Dimensions | Chapter 7

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Dimensions | Chapter 7
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École Normale Supérieure de Lyon (ENS-Lyon)

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is it he circles in space better
arranged so as to create beautiful ornaments in order to understand better the three-dimensional sphere four-dimensional space I will show you have space With and thus create what mathematicians call a fire brochure gate by the way my name is Heinz Hall and I am 1 of the main contributors to the development of topology but during the 1st half of the 20th century look at least face filled with circles that appear to be linked let me explain this picture to you circles spheres until among the simplest objects studied by poverty but topologies tries to understand the connections between these objects I worked in Berlin Princeton and
Zurich come-ons still comes across 1 nite often in contemporary mathematics conqueror hoped Hopkins variants hope algebra cops vibration buyer let me paint my portrait for you 10 but published the discovery of my vibration In 1951 but as a waste I have to say that are not on many predecessors like Clifford for instance you see here and who worked in English during the 19th century slap world let's begin with some explanations on the blackboard well a white gold this
time what do you see a two-dimensional playing well yes and no this is indeed a two-dimensional plane but it is
a plaintiff complex dimension or another a space with real dimension don't make an at each point in his playing is determined by 2 coordinates but each of these 2 coordinates it is a complex number which remember itself defined by 2 real numbers each EXEC is a complex line so that each point only has 1 coordinator which is a complex number for instance Hey you see the point to minus ally on the 1st access earlier the same is true for the other acts the Y axis here we can see the . 1 1 2 only sex now what borders magical but not enough to be able to show us the 2 planes simultaneously if we try to depict them in three-dimensional space they will intersect along a line but
in four-dimensional space they
intersect only the origin
after all they taxis In what do
you say a so-called yes and now what you see or around what you should imagine if a set of points in four-dimensional space Dederick distance 1 from the origin In other words this is nothing out of the 3 but s for a world of course you need to have a little imagination let's try to see at least half this fate intersects the 1st acts a three-state intersects the 1st access in a set points on a which are distant 1 from the origin if you see the three-phase intersects the 1st Texas in circle the same is true for the 2nd axis which intersects the 3 space in a circle as well the Blue Circle that now what is true for the horizontal line a vertical line is equally true for all lines going through the origin Hey you can see with equation said to quitter minus 2 said but we could do the same with any 9 said to he quoted at a times said 1 for any complex number In this man the 3 four-dimensional space is filled with circles 1 for each complex line going through the origin in Upland of complex dimension but careful note In the picture you get the impression that the red circles intersect each other but this is not the case in the reality of the mentioned for London's only me to the origin of their intersections with the units don't intersected told in fact I was the 1 who discovered it decomposition of the sphere into circles and ever since it is known as the hope operation wife vibrations well you should think of the fibers of fabric we're going to look a all that using started graphic projection imagine that we project the threes From the North
Pole on the a tangent space but the South Pope which is three-dimensional space here is the projection of 1 of the circles which as we have
seen Is the intersection of 1 complex none of the 3 states but there are many such circles 1 for each complex line going through the origin each complex number we can consider the Lund said To be quit 8 times said 1 and its associated circle that various number or what would amount to the same thing letters were take its 9 in order to see how the circle changes noticed that sometimes the circle appears to be a straight line but this is simply because it passes through the North Pole evolved freeze let's look at 2 of the East Coast simultaneously in the lower left-hand corner they are the 2 leading complex points 1 reads of the green you can see the circles associated to the red and green points these 2 circles are linked together like to links of the chain it is impossible to separate them without breaking them for the summer let's consider 3 said look at the dance these training circle at all if you have now now let's take many more complex chosen randomly let's look at the old 1 the circle of space and none of the this is an example of the here a woman you have to hang around while on a quiet let's try to
understand this bed said returning to the bold for know we have a hope circle for each line each 1 of these the liquidation of the form is said to be quarter 8 times at what is a complex number the slope of the plant and is indicated by the Red point moving on the Green Line actually the vertical axis did not have such an equation but in case we may see that raising and don't forget that is a complex number the Green Line is complex none so it is a mail plane so summing up the complex lines that we're interested in a complete he described by a point on the Green Line and an additional point its Infiniti a a a a a morass
but we already so 1 a pointer to infinity to the complex we get the usual tourist once more this is a demographic projections the so so the complex lines its interest are described by points on the Yellow Sea the 2 dimensional space they so we have a circle for each point on the Tuesday but Sutcliffe of dimension 1 isn't it Foley said but the trees each point on three-state belongs to a single circles and therefore was funds a point on the to say In this way we get a projection from the 3 states To the complicated isn't it mathematicians say that above any point to the place too there is a fiber which uses circles want and the total space this vibration Is this and 3 I'm very proud of my for aggression all the more so because it has become a fundamental object into poverty uh you come down on the use of
a I