IC16: Visualizing Mathematics With 3D Printing: Augmenting A Traditional Book With New Media & Editing Spherical Video With Möbius (And Other) Transformations

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IC16: Visualizing Mathematics With 3D Printing: Augmenting A Traditional Book With New Media & Editing Spherical Video With Möbius (And Other) Transformations
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I will talk about the development of a new popular mathematics book, coming out in September 2016. Most of the figures in the book are photographs of 3D printed objects. Readers can visit the book‘s website http://3dprintmath.com to explore virtual versions of the figures, download to print themselves or order online. This allows me to introduce topics which are not generally covered in popular mathematics books, presumably because of the difficulty of conveying truly 3D content. Then, I will speak on two further topics. First, the technology of spherical video, and how it differs as a medium from ordinary flat video. Second, some techniques for editing spherical video, using Möbius transformations and other conformal mappings of the Riemann sphere.
Mathematics Sphere Transformation (genetics)
Mathematics Mathematics Many-sorted logic Model theory Content (media) Figurate number
Logical constant Theory of relativity Angle Symmetry (physics) Many-sorted logic Different (Kate Ryan album) Direction (geometry) Order (biology) Content (media) Object (grammar) Perspective (visual) Dimensional analysis
Group action Symmetry (physics) Many-sorted logic Model theory Figurate number Object (grammar) Grothendieck topology Tessellation
Polytop Celestial sphere Plane (geometry) Projective plane Model theory Duality (mathematics) Acoustic shadow Solid geometry Dimensional analysis Spacetime
Point (geometry) Torus Three-dimensional space Multiplication sign Mereology Graph coloring Dimensional analysis Goodness of fit Hyperbolischer Raum Plane (geometry) Many-sorted logic Different (Kate Ryan album) Design of experiments Partial derivative Covering space Process (computing) Surface Weight Model theory Knot Maxima and minima Curvature Network topology Iteration Table (information) Spacetime
Computer programming Many-sorted logic Logic Compass (drafting) Term (mathematics) Surface Combinatory logic Model theory Function (mathematics) Rule of inference
Derivation (linguistics) Goodness of fit Link (knot theory) Many-sorted logic Model theory Grothendieck topology Physical system
Geometry Slide rule State of matter Real number Multiplication sign Image resolution Water vapor Product (business) Mathematics Sign (mathematics) Many-sorted logic Different (Kate Ryan album) Term (mathematics) Circle Arrow of time Directed graph Standard deviation Process (computing) Projective plane Sphere Celestial sphere Factory (trading post) Triangle Game theory Table (information) Spacetime
Statistical hypothesis testing Link (knot theory) Direction (geometry) Real number Mereology Mathematics Many-sorted logic Network topology Different (Kate Ryan album) Analogy Energy level Circle Data transmission Geometry Scaling (geometry) Generating set of a group Projective plane Content (media) Rectangle Connected space Mathematics Celestial sphere Root Game theory Directed graph
Rotation Point (geometry) Covering space Complex (psychology) Group action Functional (mathematics) Scaling (geometry) Quantum state Divisor Multiplication sign Forcing (mathematics) Projective plane Rectangle Sphere Celestial sphere Plane (geometry) Many-sorted logic Table (information) Data transmission Spacetime
Point (geometry) Plane (geometry) Square number Translation (relic) Numerical analysis Directed graph
Celestial sphere Process (computing) Film editing Multiplication sign Moment (mathematics) Physicalism Estimation Thetafunktion Rectangle Data transmission 2 (number)
Group action Functional (mathematics) Angle Natural number Independence (probability theory)
Musical ensemble Sphere
Mathematics Imaginary number
so what about all the things you you you you heard my name in the I introduced myself but somebody else was introduced me I guess older start so and so I many Simon I I guess I'm starting the the the next session which will then afterwards I'm well aware that I'm keeping you from lunch but after lunch will continue with with more talks and so this is actually gonna be 2 talks and so the there well will get to the 2nd talk in a minute and in the 1st always about set this book that I've been writing and so but it's a cold visualizing that meant to the printing and
I'll talk little bit about so what is doing this you and and sort of how it interacts with sort of new new media that have been used maybe in the context of book so and it's popular
mathematics book and at the sort of main twist is the most the figures in the book of photographs of sleep and things and so there's a sister website of the penn matter where you can see a virtual model on-screen rotated around you can order prints online we can download the file print yourself so this is what the website looks like and so if I have Wi-Fi than I can that click through to the chapter and then he is a and the figure from the book and again assuming the Wi-Fi is good enough so there's for example would the 3 D have built-in and this works well on friends and I can't and so on and so I mean this is sort of the uh seems like to me like this is kind of an obvious idea if you got a three-dimensional content that you want to talk about you should have a three-dimensional figures why doesn't the biology textbook come with a 3 D model of DNA all the shape of the heart and or whatever it might be and so on
so I thought I'd do is just show a few pictures from from the book and a little bit of the content and then talk a bit about sort of the technical issues in making morals and and getting them available for people to use and so the 1st step is about symmetry in he II so to try to take a different tack on some well-worn ideas but seen in a different way so so this is a q but only sensitive trying to take pictures of from different directions and ask the question how many ways are there well what are the different ways to take a picture of the Q and and this has simulation symmetry this is the and the set up that I used in order to take perspectives so and I've got a timetable here in this sort of clam stand allows me to get whatever angle I 1 on 1 object attached constants article is photographs and put them together
and this is the sculpture by Batsheva Grossman and which is actually this is this was used in and another excellent book this images of things by by Conway and bejel and command Strauss was all about symmetry in this was 1 of the examples and they said you know it's it's it's so difficult to see but even from these 2 photographs of the same symmetrical object that the same object and so this was on my attempts to to show what the relation is between those 2 different views you can rotate around and so there is some irony here I'm reading a book uh a two-dimensional format and in which I'm saying look this other format is really cool is better for seeing things but I tried to go as far as I could just in purely in 2 dimensions to to that the two-dimensional page 2 see see these three-dimensional ideas and there these are collections of objects with all the different uh
polyhedral symmetry types 1 of the side effects of of of writing this is that there is now a collection of these models that's out there for
anybody to download or to to print themselves to to buy online and see can that make a little three-dimensional museum just out of out of these figures and moving on to Paul heater these and these snap tiles which designed by Lord tone so and I'll talk about that thing Iverson in a 2nd this is the sort of sharing site for the files and so I used her designs she'd made already I didn't need to make it but I can reference uh these objects that polly he drove
costs are canadian are solids there lots of lots of cool models to make and then moving on to talk about the 4th dimension using shadows projections from a high-dimensional space as to low-dimensional spaces at this picture about is now appearing in all of my talks even some other people's talks and so he not stereo projection move mapping from the sphere the
plane using the same thing 1 dimension up to to get a picture of a hypercube and this is talking about the duality between the four-dimensional polytopes and and that you have a regular polytopes that talking about
curvature so to these a colored according to the curvature of the surface at each point blue is negative curvature is positive curvature it and this is
the I got these are on the table as flexible as sort of fabric like and 3 D printed hinged material that that shows so 1 way to see negative curvature space the there's a sort of more geometric weights to see covered negative curvature space the hyperbolic the plane many different models of and many of these a joined with what to do for the people have collaborated with and this is a moving up a dimensions three-dimensional three-dimensional hyperbolic space in the timing of three-dimensional hyperbolic space and moving on to sort topology not and services and this I should mention that well enough about this at a later again and all these models are like printed using 3 printing service Shapeways and and and they said they had just started introducing sort of fancy ceramic 3 D printing process unfortunately they don't yet have a partial ceramic partial DOE out material is really they should more from the ceramic coffee mug have to the dough FIL doughnuts and so this would be half ceramic and halftone foreshadows exist yet and I'm not stores knots at a large mentioned that the color here is is is a cheated stunning post-processing and there isn't yet a good enough color material that that really works but the files and therefore when there is so that when when that and technology comes along something will be able to print out there and they're really does look like this rather than cheating and the flat torus and the Klein quartic and let me just not say really anything about this other than to play you this very shorts and animation was part of our process in trying to come up with the shape and some sort iterative process to to try and get a good shape the Klein quartic minutes later I'm and soap films minimal
surfaces a little of interesting things you can do with 3 prints you can show of ideas using these he's using these models
so on to whether they come from what is that the software and all the different and programs and so on so I have to my me 3 program is called rhinoceros align 3 D . com and is commercial and CAD software it's not free unfortunately and that is generally agreed by the mathematical sculptors I know to be the best sort of after doing mathematical logic you can use a graphical user interface rather than just writing code and if you do a lot of sort a ruler and compass constructions and it does have a lot of scripting uh built as well so I I do a lot of the models come from that scripting in Python sometimes a combination of scripting and doing things by hand in the user interface available for both the Mac and Windows so how did I get these these printed so I use ship ways are and a online 3 D printing service you upload your file you click on onto constant in a week or so later your finger arrives in mail and quality is good the service actually excellent as is 1 of the reasons I think that they're doing well and equal to any company I've ever dealt with in terms of getting back to you when things have gone wrong and it's relatively inexpensive and so how you make them available online so there's a various different companies I mentioned shapeways already and so I well so I paid somebody to do that the website design but if you wanna have a really nice website that will an the present to use and and connect up with and everything else in the book and so so I mentioned viewing in on-screen there's a comical sketch fat and has this is just a very user-friendly and for the end user and for the creator output if we model online and then it works on laptops it
works on phones it works on really crappy phones it downgrades very gracefully
to to a less good systems and you can buy things online from Shapeways if you if you want to buy things good quality rather than Frenchman them yourself you can do that and ends the versus an excellent site for the sharing of files so on also this isn't quite done yet but also have just the wall files available to download from my website of us has a community built around its of people who have 3 printers or modify files or whatever it might be it also as excellent as citations sort of built into it if you make uh a derivative of somebody else's then when you upload your version of it you can say this came from that model and then link schools will happen within sciences really very nice yeah and 1 last question why open source I mean maybe it's sort of an obvious thing to do to make all the files available
when you put them on Thingiverse yes you can add some sort of license the base your saying this is out there in the world I could have hidden these things away but that would of course mean fewer people would have morals in their hands and then there's another aspect here thinking in the very long term and if you don't make these things available you know in 20 years by website may go down but the files will still be out there because once the on the internet nothing ever leaves the internet and and what those may be a common to make here there's a lot people using you at doing things with interactive and a website and so on which is fantastic and the business real problem of all they can survive as time goes on luckily for me but that the 3
files formats is so simple that and that's not an issue but when you have interactive concentrating its gets a lot more difficult but if that was the 1st to walk and then he took them the 2nd talk so on so that's answer spherical video so that maybe could switch to the guy had briefly and so on so this is a spherical video camera video now notice sitting about all these wonderful memoir things and that's me and and so this is in real time sort of connecting by Wi-Fi from the from the the phone to that I have here and then switch back that's the laptop please so so i'm gonna talk so so this is sort of in a new medium on least Cilicia yielded more this photograph active yesterday and so what you get from the camera itself if you downloaded to your to your computer is that for a triangular projection as somebody mentioned yet recount rectangular production of that was 1 of the games you could have played so you will be seen what is this this is unwrap the sphere latitude longitude so and so this is what you get from from the camera he has said this sort of you is that you get with the you can download this for free and then you can look around what was going on in the scene so the the so at the 1st talk about so water what are the differences so that the qualitative differences how does it differ as a medium from factory so i is we haven't had that many talks from people doing video has lost people doing and mathematical videos putting them on you tube and should you be using spherical the is gonna be a thing in the future what are the differences and so on 1 of 1 of the obvious differences this is that it's interactive you get to choose where to look I should mention that you don't need a fancy special viewer to do this the use you back on your phone you can just sort of tilt your phone around in using the gyroscope on the phone you can look around a scene and a circle video and spherical he was already supported by UT so there's already people uploading and and everybody has you to wrap on the front just just look at it and so on this causes some problems perhaps how does the videographer make you look at interesting things you can look anyway you want what you do about this you don't have control more and well so the solution demands he she will you pointed things you you assume that they're gonna start looking at you know uh what you can control where whether going to be looking at the start of the year and then you know don't sort of Teleport somewhere else to do a job that you sort of walk somewhere arrow points and whether you want to look at something and and adventitious this I think there's a lot more visual space and so so this isn't quite possible yet with current technology but when the resolution improves and you could if you just doing a sort of standard con enemy kind of a video your writing something down on a whiteboard and talking to the camera you can have multiple whiteboards arranged around the camera and then you can be running over here and then later on you running over here and if and if the viewer wants to look back at what you what he did they can do that they can just turn to look at justice and this is really the difference Wiener blackboard talk and slides talk in the slides talk you can go back to look at and see what they did before but once you have this ability to look around in a circle that you'd be able to do this again when the the resolution improves then you you can have that ability so you can put the viewer inside of interesting geometrical shapes as a sort of uh well the the 2 that the standard thing that I do this any time of someone with a big mathematical sculpture as he put it inside and you see what it looks like an inside get something interesting about and this is maybe not so relevant in terms of mathematics video but have other kinds of in the signs videos so geology you'll also know something you put the camera in a location and then you're asking what is the interesting feature in this scene and because you can look around you can actually do that that the so other aspects there's no frame and there's no framing and so and it is this has a relevance and state say you're a parade or protest and you can propagandize by cropping to show to make it look like the crowd was small than it is to look like the crowd was bigger than it is with circle but you can't do that and so in that sense it's more honest but from the sort of earth filming something aspect there's nowhere hiding can be behind the camera everything is always inside you always have to clean up your Parliament before you recalled anything and but you don't have to worry about framing because everything is inside if you're doing something on a table and the warring you know you don't have to worry if I did I do that thing out of short it too was in shock than but the
more so on the emotional reaction I think maybe to a that to a video there's a much greater sense of presence maybe maybe you you sort of got the sense here and maybe it's better if you have control of me controlling but it really feels like we would then and and rather than it sort of being a picture on a wall and so when you're talking to somebody and explaining something and you watching this inside of a spherical video it really feels like you're sitting on the couch across the room from it doesn't feel like they're talking head on television and so there's a there's a there's a better emotional connection and I think that the and I think in the U 2 generation thing is that there's a there's a tendency or direction towards the personality of the person doing the video is important that people watching want to have a uh an emotional connection with them and being in the same room is much more powerful
OK and so so that's sort of some differences I mean and I'm not sure that in a couple of um math educational videos that have appeared in spherical video I mean I think this is sort of the direction that things are going to go in the future and in particular I this all links and I should mention with the with virtual reality and Facebook bought Oculus Rift for 2 billion dollars last year surely lots of people will soon be having VR headsets and circle the was another source of content that people often think about computer games but taking view of real life things and then putting it inside of the I had said the same sort of thing and the other thing I wanted to talk about is um as something I've been thinking about the last 6 months or so which is transformations of stroke of India so and as I mentioned before you get this kind of thing out of your camera and you get the EC rectangular unwraps projection what you do with it you can put this into Photoshop or taking video you can take put it into your video software and and then you know you can change you can play with the levels of is too dark and so on that's all saying what about the transformations at 1 of these you might wanna do is rotate the image but it's based on a sphere but your photoshop isn't gonna be able to do that it doesn't matter how to do this and here's another question and in fact video you can zoom in on small some small part we know what that means we take the picture you scale it up and then you then you crop down is there an analog of zoom or scaling in a circle it so it starts right to spend a few minutes talking about and some investigations is an interesting effects and mention this is my sort of test image for all of all the users transmissions this is people may be familiar with phi hearts and and by heart again and books Lee and MEI further apart of LOD on which has they would be research and service reality video abdication sentences and that's that's just scary here and show that image on the sphere so here you are inside of their office and you can look around and see see what's going on and I said it appeared in every talk and here it
is again and so the 1st thing honeydew transmission almost the sphere the very natural thing to do is the 1st get onto the plane so again hysterical projection you get images that are almost the only put them on the plane on the and then you treat that as the complex plane and then you can do all kinds of complex transformations to the plane and then put it back on the sphere so let's see an example so this is and here's that same picture again on the sphere and then here it is projected to the plane and then I got this sort of Ray that I put in here shows shows what's going on and now suppose I just scale or the image on the plane by a factor of 2 so I just scalar by 2 and then put it back on the sphere and well you and then you have some transformation sort of like as if it also makes this is the right context for doing that rotations as well and so he is the effect and you can see the Accra time and effect it's been sort of everything's been moved upwards so all we've moved near the floor and what does that look like so here
we are we are now very shorts advisory tall where where it feels like we're very close to the to the force the and In all we could do something else any other complex transmission this is if you are applying pull back and the C. squared function then you end up with 2 copies of everything I and there's the ACC rectangular view and what does this look like inside the sphere it's a surprising effect looks very ordinary except that this is even more copies of everybody and something very strange happens if you look straight up and there's a branch point in the ceilings no 1 on the floor the following so ionization of so that the knife have a bit more space so I upgraded to a twofold French cover of my common so if you'd sell just took it myself and so there's the there's the 2 branched covering the ceiling and you know as I now have to is 1 over here and 1 over here else out to coffee tables and and see if I step forward and a little bit I'm going to go over go there and pick up my laptop from the catch and so and then I forgot my by my book over there on the coffee tables and we have and moving on so here's a and another juggling appears again this was actually more interesting the unwrapped rectangle you so if I go here that the better than the that his and will be the 1st of all this effects of and this is this video is just this view of the kinds of the and you have to have you know that was 1 of of the things it and that's the but I don't think I really have time to go through these in great detail how many people will be familiar with the the Jost facts and there's a lot involved here and the reason well this is something else well told about this and the something strange going on that and so here's the original picture again reason we have a frame appears because we wanted to the just affect the effect of the at picture that contains itself and so you may have wonders and I can answer the question if you inside of a effect picture and you turn around and look backwards what do you see was this sort of strange Paul floating in the middle of the of the office and it has to be there I I could explain but once whites that they pass version of me so I was saying 1 and this is a spherical josta video so it so slowly zooming this way around refrain here coming the this way and over there is a sort weird paddle portal in the middle of my this is the future so as future versions of nearly that and this is the past so I had off to pass version of the at to explain again what's going on they pass version of me so I was saying 1 is the same and this is a spherical josta video well this goes on like that for a while and I think I will so there's a lot more I could say but I'm I should let you get to lunch so thank you very much thank FAQ like this and it was formed in the are there any questions yes please on hi how could you explain it is going to the ECOLEAD but the idea to move when you go from the view of so thought to the other side of the when you your uh living room going in this 1 the uh the even in the 1 you particularly in the other of is this achieving the idea and the other is not canonical and that is the
magic growing keep going yeah
so so I I took the view of the anti room and I just announced
in a video editing software the copy of me that's Thomas over and uh yeah so it's just using in the where there is a the the scene between the 2 very rooms there's no seems as with the really nice thing about this kind of translation I know it's it's the transformation is you go to the complex plane and the square all of the numbers and then that doubles up everything the yeah this there's there's no marrying it's it's it's it's really very natural when you just apply complex transformations that the point we we can talk about it another question and like whenever and so what kind of video editing software lets you do that allowed you banks and yeah so it's um that yet I write I write the code and myself and so it's written in Python and so from the the the the the video that I take his just the source you can put it into I use premier per and then export all the frames to a directory and then run through them 1 by 1 in Python and I was talking to people about a really great to have this running in real time with uh a video stream from the device and Cinderella looks like it may be able to do that so maybe in the future will
have an aversion it's running in real time at the moment it takes a few seconds per frame depending on what the transmissions and the question is the whole many pixels are then just with me so uh this camera I should say Rico theta ESTs about 350 dollars so the price of a Compaq cameras not super expensive this takes video when you in rapid in the EC rectangular cuts in 1920 by 960 pixels so that sounds like H but once you're looking at a small portion of the entire sphere that's not that much time it takes photographs of 5 thousand 300 also by whatever half of as and so you know that this is the processing speed at the moment that will and they are expected to release another 1 later this year which is expected to be 4 K video at even a k yeah yeah it I have you tried to see this kind of leaders in these 2 reality that the divide of yet the are and yet it works the it's it's very very massive and I mean those and I underline that the physics of it is what the process going like 0 how do you how you view it I can
you actually viewed it on a on a hillside as the labeling when you have like these doubled space of
the others and has to play with your senses right and I guess some so this is interesting thing I mean when you this video there's a question so how finals in then the little bit of this ocean anyway but it feels kind of natural and that's that's independent of doing any of these transformations it
it feels very real and the transformations of all conformal but preserve angles so as long as you don't do too crazy transformation it feels very like was something strange going on but you know everything feels kind of recognizable and so we identify the uh very clean answer that but as you make your the functions we had gets more like uh a visual effect unless like there's something strange going on with my world and but when you're just doing a little bit like zooms and so on and it feels really quite natural this
scandalous music video but yes here yeah if anybody knows OK go with the 1 into a spherical video OCR that's another thing I have to say that people usually call it 360 video and I hope we can all
agree that this is incorrect if anybody ever tells you all mentions 360 Video when it's actually a sphere please correct them I
and do we have another question yeah OK let's thank the speaker
again that