IC16: Around Wild Knots

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IC16: Around Wild Knots
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The process of elaboration of the material of the project Wild Knots, which include computer generated animations, still images and some interactive software, has been interesting from several angles: the selection of the subject, how to recreate an infinitely complicated object with a computer, which media should be used, and which is the audience they are directed to, for instance. It seems that these questions appear in any project of producing mathematical content for the general audience. In this short talk I would like toshare my experience around this project.
Area Mathematics Phase transition Knot Imaginary number
Mathematics Mathematics Circle Projective plane Knot Arrow of time Spacetime Circle Imaginary number Spacetime
Point (geometry) Block (periodic table) Multiplication sign Projective plane Knot Knot Line (geometry) Mereology Sphere Flow separation Numerical analysis Element (mathematics) Mathematics Many-sorted logic Circle Finite set Order (biology) Spacetime Diagram Circle Mathematician Diagram Spacetime
Point (geometry) Dot product Matching (graph theory) Order (biology) Matrix (mathematics) Mereology
Group action Functional (mathematics) Sine Surface Model theory Knot Line (geometry) Mereology Sphere Numerical analysis Theory Dimensional analysis Product (business) Hyperbolischer Raum Many-sorted logic Term (mathematics) Mathematician Spacetime
Group action Transformation (genetics) Surface Projective plane Model theory Knot Transformation (genetics) Sphere Frequency Different (Kate Ryan album) Phase transition Quilt Spacetime Spacetime
Network topology Phase transition Transformation (genetics) Sphere Sphere
Point (geometry) Reflection (mathematics) Infinity Right angle Mass Limit (category theory) Sphere Tangent Spacetime
Hadamard matrix Mereology Sphere Sequence Numerical analysis
Greatest element Mathematics Process (computing) Iteration Sphere Numerical analysis
Process (computing) Reflection (mathematics) Model theory Knot Similarity (geometry) Point cloud Point cloud Knot Volume (thermodynamics) Sphere Measurement Iteration Average Natural number Forest Square number Iteration Spacetime
Divisor Euler angles Tournament (medieval) Algebraic structure Point cloud Solid geometry Imaginary number Sphere Flow separation
Calculation Water vapor
Finitismus Knot Fraktalgeometrie Numerical analysis
Computer programming Knot Connected space
Mathematics Imaginary number
so the people of the area and that was the 1st of these this is very have to be to be here with the imaginative but that's fantastic in the and learn about the depraved because of from there and I think 2008 in a way and he told me about this huge phase
local from of communication and I get very excited and when after that have done similar things a I want to talk to you about the projects of why not and the side in realizing that everybody knows what all while not use so I will try to explain you in a little bit what it is and then show you some pictures and then more pictures and that's it so as we and their lives so the this any method and used to give month the Conference notes for general public conference so it's quite complicated so I we started like mathematical company conference a I don't know if you all know what a and not use in not this is not the thing that you have on your shoes it's a way to
put this circle inside this space and the image of the circle inside the space so there complicated
words like images and 3 D space the what I saved by the month is what everything has in mind you have to draw a circle in this space without but in your pencil all about well in the 4 they mention I'm in its has to be continuously and the other part is that the space that the space it depends on what I considered by to the space where the what I'm going to say is true or not so 1 can think that the they OK the entities they there I would have won the will leave on but for everything I say is true 1 has to consider the three-dimensional sphere that adding just 1 point at infinity does the donor after that the rest will not be it like you right it's so not whole how much not at the time that can be well it's quite complicated they have 2 examples they have weird names 9 uh the 35 of the 9 sort of 75 0 that been so and they not that's too much not that are not it's been a question for several years and coho not the whole and that can be and they they started to do diagrams of the notes another say it explain you what the while not is I need to use this all of the uh image that at some point it start to the they begin to study knots thinking that the elements of the real war the chemical elements with some kind of knots in there at the they realize that wasn't true but they started to make a periodic table of knowledge and they are an asset alphabetical began order that they maybe this drawings user and count the times that did not crosses with himself and in the interstellar projections In these awards all these notes the classification of not set of all the nodes that you can draw by hand using like really drawing isn't enough find a number of uh polygonal lines 1 contains some some of these here my by polygonal lines and if you use a finite number of them you obtain at not so they tried to make this huge classification of knots starting by the the idea that the Knot can be drawn by a finite number of segments if you cannot make these diagram for not you've been what is called a wide that's and if 1 tried to figure out what the white nodes will hold well I wise not looks like when or even doesn't have any idea what over over goes to the this is an example of of the wise notes you start with a rope maker as monetary for a lot and then mothers and animals and a matter this the seas of mathematician that try to that put another thing to together in 1 point like this discovered when you obtain and that you cannot draw with a finite number of lines but it that's that word
because a more sophisticated not cannot be
seen in the wide screen with black dots let me try to 10 of the song and this is a I think they had they 20 millions of points but the thing that was created in the match matrix of 2 thousand by 2 thousand well it did look like at and
not what it's a part of another not to has to we completed because in order to see the detail I have to put it out but 1 can see the repetition of these small the for announced here and here and here like it or not like a fractal and what I'm
going to say to you is a recipe to build this this stuff so actually drawing
a not is not so easy the story they're not a fantastic because 1 has this it they are nomadic function that allows us to make some numbers of in 1 side of the torus and and another the number of terms in the other side and for instance does when 1 gives the 3 for a lot with this not so difficult but I'm at the station can you imagine the parameterization of these 3 nodes this is the 40th not with 9 crosses the 144 not with 10 grasses & there 165 not within crusts but but that's that's a probe on there is a fantastic phrase software that allows to the build this surfaces as 3 models is could not but is not free will but you can the need for the for the i and it's fantastic you can you have all that but that that almost all the knots of to the then uh then crosses classification and you can do most of the stuff that put I will tell you there what I call the mask ingested B for a while it's mask is there's a mathematician of effective matematician that sort of the theory of Canaan groups and then in his book at some and some parts of the page 1 100 to 120 3 years ago remember exactly has 1 1 line saying how to build while not using the theory of the actions of gaining groups on the dimension and hyperbolic space or something and I learned that by a product of a some people in Mexico and that of course can really in a that they were trying to use this recipe not to make not so while noting the 3 space but to tie not fair and so it to not at 2 dimensional sphere in the 4th
phase this book played we're project but I learned from them this procedure and
that and at the end they will show you that these recipe also works for some other stuff that are not not it the and the key ingredients is the following you can it if you have a mathematical mind again consider the invention you you may remember the definition of invention of our with respective spheres if you're not yeah surely have seen this there spherical mirrors that you can see yourself inside that is model of all of the period is the deformed this effect but they built the Poynting Coleman of distal transformation is that the spheres are reflective or transform it into another spheres inside or outside the sphere the difference in the big difference is like in a mirror all the cities and other spaces reflects this brick printed on the surface of the sphere in the in mathematical invention
everything outside the sphere goes inside like a tribute Trevi you compute the keep keeps his tree
for shape for instance and the but anyhow if you have a sphere this phase it goes inside assimilative and that the key ingredient to build this a of these and must address the so let's just do all this share that we show you some pictures consider this fear and when 1 1 can a think that this is a 1 of these mirrors and you can see that and other spheres and as a parent once you have
2 of them you have some infinite
procedures you have the image of the rights feeling inside the left view and the image of the masses and so on it's easy to to to the thing that the only that their limit points of this considered all this sphere community in only 2 points in this space what happens if you make the still sphere tangent when the point in the in the sphere doesn't move In that doesn't always end with respect of any of those the reflections the and you can see this repetition they much
of this is that and as a the and that's the key
that's the key of of Muscat visited because what if we consider the 7 of these uh we consider some some number of these spheres and see what's in the inside for instance 1 can take this quarter made of spheres I think there 27 and we look at that as a as a mere mirrors and 1 can see for from from part of that inside each of these 2 spheres the rest of them that that instead of making this smaller necklaces smaller and that's the key however very early images is not what is in in this office is that sequence of of Hadamard there is a the the instead of inside this blue blue spheres has the images of all
the spheres in the corner what are the like this
and when the the the bottom of this construction is that the numbers sick of spheres grows what article if you step on with 30 that is the of the in the 1st column the next iteration we have 32 square minus 30 that's a large numbers and this at the next iteration of the process well you can do the math but there it's difficult for a computer to remember all this is on these spheres so it's a it's a nice trick the thing to use this really the user
similarity between reflections and inventions and what what 1 can put this ray-tracing problem this here in this in this is fit to produce a more complicated not however that the spheres in if you would use mirrors in in an empty space 1 can 1 have where what does and find that this is so to the measure would have to make some background to reflect on the sphere and feel that the has some volume arose on the in 3 D shape so we were uh we're wondering how to to make a clouds or something and I found out this very beautiful it proceeded to draw clouds I don't know the I think it's known consider Maddox like an image and select four quarters whatever you whichever you once and put it on the on the other corners of the of the picture then did that speed the square and take the middle points between the the vectorX of the of the square and sentiment on each side of the of the square you take that the the average of the corners you selected before and in the sense that you choose at random cornered from the from the colonists and repeats again here you think that the average from from these to these that there and select another random going the and to do it in the in the forest is worth and continue and this fantastic because we then approbate choose of course in the garden that like relax being rule and black or white white white and gray you can produce some very nice clouds that allows us to to make this 3 D models for the for the and it wasn't so I will show you 1 of these it's all the same picture as before but with with volume and you can see that the image of the necklace original necklace at some in the idiocy that the reflections of the race but to except on this is the of every move the the last year just to see the that the reflection of the rest of the necklace in here the it's like it it goes public in what we want a we have ever reproduction of these 3 for that instead he what and we can do that in every In each of these that the sphere of of the of the necklace and we obtain and and 1 can feel that these this given this process will lead you to work some very complicated enough the however the initial U. S. so mean of the 2nd iteration of a particular sphere the if you here the last inside the sphere 1 has the 1st 2 shape of the not and this is the iteration of the rest of the nodes in the 2nd the 2nd 1 1 cannot see from here is small not Professor and I think there's the small a 3 4 knots In here the well because they you can find this well-known to nature was not precisely nature but you can find it in the is several out values in Paris and the syntactic because this guy is a the find the find it at the same the same man I would say that this is not the from another way they doing big jury and he find out that these the same the same must address a B appears in in in and he said the for the but just finishing dominant what the 1 that would raise their 1st if we use instead of
columns and other shaped like platonic
solids we put the spheres in several but the excel it's if 1 chosen not that he there 1 high visited this a fantastic patterns but also we had we need this background for the for the for the shape so we use out is that the flouts and 1 can use for instance Kandinsky pictures they came from the internet for them him and then we can see from inside or and the 1 attitude with a should with the the spheres are in black and white so it's complicated or what what 1 can use the anchors MIT the and 1 can see from inside this yeah by principally in the structure of these factors and that's it think tournament the so questions
so I saw your while while not video on e-mail homepage and I was wondering how do you make history the visualization a and get it
I've I was honoring how you make us 3 D visualization of does well those How do I did that did that this the the I use there's some stuff on life the history of for the last rendering and they use a lot of python scripts may myself to put the spheres in
place and calculate everything so that it's a mix of our software and and so mates of water in the emergence of a the the so thank you run much
entry love appreciated it there but especially over the last 1 with the bowls and mean it's we talk about software but
this can be made with the real Christmas bowls and it's a very fine to mix them the tools are and to see real fractals uh which are not done which are not found in nature and as such and but which can be
seen really and there are and 2 seats on computer realizations of when you had the slot in these knots with some numbers we using from Jones list of breaks finite
classification that the first one the don't you know you had 3 knots was numbers and tend to 141 and solace is use with the use of the of the I forgot they states they were brought to us and they were have OK because I knew from Jones I was was from John when he worked his his
break words out and in the yes and he gave me his lists and is also in our program so we control all the food all find Jones knots up 10 crossings and helpful and well another connection here good thank union and the