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Perfect Shapes in Higher Dimensions - Numberphile

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00:00   |   Mar 23, 2016

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Perfect Shapes in Higher Dimensions - Numberphile
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  • [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...

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Description

Carlo Sequin talks through platonic solids and regular polytopes in higher dimensions.
More links & stuff in full description below ↓↓↓

Extra footage (Hypernom): /watch?v=unC0Y3kv0Yk
More videos with with Carlo: http://bit.ly/carlo_videos

Edit and animation by Pete McPartlan
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