5th HLF – Lecture: Art, Mathematics and Computer Science
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2
00:00
Internet forumLogicMathematicianSelf-organizationPresentation of a groupLecture/Conference
00:41
Bit rateInternet forumPresentation of a groupDivisorLevel (video gaming)Interpreter (computing)NumberMathematicsSquare numberMathematical objectBuildingInteractive televisionWordRight angleSpeech synthesisData miningMereologySurfacePolygonLecture/Conference
01:48
Mathematical objectSquare numberTriangleCirclePythagorean tripleMeeting/InterviewLecture/Conference
02:22
Mathematical objectSquare numberLecture/ConferenceMeeting/Interview
02:50
Internet forumBit rateSolid geometryCubeMathematicsOctahedronLecture/ConferenceMeeting/Interview
03:16
IcosahedronMathematical objectNatural numberWordCategory of beingLecture/ConferenceMeeting/Interview
03:59
Mathematical objectKey (cryptography)HypercubeLecture/Conference
04:16
KnotPolygonDepictionMathematical objectSolid geometryRight angleLecture/ConferenceMeeting/Interview
04:39
Right angleCircleKnotDepictionMathematicianLecture/ConferenceMeeting/Interview
05:06
Internet forumSphereKnotDisk read-and-write headSurfaceMathematicsLecture/ConferenceMeeting/Interview
05:25
Internet forumExponentiationData structureMathematicsRight angleMathematical objectClient (computing)Connected spaceLevel (video gaming)SurfaceMeeting/InterviewLecture/Conference
06:18
PentagonDiagonalFigurate numberRule of inferenceWell-formed formulaProjective planeGreatest elementMathematicsAcoustic shadowCircleRight angleTesselationLecture/Conference
07:04
Internet forumTesselationGroup actionTheoremMathematicsProof theorySymmetry groupSymmetry (physics)QuicksortLecture/ConferenceMeeting/Interview
07:49
Penrose, RogerTesselationRight angleMultiplication signSymmetry (physics)PentagonLecture/Conference
08:15
Perspective (visual)Multiplication signMathematicsMathematical structureRight angleRule of inferenceProjektive GeometrieLecture/Conference
08:31
Perspective (visual)Right angleView (database)Disk read-and-write headAdditionReal numberDistortion (mathematics)Meeting/Interview
09:18
View (database)Point (geometry)Student's t-testMathematical structureMathematical object1 (number)MathematicianRight angleRadio-frequency identificationLecture/ConferenceMeeting/Interview
10:30
Mathematical structureEuklidische GeometrieNon-Euclidean geometryPlanningEndliche ModelltheorieHyperbolische GeometrieRight angleLecture/ConferenceMeeting/Interview
10:57
DepictionEndliche ModelltheorieHyperbolische GeometrieShared memoryTesselationHeptagonPlanningLecture/ConferenceMeeting/Interview
11:31
Internet forumPixelRevision controlLevel (video gaming)NeuroinformatikComputer scienceInteractive televisionMathematicsTouchscreenSolid geometryComputer programmingLecture/Conference
12:08
DodekaederQuicksortStorage area networkFerry CorstenSolid geometryMathematical objectLecture/Conference
12:33
QuicksortBounded variationWeb pagePlatonic solidCellular automatonCASE <Informatik>IcosahedronMeeting/Interview
13:03
NeuroinformatikRight angleComputer programmingDodekaederTask (computing)Meeting/InterviewLecture/Conference
13:32
Protein foldingLecture/ConferenceMeeting/Interview
13:55
FraktalgeometrieFigurate numberPythagorean tripleRectangleMereologyEmbedded systemCodecRight angleLecture/ConferenceMeeting/Interview
14:28
SpiralRectangleTriangleSierpinski triangleFrequencyFamilyCASE <Informatik>Lecture/ConferenceMeeting/Interview
15:08
TriangleNatural numberSpiralFraktalgeometrieLecture/Conference
15:26
Circle1 (number)SpacetimeInsertion lossDivisorSphereConvex setLecture/Conference
15:52
List of unsolved problems in mathematicsTablet computerSphereCircleLecture/ConferenceMeeting/Interview
16:10
CircleRight angleMultiplication signMathematical objectSeries (mathematics)MathematicianReal numberNeuroinformatikFatou-MengeMereologyTerm (mathematics)SummierbarkeitLecture/Conference
17:11
NeuroinformatikMandelbrot setReal numberLetterpress printingSet (mathematics)Lecture/Conference
17:31
Computer scienceNeuroinformatikCASE <Informatik>MathematicsDifferent (Kate Ryan album)Real numberDimensional analysisVideo gameEmbedded systemFraktale DimensionData storage deviceNatural numberIdentical particlesFrequencyDrop (liquid)Lecture/Conference
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Lattice (order)TesselationAlgorithmStudent's t-testSphereDodekaederQuicksortWordLecture/ConferenceMeeting/Interview
19:50
Lecture/ConferenceMeeting/Interview
Transcript: English(auto-generated)
00:00
Good morning. It will be presented by Pier Giorgio Odifredi from Italy. He's an Italian mathematician, logician, and popularizer of science, with more than 400 appearances on national TV in Italy. So we're very happy to have him here. And I ask Pier Giorgio to step up here.
00:22
And the floor is yours. I'm very pleased to be here, and thank you to the organizers.
00:40
And the reason I'm here among all those beautiful minds is because I helped organize the exhibition on mathematics and art that is just outside this building past the square that some of you have seen already. In particular, Sir Michael Attia yesterday, who improvised our whole lecture of explanations and interpretations.
01:04
So I proposed to him that we switch our presentation. We switch our presentation. But he refused because he said that he could easily give mine, but I could hardly give his. So we'll stick to this, right? And what I'm going to do, I only have 15 minutes,
01:21
so I'll divide my talk in three parts, five minutes each. And the first part would be to give an idea of the interactions between art and mathematics at a very surface level. In a sense, you know, it's manifest. It's the level in which you actually see mathematical objects inside the works of art.
01:42
So I just go through a number of pictures to give you examples. We start with polygons. We go to India, for example. This is the Sri Yantra that is still used nowadays, and it's been used for millennia. And you see, you know, it's all very geometrical. We go to France, for example, and we visit medieval cathedrals.
02:03
We look at the rose windows, and sometimes we see the Pythagorean star or other mathematical objects depicted. We go to China, for example, 1700s, and we find, so to speak, modern painters that depict circles, triangles, and squares, and that's the title of this work.
02:22
Or we go to Russia at the beginning of the 1900s, and we find the supremacist Malevich who has this painting which is called Black Square on a White Canvas, and he has many of this kind. But then, of course, you know, if you go to museums and you look at the Bauhaus and the steel works,
02:41
in particular Kandinsky here or Mondrian, you really see mathematical art. The objects are mathematical objects. And if we go to solids, then of course, you know, we even go back in history 4,500 years ago, the Great Pyramid in Giza, and of course, you know,
03:01
it's half of a platonic solid is, half of a octahedron. If you go to Mecca, you have to be a Muslim to do that, you know, but at least, you know, you've seen pictures. This is the famous Kaaba, which in Arabic means just cube, right? So again, you know, mathematics, where you wouldn't expect it to find it. And this is a painting by, actually a drawing by Leonardo,
03:25
who depicted many mathematical objects, 60 of them, as illustrations of a book of the 14 and 1500s by Luca Pacioli, which was called the Divine Proportion,
03:40
the Divina Proportione. And this is an interesting object, because many of you will know it, that's the truncated icosahedron that in nature comes up as a molecule of fullerene, and it's used, actually kicked all around the world, you know, as a soccer ball. And that's a more interesting mathematical object.
04:02
This is a surrealist painter, Salvatore Dali in Spanish, and it's a three-dimensional unfolding of a four-dimensional object, namely a four-dimensional hypercube. So you see that you find many things like this, polygons and solids, but also less usual objects,
04:22
like knots. So these are very old depictions of knots. On the left, you see Leonardo. He had six drawings of very intricate knots. And on the right, you see Dürer, a German painter and an architect. And that's a more recent one.
04:41
It's called the Annot, because that's what it is. It looks like a very complicated knot, but you can actually unknot it, and it becomes just a simple circle, right? And here you have on the left, Escher, with a depiction of the trifoil knot, and on the right, a sculpture by a mathematician
05:01
and a sculptist named Ferguson. And again, you know, the same knot. And that's the first picture we see from the exhibition. It's called the contortionist, but it's just really a knot with a sphere that it's supposed to be ahead. And of course, surfaces are subject of mathematics
05:21
on one side and of art on the other side. So you see here a work by Max Bill, who was a famous exponent of the Bauhaus in Germany, and he did many of those sculptures. They're scattered around Germany, but also around the world. And you see here two works,
05:41
two more works from the exhibition, with a more sophisticated topological surface, which is called the Klein bottle. It's a simple Klein bottle on the left and a triple Klein bottle on the right. But there's a deeper level that we can go and see in the connections between mathematics and art.
06:03
And it is not just depicting mathematical objects, but using mathematics to build the structure of your artistic work. So this is just one of many, infinitely many, probably, examples of the golden section. Again, an unusual one.
06:20
You see on the left the painting again by Dalí, which is called the Atomic Lida. And there's nothing mathematically, apparently, that you see, except for the little ruler in there, which is floating in the air and projects a strange shadow. But if you look at the sketch that Dalí gave,
06:41
you actually see that lots of mathematics in there. You know, he built a circle. Inside the circle, there was a pentagon. Inside the pentagon's the diagonals. So you build a Pythagorean star. And inside the star, you actually fit the figures that you have on the canvas to give them proportions. And there's even the formula for the golden section
07:01
on the right bottom. Tessellations, for example. You go to Granada, and you visit the Alhambra, which is one of the great monuments of the Moorish art. And on the wall, all over, you find tessellations like this. And you don't see much mathematics at first sight.
07:22
But each tessellation has a symmetry, so you could look for the symmetry group. And as we know from the classification theorem of Jodorov at the end of the 1800s, there are actually 17 of those groups. And it's interesting that in Granada, at the Alhambra,
07:42
you find all 17 of them. So it's a sort of proof of the classification theorem by examples. But even more interesting, this is 500 or 600 years ago. You go to Central Asia, and you find, again, an Arabic tessellation with pentagonal symmetry, which is supposed to be a recent discovery
08:02
by Sir Roger Penrose. And of course, also in chemistry, quasi-crystals. But again, they were well known by the Arabs. And obviously, every time you see a painting that is done with the perspective rules,
08:22
well, that's a mathematical structure, right? Because perspective is nothing else than projective geometry, right? But sometimes the artists use perspective in a very artistic and sophisticated way. So if you look at the left, for example, this is Mantegna's deposition. This is Jesus Christ on his deathbed.
08:42
But it's not depicted with the real perspective. And you see it on the right. This is a picture of a real person, right? And you see that the feet should be much bigger and the head should be much smaller, which means that Mantegna used perspective, but he actually divided the painting into strips.
09:04
And he used different viewpoints farther and farther from the subject to give a more balanced view. Actually, it's a correction of perspective, because if you use the real perspective, it will be all distorted. And that's an even more impressive example.
09:21
It's by a guy named Manet, whom you probably know. It's called the Bar of the Folie Bergere. And you have this bar with a waitress and you're in front of the waitress and she looks into your eyes, right? And behind there, there's a mirror. So you're not supposed to see her back reflected in the mirror and let alone to see yourself
09:43
reflected in the mirror. But Manet actually depicts all these things and even does stranger things, because if you look at the bottles on the bottom left, they're not the same ones reflected in the mirror. So there's something wrong. And for one and a half centuries, people thought, well, of course, you know,
10:02
Manet was an artist, he was not a mathematician, right? So he just decided to take artistic freedom, right? But 15 years ago, a PhD student actually discovered that you can place all the objects in the picture and look at them from a particular point of view. And this is exactly what you see
10:22
from that particular point of view. So you have to discover the mathematical structures that lie behind the work of art. And of course, you know, sometimes, you know, there are even sophisticated mathematical structures like non-Euclidean geometry. Again, you know, we find a painting by Dali,
10:42
which is Saint Sebastian on the Horse. But if you look at the background, this is nothing else than the half plane model of hyperbolic geometry, which is quite surprising. I mean, because Dali was a surrealist painter, right? And that's one very well known depiction
11:00
of the Poincare and Beltrami model of hyperbolic geometry done by Escher. And that's, again, one of the works you find, you see in the exhibition, it's a flying saucer with the tessellations or one of the topics that we discussed. And you find a tessellation, a regular tessellation,
11:23
despite the appearances done by heptagons. And of course, it's a tessellation of the hyperbolic plane. And then finally, last five minutes, there's an even deeper level of interactions between art and mathematics. And it is when computer science comes in.
11:42
And computers actually produce new tools for arts, because instead of having brushes, for example, and pigments and canvases, now you have programs, pixels and screens, computer screens. So we look at solids again,
12:02
but at generalized versions of solids, which at the beginning had nothing to do with computers, of course. This is a very old one, 1420. And some of you may have stepped on it literally, because it's in Venice, in San Marco's Basilica. And it's actually on the pavement exactly at the exit.
12:21
So people just get out of the Basilica, and step on this beautiful mathematical object, which is a stellated dodecahedron. But once you start generalizing solids like this, then you can do all sorts of things. This is from a book of the 1500s in Germany by a guy named Yamnitzel,
12:41
who did hundreds of variations on the regular platonic solids. In this case, these are just two pages from the book. And you see the left page is variations on icosahedron. And if you look at the first column and the intermediate one,
13:01
you'll see in the exhibition an example generated by the computer. So the tools, the brushes, and the pencils are substituted by a program. But the computer can do much more for you. Because, for example, it could give you things like this, which is an intersection of a hundred dodecahedrons.
13:21
And of course, if you wanted to do it by hand, it would be an impossible task. But with the computers, you can do not only a hundred, but a thousand, or whatever you want. So it's really enlarging the possibility that we have. And that's, again, a work from the exhibition, is the truncated icosahedron,
13:41
or if you wish, the soccer ball, in which the faces have been inverted inside and outside. And you obtain those beautiful forms, which again would be very difficult or impossible to do just by hand. And the last example is fractals. And of course, they've been quoted in the previous talk.
14:01
And people think that fractals are really the essence of computerized art. But of course, they go back very, very far away in history. The first fractal is probably due to the Pythagoreans. You see it here is the golden rectangle on the left, and the Pythagorean star on the right,
14:22
which are self-similar figures. They have parts which are similar to the whole. And in the exhibition, you find a beautiful fractal, which is based on the golden spiral inserted in the golden rectangle. This is a famous fractal,
14:41
which is called the Sierpinski triangle. But again, centuries before Sierpinski, these things were known in Italy, in this case, by the Cosmati family, which were paving churches all around the center of Italy. And you still find beautiful things like this.
15:01
You wonder how come the Sierpinski triangle ended up in the Renaissance period. And that's, again, a work from the exhibition. It's a Sierpinski fractal, but not a triangular one, but a pentagonal one, projected on a three-dimensional golden spiral,
15:23
which is like the skull of a nautilus. And that's the last example. It's due to Leibniz in a letter. Leibniz, to a friend, said, well, why don't you take a circle and insert into the circle three maximal circles, the red ones, and then you fill the spaces with other circles, right?
15:42
And you build a fractal like this. You can do it on convex or concave circles. You could do it on spheres. And again, this has been used for centuries. Professor Mori could tell us a lot about these things called sangaku, which are wooden tablets that people put in temples in Japan,
16:02
not with prayers, but with mathematical problems, mostly related to circles and spheres and things like this. And that's an example of Dali, again, using the same trick, namely circles into circles, only this time is skulls into skulls. And that's called the face of war, right?
16:21
A very effective painting. And in the exhibition, you'll find a portrait of Alkvarism done by using the same fractal and the same trick. And that's more or less what I wanted to say. This is a final example of what computers can do for mathematicians. In the top part of the picture,
16:43
you see a Julia set from 1925, done by hand by Julia himself. And in the bottom part, you see the real set. Namely, it was a much more complicated thing that you could do actually by hand. And it is more or less like having
17:00
a slowly convergent series and doing just the sum of the first few terms. You'll be very far away from the real object, right? And that's Mandelbrot set. Of course, we cannot avoid quoting it, but if you look up on the left, you see the first printout of Mandelbrot set in the 1970s.
17:23
Which looks very different, of course, from the real thing, that computers actually can do much better today. So what is the conclusion? The conclusion is that after centuries, millennia, and decays in case of computers, mathematics and computer science
17:41
have become indistinguishable from heart. And the last two examples are this. This is a painting by Pollock. And for those of you who know Pollock, they know that he painted by a technique called dripping. It just dripped things on the canvas. And the surprising thing is that
18:00
you can actually precisely date the Pollock's works by using the fractal dimension, the Hausdorff dimensions. In different periods of his life, he had different dimensions intuitively. So there was nothing mathematical, nothing computerized in principle, but in the end, it becomes computerized.
18:21
And on the opposite side, this is a painting or a work of art done by Anne Burns, who's an artist, a computer science artist, in which everything is done by computer, but it's indistinguishable from a picture, a real picture of real nature. So the two things actually merge and exchange their role.
18:43
And that's the last picture. And it's the one, again, one of the works in the exhibition is a sort of summing up, right? You see everything or many of the things that we have seen, a solid is the dodecahedron. You see a Sierpinski-like fractal,
19:01
which is done using a different algorithm. You see a toroid in which a hyperbolic tessellation is being reflected. So you're all invited to go and visit the exhibition. Unfortunately, the artist who did all these works was an Italian scientist and artist,
19:22
died a few months ago while preparing the exhibition. So it's very sad that he couldn't be here in my place to explain his works. And the son and the wife are here. They can explain the details of the composition of each of the works. And there are mathematical students
19:42
volunteered to do the same thing. I'll also be there. So hopefully it will be a useful complement to this beautiful meeting. Thank you very much and have a good lunch.