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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:14.86,0:00:20.38,Default,,0000,0000,0000,,Today, I'd like you to join me\Nin examining what mathematics is.
Dialogue: 0,0:00:20.92,0:00:27.01,Default,,0000,0000,0000,,In elementary, middle, and high school,\Neveryone studies mathematics,
Dialogue: 0,0:00:27.01,0:00:31.22,Default,,0000,0000,0000,,and I am sure that all of you have too,
Dialogue: 0,0:00:31.50,0:00:36.42,Default,,0000,0000,0000,,but have you ever asked yourself,\N"What is mathematics?"
Dialogue: 0,0:00:37.42,0:00:41.10,Default,,0000,0000,0000,,Many people say they are not good at math,
Dialogue: 0,0:00:41.62,0:00:44.50,Default,,0000,0000,0000,,and quite a few people also seem to think
Dialogue: 0,0:00:44.50,0:00:47.89,Default,,0000,0000,0000,,that mathematics is about memorizing\Nand using formulas for computation.
Dialogue: 0,0:00:47.89,0:00:50.79,Default,,0000,0000,0000,,How about you? What do you think?
Dialogue: 0,0:00:51.06,0:00:55.50,Default,,0000,0000,0000,,There is no single answer to the question:\N"What is mathematics?"
Dialogue: 0,0:00:55.68,0:00:57.90,Default,,0000,0000,0000,,shared by all mathematicians.
Dialogue: 0,0:00:58.67,0:01:01.54,Default,,0000,0000,0000,,However, I am certain\Nthat no mathematician thinks
Dialogue: 0,0:01:01.54,0:01:05.70,Default,,0000,0000,0000,,mathematics is about memorizing\Nand using formulas for computation.
Dialogue: 0,0:01:06.20,0:01:10.12,Default,,0000,0000,0000,,For those who have such\Na misconception about mathematics,
Dialogue: 0,0:01:10.12,0:01:13.09,Default,,0000,0000,0000,,studying math may be quite painful.
Dialogue: 0,0:01:13.66,0:01:19.54,Default,,0000,0000,0000,,If you consider mathematics as something \Nfar from what it actually is about,
Dialogue: 0,0:01:19.54,0:01:22.89,Default,,0000,0000,0000,,then you will have trouble\Nstudying mathematics properly.
Dialogue: 0,0:01:23.79,0:01:26.07,Default,,0000,0000,0000,,Therefore, it is significant
Dialogue: 0,0:01:26.07,0:01:29.94,Default,,0000,0000,0000,,to delve into and understand\Nthe essence of mathematics.
Dialogue: 0,0:01:31.28,0:01:34.01,Default,,0000,0000,0000,,Today, we are going to explore mathematics
Dialogue: 0,0:01:34.01,0:01:37.16,Default,,0000,0000,0000,,by reflecting on the words\Nof Einstein and Hardy
Dialogue: 0,0:01:37.16,0:01:42.17,Default,,0000,0000,0000,,and analyzing what we pursue\Nin the study of mathematics.
Dialogue: 0,0:01:42.64,0:01:46.16,Default,,0000,0000,0000,,This will help us understand \Nthe essence of mathematics.
Dialogue: 0,0:01:47.89,0:01:49.61,Default,,0000,0000,0000,,Einstein said,
Dialogue: 0,0:01:49.61,0:01:54.55,Default,,0000,0000,0000,,"Mathematics is the poetry\Nof logical ideas."
Dialogue: 0,0:01:55.12,0:01:57.25,Default,,0000,0000,0000,,I like this remark
Dialogue: 0,0:01:57.25,0:02:01.18,Default,,0000,0000,0000,,and would like you\Nto think about what it means.
Dialogue: 0,0:02:02.43,0:02:06.17,Default,,0000,0000,0000,,First, as suggested\Nby the expression "logical ideas,"
Dialogue: 0,0:02:06.88,0:02:10.57,Default,,0000,0000,0000,,mathematics is logically rigorous.
Dialogue: 0,0:02:11.57,0:02:13.01,Default,,0000,0000,0000,,On the other hand,
Dialogue: 0,0:02:13.01,0:02:16.14,Default,,0000,0000,0000,,there is another important\Naspect of mathematics,
Dialogue: 0,0:02:16.14,0:02:20.46,Default,,0000,0000,0000,,illustrated in Einstein's remark\Nthat "mathematics is poetry,"
Dialogue: 0,0:02:20.100,0:02:25.27,Default,,0000,0000,0000,,suggesting that mathematics\Nis rich in imagination and creativity
Dialogue: 0,0:02:25.64,0:02:28.19,Default,,0000,0000,0000,,and that it is also artistic.
Dialogue: 0,0:02:28.100,0:02:33.80,Default,,0000,0000,0000,,You may think that something \Nthat is logically rigorous
Dialogue: 0,0:02:34.43,0:02:38.47,Default,,0000,0000,0000,,cannot also be creative
Dialogue: 0,0:02:38.47,0:02:41.45,Default,,0000,0000,0000,,and that logic and creativity \Ncontradict one another.
Dialogue: 0,0:02:41.86,0:02:43.86,Default,,0000,0000,0000,,However, in mathematics,
Dialogue: 0,0:02:43.86,0:02:48.96,Default,,0000,0000,0000,,the two elements highly complement\Nand reinforce each other.
Dialogue: 0,0:02:50.38,0:02:52.47,Default,,0000,0000,0000,,As we move through the rest of this talk,
Dialogue: 0,0:02:52.47,0:02:57.53,Default,,0000,0000,0000,,I hope you will start appreciating\Nthis profound remark by Einstein.
Dialogue: 0,0:02:59.82,0:03:05.14,Default,,0000,0000,0000,,Next, let's discuss\Nwhat we pursue in mathematics.
Dialogue: 0,0:03:06.29,0:03:11.20,Default,,0000,0000,0000,,You may think that mathematics\Nis the investigation of numbers,
Dialogue: 0,0:03:11.58,0:03:14.99,Default,,0000,0000,0000,,but some branches of mathematics\Ndo not concern numbers.
Dialogue: 0,0:03:16.09,0:03:18.64,Default,,0000,0000,0000,,To understand the essence of mathematics,
Dialogue: 0,0:03:18.64,0:03:22.44,Default,,0000,0000,0000,,we must look at a little more universal\Nand fundamental aspects of mathematics.
Dialogue: 0,0:03:23.61,0:03:25.61,Default,,0000,0000,0000,,The mathematician Hardy said,
Dialogue: 0,0:03:26.20,0:03:29.44,Default,,0000,0000,0000,,"A mathematician, like a painter or poet,
Dialogue: 0,0:03:29.79,0:03:32.97,Default,,0000,0000,0000,,is a maker of patterns."
Dialogue: 0,0:03:33.19,0:03:36.98,Default,,0000,0000,0000,,He also said mathematics\Nis about pursuing structures.
Dialogue: 0,0:03:37.59,0:03:42.86,Default,,0000,0000,0000,,This illuminates the foundational\Nimportance of creating, finding,
Dialogue: 0,0:03:42.86,0:03:46.32,Default,,0000,0000,0000,,and using structures in mathematics.
Dialogue: 0,0:03:48.78,0:03:50.51,Default,,0000,0000,0000,,You may not have a good idea
Dialogue: 0,0:03:50.51,0:03:55.04,Default,,0000,0000,0000,,about what kind of structures\Ncan be investigated in mathematics.
Dialogue: 0,0:03:55.56,0:03:58.83,Default,,0000,0000,0000,,Let's consider Pascal's triangle,\Nwhich is shown here,
Dialogue: 0,0:03:58.83,0:04:01.38,Default,,0000,0000,0000,,to examine several examples\Nof mathematical structures
Dialogue: 0,0:04:01.38,0:04:05.60,Default,,0000,0000,0000,,and find some of the essential\Ncharacteristics of mathematics.
Dialogue: 0,0:04:06.85,0:04:10.04,Default,,0000,0000,0000,,This triangle appears\Nin many mathematical problems.
Dialogue: 0,0:04:10.36,0:04:12.80,Default,,0000,0000,0000,,For instance, you probably\Nlearned this in school:
Dialogue: 0,0:04:12.80,0:04:19.28,Default,,0000,0000,0000,,(x＋y)² ＝ x² + 2xy + y²
Dialogue: 0,0:04:19.96,0:04:23.92,Default,,0000,0000,0000,,Notice that the coefficients\Nform the third row of the triangle.
Dialogue: 0,0:04:24.59,0:04:27.100,Default,,0000,0000,0000,,Well, let's not go into technical details.
Dialogue: 0,0:04:29.32,0:04:33.47,Default,,0000,0000,0000,,A surprising number of structures \Ncan be found in this triangle,
Dialogue: 0,0:04:33.47,0:04:37.22,Default,,0000,0000,0000,,and they can help us understand\Nmany important mathematical concepts.
Dialogue: 0,0:04:37.49,0:04:40.61,Default,,0000,0000,0000,,One of them is fairly easy to find.
Dialogue: 0,0:04:40.94,0:04:42.98,Default,,0000,0000,0000,,I wonder if you can find it.
Dialogue: 0,0:04:43.22,0:04:48.14,Default,,0000,0000,0000,,Perhaps the most obvious structure\Nis that of vertical symmetry.
Dialogue: 0,0:04:48.35,0:04:50.93,Default,,0000,0000,0000,,The numbers appearing on the left half
Dialogue: 0,0:04:50.93,0:04:53.26,Default,,0000,0000,0000,,also appear on the right half\Nin the same manner.
Dialogue: 0,0:04:53.62,0:04:57.77,Default,,0000,0000,0000,,As you will see later,\Nsymmetry is an important structure.
Dialogue: 0,0:04:59.30,0:05:04.36,Default,,0000,0000,0000,,Can you figure out how to determine \Nthe value of each element in the triangle?
Dialogue: 0,0:05:05.24,0:05:09.71,Default,,0000,0000,0000,,The numbers along the right \Nand left edge are all 1.
Dialogue: 0,0:05:10.16,0:05:14.26,Default,,0000,0000,0000,,Each of the other numbers is the sum \Nof the two numbers directly above it.
Dialogue: 0,0:05:14.39,0:05:16.63,Default,,0000,0000,0000,,For instance, we have 2 = 1 + 1 ,
Dialogue: 0,0:05:17.17,0:05:18.52,Default,,0000,0000,0000,,3 = 1 + 2,
Dialogue: 0,0:05:18.68,0:05:20.45,Default,,0000,0000,0000,,and 6 = 3 + 3.
Dialogue: 0,0:05:21.13,0:05:24.76,Default,,0000,0000,0000,,This triangle extends endlessly,
Dialogue: 0,0:05:25.17,0:05:29.22,Default,,0000,0000,0000,,but we can construct it \Nwithout memorizing the numbers
Dialogue: 0,0:05:29.22,0:05:32.26,Default,,0000,0000,0000,,but by performing simple computations
Dialogue: 0,0:05:32.26,0:05:34.66,Default,,0000,0000,0000,,once we understand\Nthe underlying structures.
Dialogue: 0,0:05:35.60,0:05:37.15,Default,,0000,0000,0000,,This simple example
Dialogue: 0,0:05:37.15,0:05:40.69,Default,,0000,0000,0000,,shows the importance of understanding\Nand applying mathematical structures.
Dialogue: 0,0:05:42.35,0:05:47.55,Default,,0000,0000,0000,,Next, let's examine a structure created\Nby the odd numbers in Pascal's triangle.
Dialogue: 0,0:05:48.29,0:05:51.71,Default,,0000,0000,0000,,This figure shows the positions\Nof odd numbers in the first nine rows:
Dialogue: 0,0:05:51.74,0:05:56.50,Default,,0000,0000,0000,,1, 3, 5, 7, and so on, in white.
Dialogue: 0,0:05:57.33,0:06:01.74,Default,,0000,0000,0000,,By focusing on where odd numbers\Nappear in the triangle,
Dialogue: 0,0:06:02.17,0:06:04.50,Default,,0000,0000,0000,,we can find an interesting structure.
Dialogue: 0,0:06:04.90,0:06:07.65,Default,,0000,0000,0000,,Using your imagination,
Dialogue: 0,0:06:08.27,0:06:14.79,Default,,0000,0000,0000,,picture a huge Pascal's triangle\Nconsisting of 128 rows.
Dialogue: 0,0:06:15.62,0:06:20.98,Default,,0000,0000,0000,,This one consists of only nine rows.\NThe triangle of 128 rows must be enormous.
Dialogue: 0,0:06:22.27,0:06:29.06,Default,,0000,0000,0000,,This figure shows in white the positions \Nof the odd numbers in the first 128 rows.
Dialogue: 0,0:06:29.84,0:06:32.68,Default,,0000,0000,0000,,Clearly, there is a structure.
Dialogue: 0,0:06:32.68,0:06:36.24,Default,,0000,0000,0000,,Can you describe it?
Dialogue: 0,0:06:37.11,0:06:39.86,Default,,0000,0000,0000,,There is one big triangle.
Dialogue: 0,0:06:39.86,0:06:41.06,Default,,0000,0000,0000,,And if you look closely,
Dialogue: 0,0:06:41.06,0:06:46.34,Default,,0000,0000,0000,,you can see that it consists\Nof three smaller triangles.
Dialogue: 0,0:06:46.73,0:06:52.56,Default,,0000,0000,0000,,And if you take another look,\Nyou will see that each of these triangles
Dialogue: 0,0:06:52.84,0:06:56.68,Default,,0000,0000,0000,,are again made up\Nof three smaller triangles.
Dialogue: 0,0:06:56.99,0:06:59.59,Default,,0000,0000,0000,,This process repeats itself.
Dialogue: 0,0:07:02.34,0:07:03.67,Default,,0000,0000,0000,,Technically,
Dialogue: 0,0:07:03.67,0:07:06.38,Default,,0000,0000,0000,,this is an example of "fractals,"
Dialogue: 0,0:07:06.65,0:07:11.62,Default,,0000,0000,0000,,in which the structure of the whole\Nis the same as that of its parts.
Dialogue: 0,0:07:11.62,0:07:13.88,Default,,0000,0000,0000,,This property is called "self-similarity."
Dialogue: 0,0:07:14.94,0:07:18.87,Default,,0000,0000,0000,,Fractals can be found\Nin coastlines, plants,
Dialogue: 0,0:07:18.95,0:07:21.62,Default,,0000,0000,0000,,crystals, and intestinal walls,
Dialogue: 0,0:07:21.71,0:07:25.24,Default,,0000,0000,0000,,to name a very few examples\Nof fractals found in nature.
Dialogue: 0,0:07:25.54,0:07:28.37,Default,,0000,0000,0000,,Remember the fractal structure.
Dialogue: 0,0:07:28.62,0:07:32.30,Default,,0000,0000,0000,,Near the end of this talk, you will\Nunexpectedly see another example of it.
Dialogue: 0,0:07:33.60,0:07:36.60,Default,,0000,0000,0000,,We can't possibly look at all\Nthe mathematical structures today,
Dialogue: 0,0:07:36.60,0:07:39.55,Default,,0000,0000,0000,,but as a mathematician,\Nthere is another structure
Dialogue: 0,0:07:39.55,0:07:41.78,Default,,0000,0000,0000,,that I can't leave the room\Nwithout showing you.
Dialogue: 0,0:07:41.95,0:07:44.67,Default,,0000,0000,0000,,If we draw these diagonals,
Dialogue: 0,0:07:44.67,0:07:48.27,Default,,0000,0000,0000,,and for each of them,\Nwe sum the numbers on it,
Dialogue: 0,0:07:49.48,0:07:54.15,Default,,0000,0000,0000,,the sum is 1 for the first\Nand second diagonals,
Dialogue: 0,0:07:54.48,0:07:56.78,Default,,0000,0000,0000,,and 2 for the third diagonal.
Dialogue: 0,0:07:57.14,0:08:02.33,Default,,0000,0000,0000,,By continuing this, we obtain\Nthe sequence of numbers shown here.
Dialogue: 0,0:08:02.64,0:08:05.63,Default,,0000,0000,0000,,This is called the Fibonacci sequence.
Dialogue: 0,0:08:05.63,0:08:08.16,Default,,0000,0000,0000,,It has great mathematical significance
Dialogue: 0,0:08:08.16,0:08:10.44,Default,,0000,0000,0000,,appearing in a lot\Nof mathematical analyses.
Dialogue: 0,0:08:11.32,0:08:13.51,Default,,0000,0000,0000,,Like fractals,
Dialogue: 0,0:08:13.75,0:08:20.02,Default,,0000,0000,0000,,the Fibonacci sequence is also effective \Nin characterizing structures in nature.
Dialogue: 0,0:08:20.62,0:08:23.00,Default,,0000,0000,0000,,For example, we arrange squares
Dialogue: 0,0:08:23.00,0:08:28.69,Default,,0000,0000,0000,,whose side lengths are set\Nto the Fibonacci numbers as shown here.
Dialogue: 0,0:08:29.38,0:08:31.92,Default,,0000,0000,0000,,They can be arranged so neatly, \Ndon't you think?
Dialogue: 0,0:08:32.78,0:08:36.84,Default,,0000,0000,0000,,Why do these squares\Nfit together so neatly?
Dialogue: 0,0:08:37.30,0:08:40.04,Default,,0000,0000,0000,,Think about it when you get home tonight.
Dialogue: 0,0:08:40.93,0:08:44.19,Default,,0000,0000,0000,,Using these squares, \Nwe can create a spiral
Dialogue: 0,0:08:44.22,0:08:47.75,Default,,0000,0000,0000,,that is effective in characterizing \Nvarious objects in nature:
Dialogue: 0,0:08:48.03,0:08:50.57,Default,,0000,0000,0000,,a seashell,
Dialogue: 0,0:08:51.17,0:08:52.28,Default,,0000,0000,0000,,a galaxy,
Dialogue: 0,0:08:52.48,0:08:55.91,Default,,0000,0000,0000,,or a hurricane, for instance.
Dialogue: 0,0:08:57.95,0:08:58.96,Default,,0000,0000,0000,,As you can see,
Dialogue: 0,0:08:58.96,0:09:03.15,Default,,0000,0000,0000,,Pascal's triangle can open the door\Nto many different structures
Dialogue: 0,0:09:03.15,0:09:05.73,Default,,0000,0000,0000,,just by exploring\Nits mathematical applications.
Dialogue: 0,0:09:06.22,0:09:10.53,Default,,0000,0000,0000,,There are a wide variety\Nof fields in mathematics,
Dialogue: 0,0:09:10.88,0:09:13.05,Default,,0000,0000,0000,,but in each of them,
Dialogue: 0,0:09:13.05,0:09:18.10,Default,,0000,0000,0000,,we create, find, or apply structures
Dialogue: 0,0:09:18.10,0:09:22.20,Default,,0000,0000,0000,,in order to understand something\Nand establish mathematical truths.
Dialogue: 0,0:09:22.87,0:09:26.44,Default,,0000,0000,0000,,We must understand\Nthis aspect of mathematics
Dialogue: 0,0:09:26.44,0:09:27.99,Default,,0000,0000,0000,,to properly study its nature.
Dialogue: 0,0:09:29.67,0:09:34.44,Default,,0000,0000,0000,,We can also find some of the important\Ncharacteristics of mathematics.
Dialogue: 0,0:09:35.40,0:09:38.76,Default,,0000,0000,0000,,By analyzing Pascal's triangle, we found
Dialogue: 0,0:09:39.07,0:09:44.01,Default,,0000,0000,0000,,the fractal, the Fibonacci sequence,\Nand the Fibonacci spiral.
Dialogue: 0,0:09:44.61,0:09:47.95,Default,,0000,0000,0000,,Many other structures can be found\Nin this triangle as well.
Dialogue: 0,0:09:48.26,0:09:52.20,Default,,0000,0000,0000,,In mathematics, we often discover\Nthat diverse concepts and structures,
Dialogue: 0,0:09:52.22,0:09:55.08,Default,,0000,0000,0000,,which ostensibly have nothing\Nto do with each other,
Dialogue: 0,0:09:55.09,0:09:59.51,Default,,0000,0000,0000,,are intricately intertwined\Nat profound levels.
Dialogue: 0,0:10:00.98,0:10:05.39,Default,,0000,0000,0000,,To understand these various\Nconcepts or structures
Dialogue: 0,0:10:05.39,0:10:08.15,Default,,0000,0000,0000,,as well as their connections\Nwith each other,
Dialogue: 0,0:10:08.73,0:10:14.03,Default,,0000,0000,0000,,we need both rigorously logical thinking\Nand a rich imagination.
Dialogue: 0,0:10:15.39,0:10:17.72,Default,,0000,0000,0000,,No matter how imaginative you are,
Dialogue: 0,0:10:17.72,0:10:19.70,Default,,0000,0000,0000,,with imagination alone,
Dialogue: 0,0:10:19.96,0:10:24.22,Default,,0000,0000,0000,,you will not be able to recognize\Nthe links between these concepts.
Dialogue: 0,0:10:24.94,0:10:26.42,Default,,0000,0000,0000,,On the other hand,
Dialogue: 0,0:10:26.42,0:10:31.41,Default,,0000,0000,0000,,you cannot come up with these concepts\Nwith logical thinking alone.
Dialogue: 0,0:10:32.04,0:10:33.60,Default,,0000,0000,0000,,I believe that Einstein's remark
Dialogue: 0,0:10:33.60,0:10:36.78,Default,,0000,0000,0000,,that mathematics\Nis the poetry of logical ideas
Dialogue: 0,0:10:36.78,0:10:39.91,Default,,0000,0000,0000,,is starting to resonate more with you.
Dialogue: 0,0:10:41.61,0:10:46.75,Default,,0000,0000,0000,,We can also find a rather mysterious\Nfeature of mathematics here.
Dialogue: 0,0:10:47.35,0:10:49.43,Default,,0000,0000,0000,,It is that mathematics
Dialogue: 0,0:10:49.43,0:10:55.79,Default,,0000,0000,0000,,is astonishingly effective\Nin describing structures in nature.
Dialogue: 0,0:10:57.22,0:10:58.64,Default,,0000,0000,0000,,Galileo said,
Dialogue: 0,0:10:58.64,0:11:02.80,Default,,0000,0000,0000,,"The book of nature is written \Nin the language of mathematics."
Dialogue: 0,0:11:03.35,0:11:05.07,Default,,0000,0000,0000,,Feynman said,
Dialogue: 0,0:11:05.07,0:11:08.60,Default,,0000,0000,0000,,"To those who do not know mathematics,\Nit is difficult to get across
Dialogue: 0,0:11:08.60,0:11:12.74,Default,,0000,0000,0000,,a real feeling as to the beauty,\Nthe deepest beauty, of nature."
Dialogue: 0,0:11:13.11,0:11:14.75,Default,,0000,0000,0000,,Furthermore, Wigner said
Dialogue: 0,0:11:14.75,0:11:19.62,Default,,0000,0000,0000,,that mathematics is \N"unreasonably effective"
Dialogue: 0,0:11:19.62,0:11:20.99,Default,,0000,0000,0000,,in the natural sciences.
Dialogue: 0,0:11:21.59,0:11:25.94,Default,,0000,0000,0000,,It is no wonder that mathematics\Nis indispensable to sciences.
Dialogue: 0,0:11:25.94,0:11:27.64,Default,,0000,0000,0000,,Now, we are going to move on
Dialogue: 0,0:11:27.64,0:11:31.17,Default,,0000,0000,0000,,to the last topic of this talk:\Nmathematics and beauty.
Dialogue: 0,0:11:31.51,0:11:34.51,Default,,0000,0000,0000,,Hardy, the mathematician\Nwhom I mentioned earlier, said -
Dialogue: 0,0:11:35.04,0:11:38.61,Default,,0000,0000,0000,,when we evaluate mathematical ideas -
Dialogue: 0,0:11:38.72,0:11:45.29,Default,,0000,0000,0000,,"Beauty is the first test:
Dialogue: 0,0:11:45.91,0:11:50.24,Default,,0000,0000,0000,,there is no permanent place\Nin this world for ugly mathematics."
Dialogue: 0,0:11:50.52,0:11:51.75,Default,,0000,0000,0000,,This remark suggests
Dialogue: 0,0:11:51.77,0:11:57.87,Default,,0000,0000,0000,,that the pursuit of beauty\Nis a paramount element of mathematics.
Dialogue: 0,0:11:58.32,0:12:00.53,Default,,0000,0000,0000,,Let's investigate this further.
Dialogue: 0,0:12:01.86,0:12:06.31,Default,,0000,0000,0000,,First, we must examine beauty itself.
Dialogue: 0,0:12:07.20,0:12:12.29,Default,,0000,0000,0000,,What do you find beautiful?
Dialogue: 0,0:12:13.25,0:12:16.91,Default,,0000,0000,0000,,Picture something you consider beautiful.
Dialogue: 0,0:12:18.03,0:12:22.23,Default,,0000,0000,0000,,Mt. Fuji holds a special place\Nin the hearts of Japanese people.
Dialogue: 0,0:12:22.70,0:12:28.24,Default,,0000,0000,0000,,But why do we think it is beautiful?
Dialogue: 0,0:12:30.02,0:12:32.52,Default,,0000,0000,0000,,It is indeed beautiful, isn't it?
Dialogue: 0,0:12:33.55,0:12:37.36,Default,,0000,0000,0000,,Distinctively, this mountain shows \Nan almost perfect vertical symmetry
Dialogue: 0,0:12:37.36,0:12:39.58,Default,,0000,0000,0000,,no matter from what angle\Nit is viewed from.
Dialogue: 0,0:12:39.82,0:12:42.65,Default,,0000,0000,0000,,Mathematically, \Nthe feature is referred to
Dialogue: 0,0:12:42.65,0:12:45.64,Default,,0000,0000,0000,,as "symmetry with respect\Nto the central vertical axis."
Dialogue: 0,0:12:46.10,0:12:49.50,Default,,0000,0000,0000,,Also distinctive is the very smooth\Ncontour of the mountain,
Dialogue: 0,0:12:49.50,0:12:51.72,Default,,0000,0000,0000,,which we can characterize mathematically
Dialogue: 0,0:12:51.72,0:12:56.84,Default,,0000,0000,0000,,using a function expressing the curve \Nand its differentiability.
Dialogue: 0,0:12:57.32,0:13:00.93,Default,,0000,0000,0000,,These are slightly \Ndifficult technical terms,
Dialogue: 0,0:13:01.12,0:13:07.72,Default,,0000,0000,0000,,but they are all mathematical structures,\Nwhich we discussed earlier.
Dialogue: 0,0:13:08.69,0:13:14.80,Default,,0000,0000,0000,,Do these features or concepts\Nhave something to do with beauty?
Dialogue: 0,0:13:16.90,0:13:21.35,Default,,0000,0000,0000,,Recently, various scientific studies\Nhave been conducted
Dialogue: 0,0:13:21.35,0:13:23.91,Default,,0000,0000,0000,,to investigate our aesthetic sense,
Dialogue: 0,0:13:23.91,0:13:27.88,Default,,0000,0000,0000,,and they have helped us better understand\Nhow mathematics is linked to beauty.
Dialogue: 0,0:13:28.45,0:13:30.47,Default,,0000,0000,0000,,One such study showed
Dialogue: 0,0:13:30.67,0:13:33.52,Default,,0000,0000,0000,,that among images\Nhaving similar amounts of data,
Dialogue: 0,0:13:33.52,0:13:39.06,Default,,0000,0000,0000,,those that were perceived as beautiful\Nhad high data compressibility.
Dialogue: 0,0:13:39.33,0:13:41.60,Default,,0000,0000,0000,,Let's think about it.
Dialogue: 0,0:13:42.45,0:13:48.17,Default,,0000,0000,0000,,First, data is compressible when it can be\Ndownsized without loss of information.
Dialogue: 0,0:13:48.51,0:13:50.96,Default,,0000,0000,0000,,Pascal's triangle,\Nwhich we've just examined,
Dialogue: 0,0:13:50.96,0:13:52.31,Default,,0000,0000,0000,,has vertical symmetry.
Dialogue: 0,0:13:52.41,0:13:57.18,Default,,0000,0000,0000,,Numbers appearing on the left half\Nreappear on the right half.
Dialogue: 0,0:13:57.59,0:13:59.80,Default,,0000,0000,0000,,Therefore, to draw this triangle,
Dialogue: 0,0:13:59.98,0:14:06.56,Default,,0000,0000,0000,,we don't need all the numbers\Nbut only those on the left half.
Dialogue: 0,0:14:07.72,0:14:11.22,Default,,0000,0000,0000,,We can copy the left half, flip the copy,\Nand paste it on the right side
Dialogue: 0,0:14:11.22,0:14:13.01,Default,,0000,0000,0000,,to create the whole triangle.
Dialogue: 0,0:14:13.43,0:14:18.68,Default,,0000,0000,0000,,In this case, it means the original data\Ncan be compressed by half,
Dialogue: 0,0:14:19.36,0:14:23.18,Default,,0000,0000,0000,,so Pascal's triangle\Nhas high data compressibility.
Dialogue: 0,0:14:24.68,0:14:27.97,Default,,0000,0000,0000,,The same is true of the Mt. Fuji image.
Dialogue: 0,0:14:28.47,0:14:34.06,Default,,0000,0000,0000,,By copying the left half of the picture\Nand pasting the flipped copy on the right,
Dialogue: 0,0:14:34.06,0:14:38.07,Default,,0000,0000,0000,,we can create a picture that is virtually\Nindistinguishable from the real image.
Dialogue: 0,0:14:38.11,0:14:41.74,Default,,0000,0000,0000,,Therefore, the picture of Mt. Fuji\Nalso has high data compressibility,
Dialogue: 0,0:14:42.10,0:14:47.65,Default,,0000,0000,0000,,and the study has demonstrated that images\Nlike this are perceived as beautiful.
Dialogue: 0,0:14:48.97,0:14:50.75,Default,,0000,0000,0000,,What makes this picture interesting
Dialogue: 0,0:14:50.93,0:14:56.90,Default,,0000,0000,0000,,is the horizontal symmetry due to the\Nreflection of the mountain on the lake.
Dialogue: 0,0:14:57.100,0:15:01.38,Default,,0000,0000,0000,,Don't you think it enhances the beauty?
Dialogue: 0,0:15:03.33,0:15:07.20,Default,,0000,0000,0000,,We've just established that images\Nthat are perceived as beautiful
Dialogue: 0,0:15:07.20,0:15:10.15,Default,,0000,0000,0000,,have high data compressibility.
Dialogue: 0,0:15:10.35,0:15:11.70,Default,,0000,0000,0000,,Now, in general,
Dialogue: 0,0:15:11.70,0:15:15.86,Default,,0000,0000,0000,,what kind of image\Nhas high data compressibility?
Dialogue: 0,0:15:16.80,0:15:19.45,Default,,0000,0000,0000,,Like the Mt. Fuji example,
Dialogue: 0,0:15:19.65,0:15:23.03,Default,,0000,0000,0000,,the data compressibility\Nof an image is high
Dialogue: 0,0:15:23.03,0:15:25.57,Default,,0000,0000,0000,,when it has a mathematical structure.
Dialogue: 0,0:15:26.81,0:15:32.42,Default,,0000,0000,0000,,This is one of the images perceived\Nas beautiful in the aforementioned study.
Dialogue: 0,0:15:33.51,0:15:37.80,Default,,0000,0000,0000,,A mathematical structure was used
Dialogue: 0,0:15:37.80,0:15:40.62,Default,,0000,0000,0000,,to draw this face very effectively,
Dialogue: 0,0:15:40.85,0:15:43.77,Default,,0000,0000,0000,,and so the data compressibility \Nof this image is very high.
Dialogue: 0,0:15:44.03,0:15:47.96,Default,,0000,0000,0000,,Can you figure out what sort\Nof mathematical structure was used?
Dialogue: 0,0:15:49.64,0:15:51.46,Default,,0000,0000,0000,,What do you think?
Dialogue: 0,0:15:52.46,0:15:56.16,Default,,0000,0000,0000,,Actually, we already discussed it earlier.
Dialogue: 0,0:15:56.62,0:16:00.46,Default,,0000,0000,0000,,It's a fractal that is used here.
Dialogue: 0,0:16:02.01,0:16:08.87,Default,,0000,0000,0000,,Considering what the study has shown,\Nwe can say that in mathematics,
Dialogue: 0,0:16:09.12,0:16:12.69,Default,,0000,0000,0000,,we pursue what we consider beautiful
Dialogue: 0,0:16:12.69,0:16:16.72,Default,,0000,0000,0000,,or those things that constitute them.
Dialogue: 0,0:16:19.32,0:16:24.09,Default,,0000,0000,0000,,A recent study in neuroscience \Nalso supports this view.
Dialogue: 0,0:16:25.00,0:16:30.40,Default,,0000,0000,0000,,The brain region shown in green here,\Ncalled the medial orbitofrontal cortex,
Dialogue: 0,0:16:30.65,0:16:36.85,Default,,0000,0000,0000,,which has been known to respond\Nto natural scenes, paintings, and music
Dialogue: 0,0:16:36.85,0:16:39.71,Default,,0000,0000,0000,,that we perceive as beautiful,
Dialogue: 0,0:16:39.94,0:16:42.92,Default,,0000,0000,0000,,was shown in the study
Dialogue: 0,0:16:42.92,0:16:46.49,Default,,0000,0000,0000,,to responds similarly\Nto mathematical concepts or structures.
Dialogue: 0,0:16:47.19,0:16:49.07,Default,,0000,0000,0000,,For the human brain,
Dialogue: 0,0:16:49.07,0:16:51.45,Default,,0000,0000,0000,,the beauty pursued in mathematics
Dialogue: 0,0:16:51.45,0:16:57.09,Default,,0000,0000,0000,,shares similarities\Nwith the beauty in nature and art.
Dialogue: 0,0:16:59.42,0:17:01.30,Default,,0000,0000,0000,,It is time to wrap up this talk.
Dialogue: 0,0:17:02.16,0:17:05.37,Default,,0000,0000,0000,,"Mathematics is the poetry\Nof logical ideas."
Dialogue: 0,0:17:06.47,0:17:11.06,Default,,0000,0000,0000,,"We pursue and create structures\Nin mathematics."
Dialogue: 0,0:17:12.01,0:17:16.50,Default,,0000,0000,0000,,"We pursue 'the beauty' in mathematics."
Dialogue: 0,0:17:16.59,0:17:20.10,Default,,0000,0000,0000,,These characteristics might have been
Dialogue: 0,0:17:20.10,0:17:24.26,Default,,0000,0000,0000,,a little unexpected or surprising to you.
Dialogue: 0,0:17:25.52,0:17:31.14,Default,,0000,0000,0000,,You may have gained a new perspective
Dialogue: 0,0:17:31.14,0:17:35.34,Default,,0000,0000,0000,,on what we perceive as beautiful.
Dialogue: 0,0:17:36.70,0:17:39.85,Default,,0000,0000,0000,,Next time you find something beautiful,
Dialogue: 0,0:17:39.85,0:17:43.64,Default,,0000,0000,0000,,you might think about mathematics.
Dialogue: 0,0:17:44.49,0:17:46.25,Default,,0000,0000,0000,,Maybe, maybe not.
Dialogue: 0,0:17:46.46,0:17:49.48,Default,,0000,0000,0000,,If you do, then don't you think\Nthat is a substantial change?
Dialogue: 0,0:17:50.19,0:17:52.22,Default,,0000,0000,0000,,Thank you for listening.
Dialogue: 0,0:17:52.74,0:17:54.89,Default,,0000,0000,0000,,(Applause)