing a cube on paper isn't really a cube, just a depiction of one in 2D space. We cannot imagine 4 dimensional space no matter how much we try, just like a blind person could never think of what colors look like, or what sight itself is like. There are also colors that exist that we cannot see, and this is the same as what is happening here. It's fairly obvious, just wanted to point it out.

Why am I watching a 20+ minute video

7:17 find the 4d corners page on his computers

That scot accent makes math a lot cooler.

3d cube reminds of the art class making the cast of the cube was so difficult for me i guess i couldn't illustrate that in my mind because i had to think in different dimension which of 4d it makes so much sense to me know

4th dimensional beings:

"_

Pathetic_"What if reality is in 4D, we can only have perception of 3D ?

Thatās numberwang!

Awesome video! It was explained very clearly about Platonic figures in 4D and 5D!

I didn't get it. I will stick with the 3D space.

watch it in slower speed 0.75 sounds easier to understand

When thanos get the reality stone he probably changed a triangle to a circle the same way you show at 1:56

And also ahmm….what mind fucker is this? 11:11

I would love to be able to see in 4 dimensions. Even for only a few moments. So I could atleast have a mental picture of what it looks like.

What if 4D doesn't exist, because of fibonacci?

Ok then… that should put the crop circle messages to bed … most of them really depict the shapes in the presentation, the two guys with the ropes and boards are from another dimension, and the Vatican has some explaining to do now, let's ask the pope shall we, after all he is the ambassador to the higher realm is he not?

Professor X in the building

16:00 RAMIEL

i wonder if they 3d printed the shapes

I wonder if there is only one number 4. I guess there are more than one but all the others are hiding behind one. From time to time they change positions so that not always the same 4 is "visible", but nobody can see that. So everyone is thinking that there is only one number 4 where there are many.

We think we have real 3 dimensional vision, but it is a combination of two 2 dimensional pictures with a slightly different perspective. For that we need two eyes plus some properly grown and connected neurons. (binocular vision is not only a matter of having two eyes).

This guy is definitely multidimensional! Yup! I know!

I wish this guy was my teacher, I might have done more then deliver pizza.

=_= I'm tired

Excuse me what did I just watch

Apeirogon

5 6 3 3 3 3

apparently there are a bunch of scientists and almost 3 million idiots practicing absurdity by trying to perceive what can never be perceived

I wonder how we will ever be able to see and/or interact with higher dimensions. If itās even possible.

I wonder if life could evolve in higher dimensions.

I guess force act in that pattern and something else work on the structure of 10 das the shape of chakras

Force act in odd structure pattern Nd something act in even structure patterns

the weird thing is in this projections like siplex all the 3d shapes is the same size like i can uderstand that but cant imagine it wihout it intersecting that's the magic of 4d

wait what, sorry i wasn't listening

TL;DR: There is no such thing as 4-D space, since a 2-D object can be folded into 3-D, but a 4-D object must be imagined.

LOOK AT THIS DUDE!!!

I feel dumber already.

This reminds me of the third angel in Neon Genesis Evangelion, well at least the Rebuild version where it changes forms to fire the laser, in some of them it seems like a 26 cell. So knowing Anno I think the purpose was exactly that, that this weird being is actually a 4 dimension (space dimensions of course) thing that we only see its projection.

23:02 This is the ideal 6D polytope. You may not like it, but this is what peak geometry looks like.

One platonic solid is missed here. Nothing is straight, no line is straight, light is bent etc.

This guy is so easy to listen to and explains things in such an engaging, easy to understand way. I have to admit, I literally don't know much past the basic 3D shapes, but the fact that he seems so happy to explain makes me want to enter the 4th and 5th dimensions.