Dimensions | Chapter 2

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Dimensions | Chapter 2
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École Normale Supérieure de Lyon (ENS-Lyon)

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now it's
my turn to show you around the geometry garden my name is Escher and
I was a Dutch artist in the 20th century geometry was a constant source of inspiration for me on a past master in the art lot of drawing fantastic finding look at this self-portrait in spherical mirror 1 of the most famous drawings shares lizards drawn on plane but managed to break out of the now perched on a contemplated previous existence as well to prepare ourselves for life in 4 dimensions we're going to use the ideas behind but buying and the little book published at the end of the 19th century Mike 8 an English clergyman cold Vice plans let's try and explain to
these beings these creatures living track forever in
applying what our everyday life in
3 dimensions is they In let's imagine 1 of these manages to escape his miserable existence for a moment the client out onto a viewpoint looking down on his How would he explained his fellow reptiles the existence of objects in 3 dimensions show as a 1st attempt he could try to pass on three-dimensional objects through his flat will here for example is
attaching region with its full faces passing through the visits plane but the flat creatures sea-green triangle appear suddenly gradually shrink away this is all they see is that sensors are entirely restricted cannot see anything outside of that playing
each time the lizard sees these Green politicians appear change the disappear again he might imagine the form of the object that is just crossed his place To see how hard it is to visualize the full of geometric body from his cross sections try to guess what is crossing the street the plane now a did
no it was a Q and Of course you have to remember that these don't have the same perspective that we have all they a sequence of public and they'll have to develop an understanding of debt In order to fully appreciate the trade of the body and now what however an octahedron would take place and you now by Cathedral it's a solid with 20 and finally the 3rd aikido 12 faces 23rd and 38 yards now we're going to show you some across sections sections and you have to guess the ploy he and hiding behind that walls attach any drug that walls you it's getting holidays is now we see now that these creatures sucking 2 dimensions have to develop good geometric intuition if they want to understand something the 3rd dimension that we take for granted we are just the same kind of problems to get a feeling for the the 4th dimension here's a 2nd method to explain body drift off my friends stopped by inflating a body heat the disease and EKG on the set of his the the exterior graphically project onto the plane of the lizards so that our two-dimensional friends mangily spectacle Of course we can spin this around and with it or to trade just as we did before With a lot of them
let's take a look at the queue and see how many the disease a cheese and faces the and now here comes on
hatred a receive covered faces look how the projections of EAG of such at the end of the year now here comes a lot closer hatred only it structure is more complicated but it's not hard to understand the visits we can see 20 faces 12 deceased 38 she can you capital finally he is a little geometric jewel the dead a decade dress a lot of trouble with Henry around noon on my own lock on the block now for some exercises but let's take ourselves down 2 dimensions and try to recognize the body from history graphic projections it's easy Islam you could see the full faces 6 8 years and for their disease but but its attention he the get
but but it what's this 1 6 places each with Florry that's right it's key do and dead the but that was part
wasn't it here the faces a triangles but find AG start at each vertex don't there are a lot of places perhaps 20 but it's a Michaels a hatred well now Ms.
looked at the dough dirty trick each face is a Pentagon if we count them there 12 faces 3 pages of the attacks I do these fights on its have always fascinated Jonathan decreed that also
attributed the magically importance to them by
associating to each of them 1 of the fundamental elements from which the world is full we call these figures the Platonic solids so we agree then it's not easy to get a feeling for the 3rd dimension when your flat there is more than 1 way to do this austerity graphic projections that is to give a good idea of what's going on now we have to get ourselves ready for the full dimensions but we're gonna have to to use our imagination a but the but there was
only 1 in the they have uh a hole