annenberg media ♪ hola, tío pedro. hola, mercedes. hola, ramón. ¿qué tal? ¿cómo está fernando? ha dormido casi todo el día. estuvo el doctor. ya habló con el especialista de guadalajara. narrador: bienvenidos al episodio 32 de destinos. primero, algunas escenas de este episodio.

...el nieto de fernando y rosario estaba trabajando en una excavación arqueológica. ha habido un accidente. icuidado! pobre papá. si se enterara todo lo que está ocurriendo... es mejor que no se entere. sí, claro. pero si supiera... es demasiado: juan que no se lleva bien con pati carlos va a perder la oficina de miami los negocios andan mal nos aconsejan vender la gavia y ahora el nieto, el que todavía no conoce tal vez esté herido o... no te preocupes. verás que estará bien. también en este episodio vamos a aprender algo sobre una de las grandes civilizaciones prehispánicas de méxico: la civilización azteca. los aztecas eran gente trabajadora y también eran grandes guerreros.

por doscientos años, trabajaron y lucharon. poco a poco, los aztecas conquistaron las otras tribus de méxico. para principios del siglo dieciséis el imperio azteca se extendía de costa a costa. y vamos a ver cuáles son los planes de este señor. tengo dos semanas de vacaciones. mujer: ¿de vacaciones? ¿y por qué no las aprovechas para ir a méxico? ¿ir a méxico? captioning of this program is made possible by the annenberg/cpb project and the geraldine r. dodge foundation.

en el episodio previo mercedes estaba en el hospital cuidando a don fernando. allí recibió una llamada de carlos. mira. hablé al hospital a guadalajara. un momento. dime. al despertarse don fernando dijo que quería regresar a la gavia. el doctor va a pedir que te atienda un especialista. ya verás cómo pronto te pondrás bien. gracias, hijita. pero estaría mucho mejor en la gavia. te prometo que regresarás a casa en cuanto el doctor lo autorice. en la casa de ramón, juan se sentó a hablar con pati.

ella estaba pensativa y preocupada pero pronto, la conversación terminó en una discusión fuerte. mira, juan. voy a tratar de explicártelo una vez más. no es que mi vida profesional sea más importante que tú. la obra me necesita a mí en este momento. yo soy la autora. soy la directora. hay problemas y sólo puedo resolverlos yo. en la gavia, pedro y ramón recibieron noticias importantes. uds. señores, ¿qué nos recomiendan? que tomen medidas drásticas. que cierran la oficina en miami que es la causa de los problemas. después, se deben concentrar en la producción de acero. ¿y la gavia? lamentablemente, nuestra recomendación es venderla.

los dos decidieron no decirle nada a la familia por el momento. más tarde, pedro logró comunicarse con arturo quien le dio otras noticias alarmantes. ¿no ha hablado con raquel? ¿raquel? ya está en méxico? sí, me dejó un mensaje en el hotel. roberto, mi sobrino, el nieto de don fernando ha sufrido un accidente. refrescadas, raquel y angela regresaron al sitio de la excavación. ¿y sabes? estoy segura de que cuando lleguemos al lugar de la excavación ya sabrán algo de roberto. cuando ya era de noche, todavía no sabían nada de roberto.

iay! ¿qué pasa? tranquila. déjenme pasar. yo antes trabajaba en las minas. creo que puedo ayudar. tengo una idea. ven, ven. con permiso. sí, adelante. no te preocupes, hija. estos hombres son muy eficientes. hola, tío pedro. hola, mercedes. hola, ramón. ¿qué tal? ¿cómo está fernando? ha dormido casi todo el día. estuvo el doctor. ya habló con el especialista de guadalajara. miren, tengo algo que decirles. hablé con arturo iglesias.

dice que recibió un mensaje de raquel. roberto, el nieto de fernando y rosario estaba trabajando en una excavación arqueológica. ha habido un accidente. icuidado! ah, sí, eso estaba viendo en la televisión. ¿y cómo está roberto? no lo sé. adelante, pero ten mucho cuidad. cuidado. angela, háblame un poco más de tu hermano. ¿sabes qué civilización estudiaba? pues, no.

nunca le pregunté. sabemos que excavaban una tumba. me pregunto qué civilización indígena le interesaba tanto a roberto. raquel se pregunta qué civilización indígena le interesa tanto a roberto. la respuesta no es fácil. méxico tiene una rica herencia indígena y varias civilizaciones ocupaban el centro del país. una de las civilizaciones indígenas más conocidas de méxico es la azteca. aquí en el centro de méxico los aztecas gobernaron un vasto imperio de más de quince millones de personas. su centro era la gran ciudad de tenochtitlán una ciudad de jardines flotantes

numerosas casas, y grandes templos y pirámides. el lugar de origen de los aztecas se llamaba aztlán. según las leyendas, aztlán quedaba al norte de méxico pero no se sabe exactamente dónde. de aztlán, los aztecas migraron al sur, en busca de una señal. según una profecía de los sacerdotes una señal indicaría el lugar donde los aztecas debían establecerse. finalmente, en el lago de texcoco los aztecas vieron la señal: un águila devorando una serpiente sobre un nopal. hoy, esta señal es el símbolo de méxico.

los aztecas eran gente trabajadora y también eran grandes guerreros. por doscientos años trabajaron y lucharon. poco a poco, los aztecas conquistaron las otras tribus de méxico. para principios del siglo dieciséis el imperio azteca se extendía de costa a costa. bajo el dominio del emperador moctezuma segundo la civilización azteca florecía. cada día, más de sesenta mil personas acudían al mercado de tenochtitlán. los productos agrícolas eran abundantes. en grandes canchas, se jugaba al tlachtli el juego precursor del baloncesto.

florecían las artes. en los grandes templos los sacerdotes seguían ofreciendo sacrificios a los dioses aztecas. y claro, las guerras con otras tribus continuaban. luego, el veintidós de abril de mil quinientos diecinueve llegó un hombre con once barcos a la costa de méxico. este hombre era hernán cortés. cortés comenzó una de las más sangrientas conquistas de toda la historia mundial. en dos años, cortés conquistó a los aztecas. el emperador moctezuma fue asesinado. el gran imperio azteca

el imperio de los hombres guerreros pasó a ser una colonia española. aquí, en esta parte del valle de méxico vivía una de las muchas tribus conquistadas por los aztecas. fue esta cultura y sus ruinas las que roberto vino a explorar al centro de méxico. hombre: iya lo encontré! tranquilo. iayúdenme! ialguien ayúdeme! ilo encontraron! ¿de verdad? sí. te lo dije que eran eficientes.

ay, luis. has cambiado mucho, ¿eh? ¿sí? cómo? quiero decir... te ves más maduro, más apuesto... si te viera raquel... ¿y cómo está raquel? está bien. trabaja demasiado. acaba de llamar para avisar que... que va a posponer su regreso a los angeles. mira... estas fotos son recientes. ¿sí?

hombre: hola, luis. ¿qué tal? ¿cómo estás? qué gusto verte. gracias. igualmente. siéntate. gracias, gracias. cuéntame, ¿cómo has estado? bien, ahora tengo trabajo aquí. me he mudado a los angeles. pero en este momento tengo dos semanas de vacaciones. mujer: ¿de vacaciones? ¿y por qué no las aprovechas para ir a méxico? ¿ir a méxico? claro, méxico es muy bonito muy romántico y... y raquel está allí. ¿la verdad? me gustaría mucho. nosotros también pensábamos ir a ver a raquel y tomar unos días de vacaciones. ¿verdad, pancho? bueno, la verdad es que... podemos ir a visitar a los parientes de guadalajara. pero de todos modos, maría, los pasajes... hay agencias de viajes que ofrecen planes muy económicos. tú ocúpate de tus cosas y deja que yo me ocupe de esto. ¿qué dices, luis? ¿cree ud. que raquel querrá verme después de tantos años? iclaro!

ise va a poner muy contenta! con permiso. pase. sí. debo irme, ramón. tengo cita con arturo en el hotel donde está. veré si sabe más del accidente. sí. tal vez ya se haya podido comunicar con raquel. ojalá. esperaré a mercedes para regresar juntos a casa. bien. vean las noticias. por supuesto. también llamaré al canal por si tienen más información. bueno. nos vemos. hasta luego. hasta luego, tío. ¿te gusta? sí. un señor con anteojos y bigotes... un poco calvo... ¿quién es? el doctor que va a curar al abuelito.

ramón: hola. hola, papá. ¿qué tal? mira. qué bonito, maricarmen, que es, ¿eh? pobre papá. si se enterara todo lo que está ocurriendo... es mejor que no se entere. sí, claro. pero si supiera... es demasiado: juan que no se lleva bien con pati carlos va a perder la oficina de miami los negocios andan mal nos aconsejan vender la gavia y ahora el nieto, el que todavía no conoce tal vez esté herido o... no te preocupes. verás que estará bien. al menos yo soy afortunado. te tengo a ti y a maricarmen. yo también soy afortunada.

finalmente, arturo y pedro se conocen y arturo le cuenta a pedro lo de la búsqueda de angel. le cuenta de cómo él y raquel comenzaron a investigar el paradero de angel en la boca, un barrio de buenos aires. esa es la calle caminito. la última vez que vi a mi hermano fue aquí. sus amigos vivían por aquí. el problema es encontrar a alguien que lo recuerde. ¿y si preguntamos en las tiendas...? empecemos por ahí. le cuenta a pedro de cómo anduvieron de lugar en lugar haciendo preguntas y mostrando una foto de angel.

gracias. estamos buscando a una persona que frecuentaba esta zona. esta es su fotografía. no. no lo conozco. ¿por qué no preguntan en el negocio de al lado? la señora conoce a todo el mundo. muchasracias. se llama angel castillo. tenía amigos aquí en el barrio. no. mmm, sí. creo que lo recuerdo... pero no estoy seguro.

lo siento. por favor, trate de recordar. es muy importante. no. al principio me pareció pero no, no lo conozco. bueno. gracias. y arturo le dice cómo por fin encontraron a una persona que conocía personalmente a angel. claro que lo recuerdo bien. era mi amigo. arturo también le cuenta a pedro de la carta que tenía héctor. está fechada en san juan de puerto rico. le da las gracias por su recomendación.

dice que no es un verdadero marinero... y que sigue pintando. a través de la conversación, pedro llega a saber un poco de cómo raquel encontró a angela... perdone. ¿qué hace ud. aquí? estoy tomando una foto. ¿de la tumba de mis padres? y de cómo decidieron vir a méxico lo antes posible... y de cómo, por un mense de raquel arturo supo del accidente en la excavación. y yo he venido a conocer a mis sobrinos y a don fernando, por supuesto. ¿raquel se ha comunicado con ud. otra vez?

no. no he vuelto a tener noticias de ella. vengan. podemos acercarnos un poco. ¿pero no estorbaremos? ve tú, angela. yo me quedo aquí para no estorbar. está bien. iqué emoción! ¿no? están a punto de sacar a roberto del túnel. ¿recuerdan lo que hacía roberto en el momento del accidente? exploraba una tumba... una tumba indígena. en el centro de méxico existían varias culturas indígenas. ¿cuál es la más conocida?

¿la civilización azteca o la maya? una de las civilizaciones indígenas más conocidas de méxico es la azteca. aquí en el centro de méxico los aztecas gobernaron un vasto imperio de más de quince millones de personas. bueno, de todas las civilizaciones del centro de méxico, la azteca es la más conocida. pero según las leyendas los aztecas no eran originarios del centro de méxico. ¿de dónde vinieron entonces? ¿del norte, del sur, del este o del oeste? según las leyendas, los aztecas eran originarios del norte

de un lugar llamado "aztlán". los aztecas se establecieron en el lago de texcoco y fundaron la ciudad de tenochtitlán. ¿saben uds. cómo eran estos indígenas? ¿eran pacíficos? ¿o eran guerreros? los aztecas eran guerreros y lograron conquistar todo el centro de méxico. bueno. no sé si la tumba que excavaba roberto era de los aztecas o si era de una de las tribus que los aztecas conquistaron. bueno, pero lo más importante en este momento es que saquen a roberto de allí. mientras raquel y angela han estado esperando noticias de roberto

¿qué ha pasado en la familia castillo? ¿saben del accidente o no? sí, saben algo, porque pedro se lo dijo a mercedes y ramón en el hospital. ¿y le dijeron algo a don fernando o no le dijeron nada del accidente? a don fernando no le dijeron nada acerca del accidente en la excavación. mercedes no quería preocuparlo. ¿y se comunicó arturo con pedro o no? sí, y pedro fue a visitarlo en su hotel. allí arturo le contó todo lo de la búsqueda de angel en buenos aires. pero arturo no es la única persona

que tuvo una visita hoy. ¿quién más tuvo una visita? también tuvo una visita la mamá de raquel. ¿quién la visitó? ¿y cómo está raquel? está bien. trabaja demasiado. luis, el antiguo novio de raquel visitó a la mamá de raquel. y hablaron de ir a méxico para ver a raquel. claro, méxico es muy bonito muy romántico y... y raquel está allí. ¿cree ud. que raquel querrá verme después de tantos años? iclaro! ise va a poner muy contenta! parece que algo está pasando. me voy a acercar para ver mejor.

( gritas ) captioned by the caption center wgbh educational foundation

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>> we live in a world -- a reality -- ruled by straight lines. our streets, houses, cubicles,

the shelves in our closets -- well, not my closet -- but virtually all of our space is parceled into rectilinear grids. mathematicians were also ruled by straight lines -- some would say imprisoned by them -- for 2,000 years. but what is a straight line? and when is a straight line not straight? could it be when you live in a curved world? these questions are at the heart of one of mathematics' most mind-bending revolutions, a revolution that created whole new geometries and whole new worlds. the world is filled with an immense variety of shapes and structure. we use geometry to describe this physical space. lines, points, angles, and numbers characterize, organize, and transform the shape of the world -- even of space itself -- into coherent ideas. the word geometry comes from "geo," meaning "earth," and "meter," meaning "to measure." and anyone who sees the pyramids

knows that the ancients knew how to measure a straight line. much of early mathematics was all about geometry as well as simple computation. and it was largely developed for trade, agriculture, and building. but mathematics was also linked to religious observances, the motions of the planets, and the construction of calendars. for the greeks, mathematics was all about the tangible: things we can see and touch, things that are real and measurable. aristotle, in fact, considered even physics to be subservient to mathematics, believing that perfect motion had to take the form of the perfect geometric figures of straight lines and circles. the paradigm of this geometric outlook lies with the great euclid. with one monumental work, euclid set the stage for how we look at and measure the world geometrically. except there was a flaw, for there are more than circles and lines in our universe.

in his book, the elements, euclid gathered all the theorems of his day into a framework of basic theory and proofs. in it, he laid down five postulates and invented a method by which we can prove geometry's most basic truths. and it all starts with two points and a line on a piece of paper, a two-dimensional plane. now, the first three postulates are simple enough. the first states that a straight line segment can be drawn joining any two points, just like that. the second says any straight line segment can be extended indefinitely in a straight line. the third states that given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. the fourth states simply that all right angles are equal. and we need this so that we're sure the space we're working in is essentially the same everywhere. but the fifth is not so simple because it deals with the nature of parallel lines. it goes something like this: suppose a line segment intersects two straight lines in

such a way that two interior angles on the same side add up to less than two right angles. then the two lines, if extended indefinitely, must meet on that side on which the angles sum to less than the two right angles. now, that sounds logical. in fact, so logical that mathematicians actually believed that it wasn't so much a postulate -- that is, an assertion about geometry that was independent of the other four postulates -- but rather, that it was a logical consequence of the other postulates. in other words, mathematicians seemed sure that all we needed to know was in those first four postulates and that anything else could be derived from them. so for 2,000 years, mathematicians tried to prove that fifth postulate from the other four, and they failed. it became known as "the scandal of geometry." but why? well, again, implicit here is the idea of parallel lines. for what happens if those interior angles do add up to two right angles? then the lines must never meet. well, maybe.

the truth is that there's more than one kind of geometry. you see, we don't live on a flat world, we live on a curved one. it's just that it's not obvious to us, at least not from where we're standing. if a man's walking on a surface, then for all he knows, he's on a plane. locally, his world is flat. but if he's really on a curved surface, how does he know he's on a curved surface? from the point of view of just looking, he doesn't. so for the greeks, mathematics was largely about the geometry that they could see and measure in a rational way. so straight lines, right angles, parallel lines, those all seemed rational and measurable and seemed to be the ones that they wanted for the flat world. and even though they knew the world was round, it wasn't something they could measure with a ruler, and so they ignored it. logically, the five postulates made perfect sense, but from time to time, mathematicians challenged that fifth postulate, even as early as the 5th

century. but it wasn't until the 19th century when three mathematicians finally proved that there were indeed worlds out there and geometries that may seem impossible but obey logically consistent laws. in other words, the fifth postulate was independent of the other four. one of those three mathematicians was carl friedrich gauss. gauss, a professor at the university of gottingen in central germany, was known even in his own time as a great mathematician. napoleon's armies spared his town simply because he lived there. today he's mentioned in the same breath with archimedes and isaac newton. in 1822, gauss was hired to conduct a survey of hanover and make a map of the german countryside that would connect to an existing map of denmark. naturally, he was faced with the challenge of making a two-dimensional map from three-dimensional data, affected not just by changes in elevation but by the curvature of the earth. because he was a surveyor, he

knew that, measured over great distances, a triangle's angles do not add up to 180 degrees. but he also knew that the 180-degree rule was the equivalent of the fifth postulate. it is at this point that gauss made one of those leaps of intuition that have often propelled mathematicians to whole new understandings of our universe. gauss knew a triangle doesn't add up to 180 degrees in a curved world. this led him to wonder if space itself were curved. what if he threw out the fifth postulate? gauss realized was that there exists a whole different kind of geometry, one that moves beyond flat, euclidean space into curved spaces. he was so unnerved by this realization that he didn't publish it. and by not publishing, gauss muddied his legacy, because within a few short years the mathematicians janos bolyai and nikolai lobachevsky both came up with a similar geometry, what they called "absolute geometry" because it was established without euclid's fifth

postulate. still, the geometry they described was complicated, and it took the mathematician bernhard riemann to think about this problem differently. he asked a more basic question: can i characterize a world where curvature is the same everywhere? and he did this using gauss curvature. instead of looking at triangles on a surface, we look at circles. if we use a piece of string of length "r" to trace out a circle on a flat surface, we know that we get a circle whose circumference is 2 pi r. but if the surface is negatively curved, we get a number greater than 2 pi r. and if it's positively curved, we get a number less than 2 pi r. the discrepancy between the circumference of a circle on a curved surface and the circumference of a euclidean circle of the same radius can be used to define curvature. with this notion of curvature, riemann was able to classify surfaces of constant curvature, whether it be positive, negative, or zero.

i'm here with daina taimina, adjunct professor of mathematics at cornell university and an expert in helping people visualize non-euclidean geometries. daina, let's try to show people the distinctions between flat space and curved space. >> here we have a flat paper, which can be a representation of usual euclidean geometry. >> right, or a map, for example, right? >> yes, it can be like -- something flat. and here we are having a globe, which is a representation of the sphere. >> right, a curved space. >> yes. and if our geometries would be the same, then we should have no problem of wrapping this paper around the ball like we are doing it as a present. okay, let's try. >> let's give it a shot. roll it around, roll it around. yes, another perfectly wrapped birthday present, all right. so we've got some places which are flat and -- >> yeah, some places are flat, but then it gets too much, and then this is not very nice,

doesn't look very nice. so can't really put it on. once we are opening -- >> so we're actually seeing something quite fundamental, which is that flat space cannot go directly onto a curved space. there will always be some problems. >> yeah, and those are the problems which mapmakers since ancient times have encountered, how to represent this curved space on a map. so it means you have to decide what you are preserving, because there are different ways of making maps. you can preserve kind of like in angles or you can preserve proportions between distances, but you can't do both. no, you can't. >> there's always going to be some scrunching, just like we've seen here. >> and that scrunching happens because this is different curvatures. because our flat paper had zero curvature and this one is constant positive curvature. >> this representation can't both be distance-preserving and angle-preserving. is that right? >> yes, because flat maps, that's a euclidean plane. and if we want to travel --

especially when we are doing air travel, so we have to take in account that our earth is more like a sphere. >> so let's think about a potential travel plan. i'm in seattle and maybe i want to go to london, for example. so there's seattle over there and there's london over there. >> in the euclidean plane, it would look like a straight line. >> a famous saying: the shortest distance between two points is a straight line. >> let's see what happens on a sphere. >> and so now the shortest path i can imagine -- think of that as a string, and i want to think of using the least amount of string that's constrained to be on the globe, is that right? >> yes, it should be on a circle, and that is an arc of a great circle. >> an arc of a great circle, which is sort of like an equator that's at an arbitrary place. you define two points -- >> and now you see the difference between -- now you have two points, your seattle and london, which is connected with two straight line segments on a sphere. >> right, right. there's the short one and -- >> there is a short one, there is a long one. so therefore we can see that shortest and straight is not the same on a sphere. it's another difference between

two geometries. >> so now we've seen two kinds of curvature. there's positively curved space, like the sphere, there's flat space, like this tabletop. but, in fact, there are negatively curved spaces, and they are very mysterious and interesting, like hyperbolic space. so let's take a look at one. >> i'm martin steiner, i'm at the university of north carolina. i'm a research assistant professor both at the department of computer science and psychiatry. and this is of course kind of like a weird association: what has psychiatry to do with computer science? what we're doing here basically is neuroimaging. >> okay, so what i'm showing you here is a movie of the average brain as a function of age. now, this is human brains, they're all healthy, and you can see that at the beginning it's age 30 and at the end it's age 70, so they're all adults. what i want to do is show what are the essential changes in the brain as a function of age. and so now what you can see is i've taken all of the actual -- the images of the real subjects -- these are real subjects, not

averages -- and put them together in this movie that shows you the amazing amount of variability you see in the shape of the brain even between healthy subjects. now, what you're looking at here is this nonlinear shape space. and the idea is that we want to compute the mean on this curved space rather than in euclidean space. and so what we have to do is have a notion of centrality that i'm depicting here, where this is the average that we're going to compute, and it's going to minimize a sum of squared distances on this curve's manifold. and that's how we define the average. >> using all of the brains -- >> using all of the brains that we've collected. >> what we're doing basically is developing tools for the analysis of medical images. specifically, we look at the brain mostly. so the brains are imaged on a scanner -- whether that's an mri

scanner or a ct scanner -- and the images are brought to us to evaluate them. so this is not something that the radiologist does by simply looking at the images, but it's about extracting information that's more than just visually can be seen in these images. so for example, a volume of a certain structure that might change over time and gives us information whether that person may have alzheimer's or not. since we're computer scientists, so we are actually programming mathematical algorithms that then read in these images, they extract a certain value, certain measurement, and then give that measurement back either visually but also quantitatively in order to be analyzed by a statistician or by a clinician. there's many things we do nowadays that have not been possible before. we started basically just looking at images of the brain and trying to understand where is white matter, where is grey matter, where is the fluid that

nourishes the brain, the cerebrospinal fluid, that were basically the earliest ways to look at the brain. and not even looking at this in any pathology, but first trying to understand: can we actually evaluate the brain -- these brain images -- with any kind of pattern-recognition algorithms that we were using back then? that's maybe 10 years ago. >> so the reason that we need the curved space is that the shape differences and the shape changes that occur in the brain occur locally. there's twisting and bending, there are small structures moving on one side of the image while things are fixed on the other parts of the image, and so it also allows you to measure these local deformations, so changes in one part of the brain twisting, bending, things that you couldn't capture with the rotation or a translation. and finally, it allows you to put essentially a measuring stick in this curved space. >> and nowadays we are going more towards actually looking at the fine details of the brain. so nowadays we have algorithms that can take out of the brain

certain structure, that have certain function, let's say the hippocampus, which is implicated in schizophrenia, in epilepsy, or in alzheimer's disease, to be different. we can measure the volume or even the shape of the hippocampus and compare that. that's something totally new that we wouldn't have been able to do 10 years ago. >> what i'm showing here is a 3-d rendering at the top, but on the bottom, this is actually a slice that's taken out of this image of the average. and there you can see more of the deep structures. >> looks kind of like an mri slice. >> exactly. >> but it's also not an actual mri slice, but an average mri slice. when one starts to analyze these images, one first tries the most simple tools, and that's usually tools that are based in linear geometry. and you can only get so far with linear geometry. because our brains are very different from person to person, you often then need to go to a nonlinear space. as soon as we want to use

something like that, we need to have nonlinear geometry. so we need to go basically from a level of mathematics that we started out from to a higher level of mathematics basically that we're using nowadays. non-euclidean geometry, or non-euclidean analysis, gives us just more detailed information. its simply not possible to do in a euclidean way because the space that we're operating on is a hyperbolic space, so it's very easy, if you just try to apply euclidean geometry, to step out of the space of valid results. for example, if you look at shape analysis, where we actually try to describe certain parts of the brain -- let's say, the hippocampus -- and we try to describe it, how is it shaped, all of that basically -- whether that's now with the description of the interior, which we use with a skeletal description, or the description of the exterior, which uses surface description -- we both need for these things

non-euclidean geometry. in the surface, we go to a spherical geometry, so we do everything on the sphere. for the medial description or skeletal description, we actually need to go to a fully riemannian space and do all our analysis in the riemannian space. so non-euclidean geometry gives us new insight because it allows us to see differences, let's say, between one person and the other, looking at the brain not just in the linear fashion, but we really need to be able to do that. and it kind of like is a must for our research nowadays. what i really love about my work is really that it is multifaceted. the other thing, of course, that's really inspiring is that we are helping people, that the goal of what we are doing is something that's towards medicine that then actually will be applicable. >> flat surfaces and lines, spherical surfaces and lines, and two types of geometry to go with them.

but the curved world and a curved universe don't stop with spherical geometry. what worried gauss the most about throwing out euclid's fifth postulate was that if he ended up with a geometry that admits triangles of greater than 180 degrees, it would also be logical that there should be a geometry that admitted triangles of less than 180 degrees. bolyai and lobachevsky came to the exact same conclusions, and we now call the geometry described by them and gauss "hyperbolic geometry." let's see if daina can help us see this world, too. >> now, those were some beautiful virtual hyperbolic planes, hyperbolic structures, but you're going to show us, in fact, that these are really tangible things as -- and beautiful. >> we can have something tangible. like, for example, we were saying something about the circle. we are having a circle with a certain circumference in euclidean plane, and then the circles -- we were talking about the circles on a sphere. now, this is a crocheted version

of a so-called pseudosphere. >> a hyperbolic surface. >> yes, that's a hyperbolic surface. this is what is characteristic for a hyperbolic plane. it's like it's really connected with an exponential growth. >> that's right. so on a flat surface, we know that if you have a circle of radius "r," then the circumference is 2 pi r, so that the circumference grows what we would say linearly, right? but here -- >> yes, that grows exponentially. now let's go back to our old friend triangle. >> right, the triangle we've been using to show differences between all the geometries. >> we know very well. now, how do we get a triangle on this hyperbolic plane, which is essentially -- think about like a piece of paper. and now to get a straight line, we can simply fold it like we would do it with a euclidean plane, so we fold it. we fold it so it's straight. so then we can fold another one. we are getting a triangle, and we see these angles. just visually we can see that these angles are much less. like they are getting -- >> they are becoming thin.

>> they are becoming thin. and even more what's happening is that if we are having a larger piece and then we are trying to fold it -- >> that's right. so now we're extending it. we're going to have longer and longer. >> they are extended longer and longer. so then we even get so-called ideal triangle. >> if we had extended it to infinity, it would be the ideal triangle. >> if we extended all three sides to infinity, we are getting ideal triangle, whereas the limit of the sum of interior angles goes to zero. >> what about parallel lines in this world, is there anything special about those? >> there is something. and that is sometimes used, and this is something which was with nikolai lobachevsky and janos bolyai when they were talking about that in a space, there is a geometry where we can have a line. here we have -- here we have a straight line, and then we can have more than one line which is parallel to the given line.

and let's see how it happens. that we can fold the line, and we can see that this line -- well, what do we call parallel? now we have to think about it now. these lines won't intersect. >> not even out in infinity is the point. >> no, that's the point, that these lines won't intersect. and these are non-intersecting lines. so that's what we call parallel lines. >> non-intersecting lines. but the point being that -- but that they all go through this point. >> yes, exactly. >> that's the important thing. >> we can see that these ones, like for example, here there are three lines going through the one line, but none of them intersect. >> and they're all parallel to this, whereas in the plane, we know that if you have one line and you take a point off the line, then there's only one parallel line that goes through it. >> that's another difference between the geometries. that's a different behavior. and this is something where our intuition -- just we don't have intuition about it. and that's about like the same thing about these other shapes, which is -- i have met other scientists, which they are seeing something in their lab

which looks like this. >> all of a sudden something clicks, is that right? >> something clicks and they say, "oh, this looks like some of the mushrooms, or there are some --" >> sea anemone, for example. >> there are sea anemones, there are nudibranch: you see the shape and you have -- so when they say, "well, we didn't know we have to use hyperbolic geometry." and that's when the visualization is of help. >> so let's summarize and sort of take our tourist triangle on these three different geometries, right? so let's see. so the flat one there is our friendly triangle, 180 degrees, no problem. now he's going to wander off into -- into positively curved space on the sphere and fattens himself up, and more than 180 degrees. and then he slouches off to hyperbolic space and he's thinned down, slimmed down to less than 180 degrees. well, daina, thanks so much. and, in particular, thanks for showing everybody how truly beautiful mathematics is and can be. >> thanks for hosting me. i'm glad to be here. thank you. >> to get a picture of the hyperbolic plane on a larger scale requires some

mind-bending ingenuity. one way to visualize this enigmatic space was developed by the great mathematician henri poincare. this is a poincare disk, a mathematical model that's a map of the hyperbolic plane, where the plane is compressed into a disk. now remember, a hyperbolic plane has points that are infinitely far away. and in the mapping of the hyperbolic plane to the disk, the points at infinity become the circumference. the effect of this is that long distances in the hyperbolic plane get squashed in the poincare disk. so as i move away from the center in the poincare disk, approaching infinity, i'm actually moving exponentially far. here's another way to look at it: let's make a beautiful pattern on the disk. if we start with a euclidean plane and tile it with a pattern, what we have is a repeating pattern that is reflected across straight lines. now let's take another butterfly

and build the analogous pattern with it in the poincare disk by reflecting that pattern across straight lines in the disk. see what we get? the model poincare gave us is obviously useful to artists like jos leys, whose work we're looking at here. it's useful because it's what mathematicians call "conformal." that is, you can measure the angles in the plane and they're the same in the disk. because it's bounded by the circle, the design can be viewed in its entirety. poincare's model offers a precise, aesthetically pleasing way to depict diminishing and exponentially increasing figures within a circle. through its effect, we see an illusion of negative curvature on this two-dimensional surface. leys' work, like that of the 20th-century dutch artist m.c. escher, who inspired him, has produced hypnotically intricate patterns of images that seem otherworldly. and in fact, the hyperbolic plane lies outside the daily

experience of the physical world. or does it? daina taimina's models of a surface with constant negative curvature not only show us what these surfaces look like, they also give us clues about how to find hyperbolic geometry in the natural world. it may have taken mathematicians 2,000 years after euclid to discover hyperbolic space, but nature discovered it millions of years ago. it's embodied in the structures of marine animals like sea slugs, flatworms, and nudibranchs. when gauss realized that mathematics could see beyond euclid, it opened myriad possibilities. it's all about seeing ourselves like a bug on a curved world. you could say that, like the bug, we are prisoners of the space in which we live, unable to break free and observe the universe from the outside. but new geometries are giving mathematicians the ability and the tools to figure out things like the structure of space and the shape of our brains or even

the shape of the universe. captions by lns captioning portland, oregon www.lnscaptioning.com

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