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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Imaginary Conference 201620 / 26
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00:00
MathematikKugelTransformation <Mathematik>MathematikHochdruckSortierte LogikVorlesung/Konferenz
00:54
MathematikSortierte LogikMathematikOrdnung <Mathematik>RelativitätstheorieSymmetrieModelltheorieDimensionsanalyseHochdruckInhalt <Mathematik>WinkelObjekt <Kategorie>Figurierte ZahlPerspektiveKonstanteRichtungDifferenteVorlesung/Konferenz
04:01
SymmetrieModelltheorieGruppenoperationMathematische LogikOptimierungStereometrieTopologieFunktion <Mathematik>VersuchsplanungIterationKnoten <Statik>ParkettierungMinimalflächeKombinatorDimensionsanalyseDualitätstheorieEbeneGrothendieck-TopologieHochdruckHyperbolischer RaumKugelMereologiePolytopProjektive EbeneTabelleTermTorusGüte der AnpassungÜberlagerung <Mathematik>Gewicht <Ausgleichsrechnung>KrümmungsmaßPunktDimension 3Zirkel <Instrument>Sortierte LogikAbschattungProzess <Physik>Partielle DifferentiationGraphfärbungFlächentheorieDifferenteObjekt <Kategorie>MultiplikationsoperatorSchlussregelMinkowski-MetrikFigurierte ZahlArchimedische SpiraleZweiVorlesung/Konferenz
09:41
Physikalisches SystemModelltheorieDerivation <Algebra>Grothendieck-TopologieVerschlingungGüte der AnpassungSortierte LogikVorlesung/Konferenz
10:47
Gerichteter GraphReelle ZahlSortierte LogikMultiplikationsoperatorModelltheorieVorlesung/Konferenz
11:30
GeometrieMathematikSpieltheorieProdukt <Mathematik>DreieckAggregatzustandKugelProjektive EbeneRechenschieberTabelleTermZeitrichtungWasserdampftafelFaktor <Algebra>KreisflächeSortierte LogikProzess <Physik>Vorzeichen <Mathematik>DifferenteAuflösung <Mathematik>MultiplikationsoperatorStandardabweichungMinkowski-MetrikMomentenproblemWechselsprungKrümmungsmaßPunktSchnitt <Mathematik>Vorlesung/Konferenz
16:45
MathematikGeometrieTopologieWurzel <Mathematik>Einfach zusammenhängender RaumLeistung <Physik>Sortierte LogikRichtungEndlich erzeugte GruppeVorlesung/Konferenz
17:42
MathematikSignifikanztestTransformation <Mathematik>AnalogieschlussÜbergangEbeneGerichteter GraphGruppenoperationInhalt <Mathematik>Komplex <Algebra>KugelMereologieProjektive EbeneQuantenzustandZentrische StreckungVerschlingungReelle ZahlTeilbarkeitSortierte LogikRichtungDifferenteSpieltheorieReduktionsverfahrenRechteckKreisflächeMultiplikationsoperatorKreisbewegungVorlesung/Konferenz
21:13
Transformation <Mathematik>GruppenoperationPunktReduktionsverfahrenForcingFunktionalKugelRechteckVorlesung/Konferenz
21:57
Transformation <Mathematik>GruppenoperationTabelleÜberlagerung <Mathematik>DifferenteMultiplikationsoperatorEinsRechteckMinkowski-MetrikVorlesung/Konferenz
23:57
GruppenoperationKugelSortierte LogikVorlesung/Konferenz
26:08
Transformation <Mathematik>EbeneMomentenproblemThetafunktionProzess <Physik>Numerische MathematikReduktionsverfahrenGerichteter GraphKugelPhysikalismusSchätzungTeilbarkeitRechteckTranslation <Mathematik>QuadratzahlPunktSchnitt <Mathematik>MultiplikationsoperatorZweiVorlesung/Konferenz
29:13
Natürliche ZahlDelisches ProblemStochastische AbhängigkeitVerzerrungstensorSortierte LogikTransformation <Mathematik>Minkowski-MetrikVorlesung/Konferenz
29:38
Transformation <Mathematik>FunktionalGruppenoperationWinkelVorlesung/Konferenz
30:15
KugelGüte der AnpassungFormation <Mathematik>Vorlesung/Konferenz
30:41
Imaginäre ZahlMathematikVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:19
Shall I introduce myself, unless somebody else wants to introduce me?
00:26
I guess I'll just start. So hi, I'm Henry Seggman. I guess I'm starting the next session, which will then afterwards, I'm well aware that I'm keeping you from lunch. But after lunch, we'll continue with more talks. So this is actually gonna be two talks.
00:42
So the, well, we'll get to the second talk in a minute. And the first talk is about this book that I've been writing. So it's called Visualizing Mathematics with 3D Printing. And I'll talk a little bit about sort of what it's doing that's new, and sort of how it interacts with sort of new media that haven't been used,
01:02
maybe, in the context of a book before. So it's a popular mathematics book. And the sort of main twist is that most of the figures in the book are photographs of 3D printed things. And there's a sister website, 3dprintmath.com, where you can see a virtual model on screen and rotate it around.
01:24
You can order a print online, or you can download the file to 3D print yourself. So this is what the website looks like. And so if I have Wi-Fi, then I can click through to a chapter, and then here's a figure from the book. And again, assuming that the Wi-Fi is good enough.
01:42
So there's, for example, the 3D built-in. And this works well on phones and iPads and so on. So, I mean, this is sort of, seems to me like this is kind of an obvious idea. If you've got three-dimensional content that you want to talk about, you should have three-dimensional figures.
02:02
Why doesn't the biology textbook come with a 3D model of DNA, or the shape of the heart, or whatever it might be? So I thought what I'd do is just show a few pictures from the book, and a little bit of the content, and then talk a bit about sort of the technical issues in making the models and getting them available for people to use.
02:24
And so the first chapter's about symmetry, and I sort of tried to take a different tack on some well-worn ideas, but seen in a different way. So this is a cube, but obviously I'm sort of trying to take pictures of it from different directions, and ask the question,
02:40
how many ways are there, or what are the different ways to take a picture of a cube? And this has some relation to symmetry. This is the setup that I used in order to take those pictures. So I've got a turntable here, and there's a sort of clamp stand that allows me to get whatever angle I want on the object that's attached to the clamp stand, so I took all those photographs and then put them together.
03:04
This is a sculpture by Bathsheba Grossman, which is, actually this was used in another excellent book, The Symmetries of Things, by Conway, Bergell, and Goodman Strauss. And that book was all about symmetry, and this was one of the examples.
03:22
And they said, it's so difficult to see, even from these two photographs of the same symmetrical object, that they're the same object. And so this was my attempt to show what the relation is between those two different views, where you can rotate it around. So there was some irony here. I'm reading a book, a two dimensional format, and in which I'm saying,
03:46
look, this other format is really cool, and is better for seeing things. But I tried to go as far as I could, just impurely in two dimensions to the two dimensional page to see these three dimensional ideas. There are these collections of objects with all the different
04:03
polyhedral symmetry types. One of the side effects of writing this is that there's now a collection of these models that's out there for anybody to download or to print themselves or to buy online, and so you can sort of make a little three dimensional museum just out of these figures.
04:22
Moving on to polyhedra, these snap tiles, which were designed by Laura Tauman, so I'll talk about Thingiverse in a second. This is a sort of sharing site for 3D files, and so I used her designs. She'd made it already. I didn't need to make it, but I can reference these objects.
04:43
Polyhedra, of course, Archimedean solids, lots of cool models to make. Then moving on to talk about the fourth dimension, using shadows, projections from high dimensional spaces to lower dimensional spaces. This picture apparently is now appearing in all of my talks,
05:03
even some other people's talks, talking about stereographic projection, mapping from the sphere to the plane, using the same thing one dimension up to get a picture of a hypercube. This is talking about duality between the four dimensional polytopes and their dual regular polytopes.
05:24
Talking about curvature, so these are coloured according to the curvature of the surface at each point, blue is negative curvature, red is positive curvature. This is the, I got these over on the table. This is flexible sort of fabric like 3D printed hinged material
05:43
that shows sort of one way to see negative curvature space. There's a sort of more geometric way to see negative curvature space at the hyperbolic plane. Many different models of that, many of these are joined with lots of different people that I've collaborated with.
06:01
This is a moving up a dimension three dimensional, three dimensional hyperbolic space and a tiling of three dimensional hyperbolic space. Moving on to a sort of topology, knots and surfaces. This, I should mention, well, and I'll say about this later again. All of these models I got printed using the 3D printing
06:22
service Shapeways and they so they had just started introducing sort of a fancy ceramic 3D printing process. Unfortunately, they don't yet have a partial ceramic, partial dough material because really they should morph
06:40
from the ceramic coffee mug over to the dough filled doughnuts. So this would be half ceramic and half dough. Unfortunately, that doesn't exist yet. Knots, torus knots, oh, I should mention the colour here is a cheat. It's done in post-processing. There isn't yet a good enough colour material that really works.
07:04
But the files are there for when there is. So when that technology comes along, somebody will be able to print out their own that really does look like this rather than cheating. The flat torus, the Klein caudic. Let me just not say really anything about this other than to play you
07:23
this very short animation that was part of our process in trying to come up with the shape. Some sort of iterative process to try and get a good shape for the Klein caudic, we'll get to that later. And soap films, minimal surfaces.
07:41
So there's an awful lot of interesting things you can do with 3D prints and you can show a lot of ideas using these models. So where do they come from? What is the software involved, the different programs and so on? So my main 3D program is called Rhinoceros or rhino3d.com.
08:01
It is commercial CAD software. It's not free, unfortunately, but it's generally agreed by the mathematical sculptors I know to be the best sort of for doing mathematical art. If you're going to use a graphical user interface rather than just writing code. So you can do a lot of sort of ruler and compass constructions.
08:22
It does have a lot of scripting built in as well. So I do a lot of the models come from scripting in Python, sometimes a combination of scripting and doing things by hand in the user interface. It's available for both Mac and Windows. So how did I get these printed?
08:41
So I use shapeways.com. It's an online 3D printing service. You upload your file, you click on Add to Cart, and then a week or so later your thing arrives in the mail. The quality is good. The service is actually excellent. This is one of the reasons I think that they're doing well,
09:01
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. So how do you make them available online? So there's various different companies. So I mentioned Shapeways already. So I paid somebody to do the website design, but you want to have a really nice website
09:22
that will be pleasant to use and connect up with everything else in the book. So I mentioned viewing in 3D on screen. There's a company called Sketchfab that has this. It's just very user-friendly for the end user and for the creator.
09:40
Put a 3D model online and then it works on laptops, it works on phones, it works on really crappy phones. It downgrades very gracefully to less good systems. You can buy things online from Shapeways. If you want to buy things good quality rather than try and print them yourself, you can do that.
10:01
And Thingiverse is an excellent site for sharing files. So I'll also, this isn't quite done yet, but I also have just the raw files available to download from my website. Thingiverse has a community built around it of people who have 3D printers or modify files or whatever it might be.
10:23
It also has excellent citation sort of built into it. If you make a derivative of somebody else's model, then when you upload your version of it, you can say this came from that model and then the links all sort of happen within the site. So it's really a very nice system. And one last question, why open source?
10:43
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, but basically you're saying this is out there in the world. I could have hidden these things away, but that would, of course, mean that fewer people would have models in their hands. But I think there's another aspect here,
11:01
thinking in the very long term, if you don't make these things available, you know, in 20 years, my website may go down, but the files will still be out there because once they're on the Internet, nothing ever leaves the Internet. And well, there's maybe a comment to make here. There's a lot of people doing things with interactive websites and so on,
11:23
which is fantastic, and there's this real problem of are they going to survive as time goes on. Luckily for me, the 3D files formats are so simple that that's not an issue, but when you have interactive content, I think it gets a lot more difficult. Okay, that was the first talk.
11:42
Let me talk about the second talk. So that's spherical video. So maybe we could switch to the iPad briefly. There we go. So this is a spherical video camera over here, and there it is, sitting above all of these wonderful MoMA things.
12:02
And there's me. And so this is in real time sort of connecting by Wi-Fi from the phone to the iPad here. Maybe we can switch back to the laptop, please. So I'm going to talk... So this is sort of a new medium.
12:24
So let me sort of show you a little bit more. This is a photograph I took yesterday. So what you get from the camera itself, if you download it to your computer, is the equirectangular projection. Somebody mentioned the equirectangular projection of the Earth. It was one of the games you could have played,
12:42
so you've already seen what this is. This is unwrap the sphere latitude, longitude. And so this is what you get from the camera. Here's the sort of viewer that you get with the... You can download this for free, and then you can look around at what was going on in the scene.
13:06
So let me first talk about, so what are the differences, the qualitative differences? How does it differ as a medium from flat video? So I guess we haven't heard that many talks from people doing video, but there's lots of people doing mathematical videos,
13:21
putting them on YouTube. Should you be using spherical video? Is it going to be a thing in the future? What are the differences? So one of the really obvious differences is that it's interactive. The viewer gets to choose where to look. I should mention, you don't need a fancy special viewer to do this. The YouTube app on your phone,
13:41
you can just sort of tilt your phone around and using the gyroscope on the phone, you can look around a scene in a spherical video. And spherical video is already supported by YouTube, so there's already people uploading it, and everybody has YouTube app on their phone, can just look at it. So this causes some problems, perhaps. How does the videographer make you look at interesting things?
14:03
You can look anywhere you want. What do you do about this? You don't have that control anymore. Well, so the solution at the moment seems to be, well, you point at things. You assume that they're going to start looking at you. Well, you can control where they're going to be looking at the start of the video, and then don't sort of teleport somewhere else
14:22
or do a jump cut. You just sort of walk somewhere or point somewhere that you want to look at something. An advantage to this, I think, there's a lot more visual space. So this isn't quite possible yet with current technology, but when the resolution improves, if you're just doing a sort of standard Khan Academy
14:42
kind of video, you're 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 writing over here, and then later on you're writing over here. And if the viewer wants to look back at work that you already did, they can do that. They can just turn to look.
15:01
And this is really the difference between a blackboard talk and a slides talk. In a slides talk, you can't go back to look and see what they did before. But once you have this ability to look around in a spherical video, you'd be able to do this. Again, when the resolution improves, then you can have that ability. You can put the viewer inside of interesting geometrical
15:21
shapes. This is sort of, well, the standard thing that I do with this anytime, somewhere with a big mathematical sculpture is you put it inside and you see what it looks like from the inside. And you can get something interesting out of that. And this is maybe not so relevant in terms of mathematics video, but perhaps other kinds of science videos, so geology or botany
15:44
or something. You put the camera in a location, and then you're asking, what is the interesting feature in this scene? And then because you can look around, you can actually do that. So other aspects. There's no frame, and there's no framing.
16:01
So I mean, this has relevance in, say, you're at a parade or a protest. You can propagandize by cropping to make it look like the crowd was smaller than it is or look like the crowd was bigger than it is. With spherical video, you can't do that. And so in that sense, it's more honest.
16:21
From the sort of filming something aspect, there's nowhere to hide. You can't be behind the camera. Everything is always in sight. You always have to clean up your apartment before you record anything. But you don't have to worry about framing because everything is in sight. If you're doing something on a table and you're worrying, you don't have to worry, did I move that thing out
16:43
of shot? It's always in shot. More sort of on the emotional reaction, I think, maybe to a video, there's a much greater sense of presence. Maybe you sort of got the sense here. Maybe it's better if you have control
17:00
rather than me controlling. But it really feels like we were there. And rather than it sort of being a picture on a wall. So when you're talking to somebody and explaining something, and you're watching this inside of a spherical video, it really feels like you're sitting on the couch across the room from them.
17:20
It doesn't feel like they're a talking head on a television. And so there's a better emotional connection. And I think in the YouTube generation, I think there's a tendency or a direction towards the personality of the person doing the video is important, that people watching want to have an emotional connection with them.
17:42
And being in the same room is much more powerful. OK, so that's sort of some differences. I mean, I'm not sure. There have been a couple of math educational videos that have appeared in spherical video. But I think this is sort of a direction that things are going to go in the future.
18:00
In particular, this all links in, I should mention, with virtual reality. Facebook bought Oculus Rift for $2 billion last year. Surely lots of people will soon be having VR headsets. And spherical video is another source of content. So people often think about computer games, but taking video of real life things
18:21
and then putting it inside of a VR headset is the same sort of thing. The other thing I wanted to talk about is something I've been thinking about for the last six months or so, which is transformations of spherical video. So as I mentioned before, you get this kind of thing out of your camera. You get the equirectangular unwrapped projection.
18:43
What do you do with this? You can put this into Photoshop, or if you're taking video, you can put it into your video software. And then you can play with the levels. If it's too dark and so on, that's all the same. But what about the transformations? One of the things you might want to do is rotate the image.
19:00
It's supposed to be on a sphere, but your Photoshop isn't going to be able to do that. It doesn't know the math of how to do this. Here's another question. In flat video, you can zoom in on some small part. We know what that means. You take the picture, you scale it up, and then you crop down. Is there an analog of zoom or scaling in spherical video?
19:23
So I just wanted to spend a few minutes talking about some investigations and some interesting effects. And I'll mention, this is my sort of test image for all of these transformations. This is people may be familiar with Vi Hart and Vi Hart again.
19:40
Andrea Hawksley and Emily Eifler are part of LOVR, which is a research into virtual reality video. They're based in San Francisco. And let's just go over here and show that image on the sphere. So here you are inside of their office. And you can look around and see what was going on.
20:06
I said it appeared in every talk, and here it is again. So the first thing, how do you do transformations on the sphere? The very natural thing to do is to first get it onto the plane. So again, the stereographic projection, you get images that are on the sphere and you put them on the plane.
20:22
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, here's that same picture again on the sphere. And then here it is projected to the plane. And then I've got this sort of ray that I've put in here
20:41
that shows what's going on. And now suppose I just scale the image on the plane by a factor of two. So I just scale it up by two, and then put it back on the sphere. And well, then you have some transformation that's sort of like a zoom. It also makes, this is the right context for doing rotations as well.
21:00
So here's the effect. And you can see there's the equirectangular version of the effect. And it's been sort of, everything's been moved upwards or we've moved nearer the floor. And what does that look like? So here we are, we are now very short and Vi is very tall. And it feels like we're very close to the floor somehow.
21:24
Or you could do something else, any other complex transformation. This is if you apply or pull back by the z-squared function, then you end up with two copies of everything. And there's the equirectangular view. And what does this look like inside of the sphere?
21:42
It's a surprising effect. It looks very ordinary, except that there's even more copies of everybody. And something very strange happens if you look straight up. There's a branch point in the ceiling and there's another one in the floor. So...
22:00
Hi, I'm Henry Segelman. So I thought it would be nice to have a bit more space. So I upgraded to a two-fold branch cover of my apartment. So if you... So I'll just talk over myself. So there's the two-fold branch cover in the ceiling. You notice I now have two couches, one over here and one over here.
22:21
I also have two coffee tables. Let's see, if I step forward in a little bit, I'm gonna go over there and pick up my laptop from the other couch. And then I forgot my book over there on the other coffee table. So I'll go over there and pick that up.
22:43
Moving on. So here's another juggling appears again. This one's actually more interesting in the unwrapped equirectangular view. So if I go here,
23:04
so because it's all, we're juggling around the camera, so we're facing forwards all the time. So it has this strange little effect here. Anyway, I wanted to show you this because this video is just this, a few different juggling patterns. Just taking this down from university.
23:21
If you happen to have only three jugglers but you wanna do a pattern that involves six, you can also use the C-squared transformation to do that. Let's see. I don't think I really have time to go through these in great detail. Many people will be familiar with the Droster effect. There's a log involved here.
23:44
So the reason, oh, well, this is something else. Maybe I won't talk about this. But there's something strange going on there. So here's the original picture again. The reason we have a frame up here is because we want to do the Droster effect, the effect of a picture that contains itself.
24:02
And you may have wondered, and I can now answer the question, if you're inside of a Droster effect picture and you turn around and look backwards, what do you see? Well, there's this sort of strange portal that's floating in the middle of the office. It has to be there.
24:21
I could explain but won't why it's there. A past version of me. So as I was saying, my name is Henry Segerman. This is a spherical Droster video. So you're sort of slowly zooming this way, or rather the frame here is coming over you this way. And over there, there's a sort of weird pedal portal in the middle of my apartment.
24:42
This is the future. So there's future versions of me over there. And this is the past. So I'll hand you off to a past version of me to explain again what's going on. A past version of me. So as I was saying, my name is Henry Segerman. This is a spherical Droster video. Well, and this goes on like that for a while.
25:03
I think I will stop there. There's a lot more I could say, but I should let you get to lunch soon. Thank you very much.
25:22
I guess I get to ask for my own questions. Are there any questions? Yeah, please.
25:46
Could you explain, not too technically, but the idea to when you go from a sofa to the other sofa, when you duplicate your living room. Oh, in this one, yeah.
26:01
Even in the one, you take your laptop on the other sofa. Yeah, there's some cheating, yes. Yeah, that is not mathematical. That is movie magic. No, keep going. Yeah, so I took video of the empty room, and then I just clone out in a video editing software
26:22
the copy of me that's on the sofa. And yeah, so it's just cheating. Where there is the seam between the two rooms? There's no seam. That's the really nice thing about this kind of transformation. Is it mirrored? No, it's the transformation is you go to the complex plane
26:42
and then you square all of the numbers, and then that doubles up everything. Yeah, there's no mirroring. It's really very natural when you just apply complex transformations to the plane. We can talk about it later.
27:01
Another question? Hand the microphone over. So, what kind of video editing software lets you do that, or do you do it by hand? Yeah, so, yeah, I wrote the code myself, so it's written in Python.
27:21
So, from the video that I take, just the source, you can put it into, I use Premiere Pro, and then export all the frames to a directory, and then run through them one by one in Python. I was talking to people about it would be great to have this running in real time
27:41
with video streamed from the device, and Cinderella looks like it may be able to do that. So, maybe in the future, we'll have a version that's running in real time. At the moment, it takes a few seconds per frame, depending on what the transformation is.
28:03
Another question, yeah. How many pixels are there in just one image? So, this camera, I should say, is a Ricoh Theta S. It's about $350, so the price of a compact camera is not super expensive. This takes video when you unwrap it in the equirectangular
28:21
at 1920 by 960 pixels. So, that sounds like HD, but once you're looking at a small portion of the entire sphere, that's not that much. It takes photographs at 5,300 or so by whatever half of that is. So, you know, it's just the processing speed at the moment.
28:41
That will, they're expected to release another one later this year, which is expected to be 4K video or perhaps even 8K. Have you tried to see these kind of videos in a virtual reality, like with the Vibes?
29:02
Yeah, yeah. Yeah, it works, it's very, very immersive. I mean, so. No, I don't mean the 360 videos, but the processed one, like. Oh, how do you view it? Like, how do you actually view it on a headset? How does it feel, like when you have like this double space? Oh, yeah.
29:20
It has to play with your senses, I guess. Right, I mean, I guess, so there's this interesting thing. I mean, when you're viewing this video, there's a question of sort of how far you're zoomed in, so there's a little bit of distortion anyway, but it feels kind of natural. I mean, that's independent of doing any of these transformations, it feels very real,
29:41
and the transformations are all conformal, they preserve angles, so as long as you don't do too crazy a transformation, it feels very like, well, there's something strange going on, but, you know, everything feels kind of recognizable. So, yeah, I don't know if I have a very clean answer to that, but as you make the functions weirder,
30:03
it gets more like a visual effect and less like there's something strange going on with my world, but when you're just doing a little bit like zooms and so on, it feels really quite natural. It's gonna be a good music video for a while. Yeah, yeah, yeah, if anybody knows OK Go,
30:21
if they wanna do a spherical video, ah, that's another thing I have to say. People usually call it 360 video, and I hope we can all agree that this is incorrect. If anybody ever tells you or mentions 360 video when it's actually a sphere, please correct them.
30:45
Do we have another question? OK, then let's thank the speaker again. Thank you. Thank you.