Dimensions | Chapter 3

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Dimensions | Chapter 3
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École Normale Supérieure de Lyon (ENS-Lyon)

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my name is Ludwig Schläfli i am a swiss geometer
I lived in the 19th century I'm going to open the door to the 4th dimension the even if I say
so myself I was a visionary I was 1 of the very best
to understand but spaces
with a high number of dimensions really exist the and the geometry could be and believe that creatures it is living in a plane can understand three-dimensional and why should shouldn't we understand fully trained but 1 of my main achievements was to describe all give fully dimensions what is the 4th dimension a lot has been written on the take science fiction lots never talk to dear I'm going to explain things on the blackboard he will see the black board has magic capacity what's important is prepare yourself forget about the world which is familiar to and to imagine a new world not since have direct access will have to be smart just like the lizards work I'm going to climb on top of the viewpoint that unfortunately you cannot see and I'll try to describe what policy from there but before we begin on draw a straight line on the bold let me just mark the origin here each point on this line Camila communicated by its distance from the origin with a minus sign if it is on the left and a plus sign if it is on the rise usually the number is denoted by 6 underscore the abscissa since the position of a point on a line can be described by a single number we save line has the mentioned 1 now I draw a 2nd axes perpendicular to the first one 8 point in the blackboard playing is now completely described 2 numbers usually denoted Banks and he had to and the orders the plane has dimension to if you had to explain to some being living on a line what it is to be appointed the plane is unknown to him you could simply a point in the playing is just a pair of numbers let's go into the 3rd dimension the chalk no rights in the year and draws the 3rd axis perpendicular to the 2 previous ones a point in space is described by 3 numbers X said 1 could say to the red reptiles that security to know about how a point in space is just 3 numbers let's go to the 4th dimension 1 could try and draw for faxes perpendicular to the others but that's impossible so we have to do something else instead of course we might just a point in the 4th dimension is nothing other than for numbers of fixed fly that doesn't help us a lot in spite of the difficulties we are going to try and get a feeling for this geometry as the 1st attempt at understanding we should proceed by analogy here's a statement and an equilateral triangle look and finally a regular train Yukon look on magical blackboard enables us to draw in space but How come we keep this up in 4 dimensions observed the the triangle a tetrahedron have 2 3 and 4 vertices respectively therefore we can try to continue with 5 vertices let's go for the segment the triangle the patron each pair of vertices so we have to connect the fire vertices impaired we count 1 edge 2 3 tho 6 7 knowing and 10 In the Tetra he drew and there is a triangular face a
trickle of voters is we proceed in the same way which
gives us 2 3 10
faces but if we keep
going by analogy we have to either Tetra he face for each forethoughtful vertices verify them that's it we
constructed of four-dimensional object With college the simplex you let's spin it around in space as we did with the federal patron Of course you have to to imagine the simplex spinning in a four-dimensional space what you see His only projection on the blackboard such complicated Is faces get tangled undertaken cross each other well some experience is required to
see if we're going to take the a simplex which is in four-dimensional space and moving gradually so that different cross sections of meat Our three-dimensional space in the same way that reptiles could see a polygon appearing and disappearing we will see a three-dimensional folly which appears changes shape and then vanished is simplex passing through our three-dimensional space we're now going to make the ordination of Polly he drew passing through our own three-dimensional here is the whole attitude a member of the family that starts with the segment and continues through the square and the cube please it has to be said that getting a feeling for the geometry of the slice method like it's rather tricky I discovered the analogs of the Costa he joined the dough he tried they have complicated pathologist them 120 sale 600 since the former has a 120 faces basis and the letter 600 look at the 120 self the it's just passing through our space Gore the and now he's 600 Of course when I say the four-dimensional pulling heated has 600 faces I mean three-dimensional these 600 faces a 600 as for the 100 20 sell it consists of a 120 a day In a minute will see how we can get to know them better to observe these full dimensional objects we three-dimensional we can look at bay shadows the objects as demand space but their projected on 3 space exactly like a painter much projector landscape on campus we've already done just days with the simplex the here is the hypercube the Of course spinning in space so that we can appreciate all the details notice for instance the hypercube has 16 vertices all Lucy he is a little newcomer the it's the most beautiful of my discoveries an object that cycle with 24 cell it has absolutely no analog in dimensions rate is a purely four-dimensional creature so I'm very proud of my discovery look how wonderful it is 24 96 edges 96 triangles and 24 octahedron realist GM and here is the shadow of the 120 In all its majesty a rather complicated majesty you have to agree the yeah the there let's get inside and have a structure the and now 600 to 1 thousand edges start a teacher but completely regular structure although it is only to play the same role but it's a pity that projection pranksters symmetry that's work your imagination of imagine object to in which a huge group of rotations meets all these vertices and edges
champion is the 600 self like a gigantic macromolecules With its 700 128 years and 120 vertices and 1280 starting from each you both it Our exploration of four-dimensional poorly he drew went stop here as this demographic projections bound to give us a better feeling for the geometry do you know do not
need to be on the way you see EU ahead he has do you think about too