How geometry created modern physics - with Yang-Hui He

What's the story behind the five axioms of Euclidean geometry - and how is post-Euclidean geometry linked to modern physics?
Watch the Q&A with Yang-Hui He here: • Q&A: How geometry crea...
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From geometry’s classical beginnings, via the Renaissance and the Enlightenment, to the present day, Yang-Hui He takes us on a journey through time and space, culminating in our understanding of spacetime itself. In the 19th century, mathematicians such as Carl Gauss and Bernhard Riemann considered what would happen if we relaxed Euclid’s axioms. The result was the explosion of post-Euclidean geometry, which paved the way for Einstein’s theory of relativity and the birth of modern physics.
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This Discourse was filmed at the Ri on 24 February 2023, in partnership with the London Institute for Mathematical Sciences (LIMS).
Discourses are one of the Ri’s oldest and most prestigious series of talks. Since 1825, audiences in the theatre have witnessed countless mind-expanding moments, including the first public liquefaction of air by James Dewar, the announcement of the electron by JJ Thomson and over 100 lectures by Michael Faraday. In more recent times, we have had Nobel laureates, Fields medal winners, scientists, authors and artists - all from the cutting-edge of their field. Discourses are an opportunity for the best and brightest to share their work with the world.
Steeped in nearly two centuries of tradition, a Discourse is more than just a lecture. To keep the focus on the topic, presenters begin sharply at 7:30pm without introduction and we lock the speaker into a room ten minutes ahead of the start (legend has it that a speaker once tried to escape!). Some of our guests and speakers dress smartly for our Discourse events to add to this sense of occasion.
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Prof. Yang-Hui He is a Fellow at the London Institute, Professor of Mathematics at City, University of London, Tutor in mathematics at Merton College, Oxford, and Chang-Jiang Chair of physics at Nankai University in China. He obtained his BA at Princeton, where he graduated summa cum laude and was awarded the Shenstone Prize and Kusaka Prize. He did his MA at Cambridge (Distinction, Tripos) and earned his PhD at MIT. After a postdoc at the University of Pennsylvania, Yang joined Oxford University as the FitzJames Fellow and an STFC Advanced Fellow. He works at the interface of string theory, algebraic and combinatorial geometry, and machine learning.
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[@theccieguy]

I am still amazed by the fact we get these great lectures for free, anywhere we want. Thank you RI.

[@RussellChapman99]

Without geometry, life would be pointless.

[@jamieanderson7757]

I like your line of argument.

[@seancharles1595]

I am always going off at a tangent.

[@OfficialGOD]

Life would be infinity

[@phoenixfire8978]

That was a BAD pun, take my like.

[@MiroslawHorbal]

Thank you for the chuckle 😄

[@kennethemmanuel3065]

This is one of the best Royal Institutions Lectures I have watched. Showing how mathematics has evolved over time - and consequently physics as well. For the first time I finally got to see how mathematics is the beautiful language of the universe. I loved the professor’s enthusiasm and his way of communicating these ideas. Beautiful video.

[@IsomerSoma]

@Winged Jupiter ?

[@IsomerSoma]

@Winged Jupiter Has anyone ever taken your uneducated and incoherrrent ideas seriously?

[@chanrasjid8688]

Sad to dampen any euphoria about physics going non-Euclidean. God created the world in His Image. It is not certain God accepts images other than the Euclidean. Chan Rasjid, Singapore.

[@RD-sk8cx]

Yang-Hui He is such a good communicator, would have loved to be in his class. This talk brought back what first kindled my interest in mathematics in school - Euclidean proofs. And then he builds up from there. Brilliant.

[@mickybadia]

Hmm, I was just thinking, I find his style a bit too narcissistic to be really pleasant.

[@RD-sk8cx]

​@@mickybadia being pleasant was not a trait I would associate with even one of my mathematics teachers or professors. What we appreciated was mastery of the subject, the ability to transmit knowledge clearly, solve complex problems logically and methodically. We never really cared whether they were pleasant or not. I don't find Yang-Hui narcissistic, you do, perhaps this variance reflects our individual personalities rather than his.

[@TazPessle]

I remember in high school (about 14) we were all told to make a triangle out of paper, tear off each corner and place them on a line in our excercise book. It was the only math lesson that we used glue sticks, but something about that practical aspect allowed me to start manipulating geometric problems in my head to make the solution more apparent.

[@Niyoriety]

Nice, I remember my teachers following a education system to teach us kids and it was nowhere near this level of creativity and inspiration. I learned more in 1 hour than about 8 years of school.

[@brianlee9310]

I could watch these brilliant lectures all day and night long and I have . I AM

[@richardyoung5293]

you are what homie

[@MrElvis1971]

​@richard YOUNG possibly... "I AM" doing what he said he could do. As in, in the present tense... verb

[@muthukumaranl]

Thank you so much to Prof. Yang-Hui He and RI for this amazing lecture!

[@mariavm9178]

What a phenomenal lecture! Thank you, Prof. Yang-Hui He and Ri. Looking forward to reading your books!

[@lukschs1]

En serio a esto le llaman lectura??

[@tehdii]

I salute him for making "A compact history of infinity" by David Foster Wallace and making it digestible and clear in a 60 min lecture... Bravo!

[@armchairgravy8224]

Linear algebra somehow clicked for me in a way calculus did not. Seeing R3xR really helped. I feel so lucky to see RI lectures like this.

[@F_L_U_X]

I forgot how amazing Ri videos are. Been falling asleep to The History of the Universe lately. Daniel Tammet comes to mind when listening to this guy explain his passion. I would love to see them hang out.

[@raphaelkaume]

What an amazing teacher! Such a joy to listen to! I mean these are fearsome topics for most but he had me smiling and laughing several times. His students are most fortunate and now so is the rest of the world able to access this. Thanks Yang-Hui He!!!😃

[@danielparsons2859]

I absolutely loved this lecture. I struggle with being told what to do as opposed to what to think about which engages me.

[@photon434]

You create an infectious excitement in all your lectures. Amazing!

[@primajump]

One of the most riveting experiences for me. Very interesting in deed! I am grateful that these talks are available for us. Thank you !

[@koenth2359]

The 5th axiom as given at 7:16 would only hold in 2 dimensions. Later, at 44:41 it is represented in a better way, using the word *parallel,* which is also valid in general for Euclidian spaces of higher dimension.

[@ktiwari31]

I loved it! It’s one of the best lectures I have found in a long long time! Yong-Hui is amazing!

[@jjson775]

Great presentation and kudos to professor He for dressing so well in this revered venue.

[@Danny-hb1zb]

Love whenever Yang-Hui He does a presentation it’s always brilliant 👏🏻

[@as-qh1qq]

Such a beautiful lecture, like a well written screenplay.

[@freeheeler09]

I would have liked to take a course or seven from you! Your love of mathematics is inspiring!

[@mastershooter64]

This guy is awesome dude i watched his RI lecture on string theory

[@otbricki]

Superb presentation. Just one comment tho - Al Kwarizmi got some of his ideas on algebra and numbers from India. His most famous book is preserved only in a Latin translation, Algoritmi de numero Indorum (“Al-Khwārizmī Concerning the Hindu Art of Reckoning”)." It included the manual method of taking square roots that has been taught to school children for centuries. This is the first use of the word algorithm. Al Kwarizmi made other important contributions as well, including the most accurate trig tables available at the time.

[@shankarvk922]

Well said.

[@Syntax753]

One of my favourite RI presentations - but biased as Euclid's Elements is one of the only books I own. Fantastic!

[@5Andysalive]

still waiting for the movie.

[@Neilhuny]

What an inspiring lecture, very infectious enthusiasm

[@atulisrockin]

This was a fantastic lecture, presenting some fairly complex ideas spanning centuries in a coherent and easily digestible narrative.

[@dag410]

Awesome job. Great video!

[@andywong2135]

This lecture is so good !

[@taohidhasnat7093]

There is art everywhere in the universe and geometry gives the highest perfection to art.

[@chyldstudios]

Wonderful tour through the history of the authors of mathematics.

[@theeyeguysBR]

Beautiful lecture!

[@rajendrapatodia]

this was awesome. mesmerizing.

[@moinmalik1320]

A truly mesmerizing lecture.

[@VHenrik007]

Great lecture, thank you!

[@dosesandmimoses]

Great lecture! Gratitude

[@joebloggs396]

I watched some of this and found him very jilted, so surprised there is so much praise.

[@s.c.r.e.a.m.forp.e.a.c.e.3541]

Excellent lecture!

[@greymonwar9906]

It is a pleasure to go through everything in the presentation. Interesting and informative, the last bell ring is kinda mean.

[@andrewharrison8436]

That was erudite and entertaining. A very enjoyable and far ranging hour.

[@iiitokyo]

Great and Generous Insitute of the World! Great Lecturer!

[@Greg41982]

This lecture was amazing. Enough said.

[@alsaba5203]

Bravo. Thank You.

[@prarthananeema9774]

this is amazing i want more content on geometry

[@theccieguy]

Great Job

[@haroldmiller9559]

Thanks!

[@hassannabil9792]

Super lecture

[@as-qh1qq]

34:10 At this point, it should be mentioned that new, nature inspired axioms have entered the picture, thus creating physics

[@edgarcastillo2804]

Wow. This is so nice.

[@pierorivera7162]

Thanks

[@TheRoyalInstitution]

Thank you! We're so glad you enjoyed the lecture

[@HughChing]

Hi Prof. He, Good talk. I had never missed a geometry problem in school and never had to take finals. The last problem I solved was Steiner-Lehman Theorem. But today, I realize that something is completely wrong, for which I invent The Obviousness Theory of Proof based on the 16 Methods of Reason, which says that Obviousness is different for different way of reason. Warmest regards, Hugh

[@keep-ukraine-free528]

Student to teacher: "I don't see any point in geometry. It seems infinitesimal, approaching the pointless. Maybe I've reached my Limits!"

[@PawlTV]

Love this guy

[@bombyxmori3431]

this was so good

[@ronpaulrevered]

I have many questions about epistemology. What is the epistemological standing of logic and geometry and math? Is empiricism the method for all claims of knowledge? What are the domains of logic, and empiricism and how do they coexist?

[@baoboumusic]

Aside from everything else, that was pretty darn impressive pronunciation of the Greek text.

[@ProPollys]

Brilliant! Love that he's wearing black tie 🧐

[@zack_120]

49:00 - Actually these right angles don't exist in the absolute sense since any lines of any length on thus surface are not straight but curved and therefore can't define an angle. So it is only relative in the practicle sense.

[@tradetor]

I believe most people going to a RI meeting did come to see a math proof 😂

[@zholud]

So how exactly did you prove step 2 using axioms?

[@keep-ukraine-free528]

It's suggested, in life & in mathematics: When taking a _tangent,_ stick to the *_original point._*

[@zelmoziggy]

Why is it that if the yellow angles and/or the blue angles are unequal, the lines must intersect?

[@mrp8811]

every single royal institution presentation so far i could do. it should be debate

[@jonvalen440]

I was hoping I would see how geometry has influenced “modern” physics, such as how geometry shaped string theory or other frontiers in physics

[@mhunt25]

Ahhh, I'm glad I saw this

[@KaliFissure]

Q:. Why do we imagine the universe is spherical when we know that gravity curves space? Shouldn't we use a cardioid since it shows the full range of tangent angles?

[@paulrattray8121]

Such a good lecture. So much of this I vaguely remember but could still follow.

[@jonvalen440]

Bummed to see that it is minute 44 and we have yet to talk about modern physics……this was more like “a history of geometry”. Even then, it would’ve been necessary to talk about algebra and physics as those were major turning points for geometry. Algebra for obvious reasons, and newton mechanics/calculus because it calculates areas of tons of shapes

[@koenth2359]

54:56 Maxwell's middle name is written 'Clerk' but pronounced as Clark.

[@Saurabhmaths1999]

Love from India

[@staffanlantz3876]

20:45 Did the Oxford calculators investigate acceleration by Rolling a ball down a slope, hence a triangle?

[@dougr.2398]

How could Archimedes have been omitted? Heiberg’s 1906 revelation of Codex B as a palimpsest showed that he (Archimedes) used calculus in a non-rigorous way to determine the volumes and surface areas of the sphere, cylinder and cone. Lost (maybe) until 1906, with the unknown exceptions of Codices A & C (also list at unknown dates).

[@amittiwari8]

if intresested u can go through bbc documentry of macus du soutoy. it will let you all know about eastern contribution to mathematics. to name a few u can learn about pingalacharya, lilavati,bhaskaracharya,suryasiddhantam,aryabhattiyam etc. and the famous ramanujan.

[@1ofuswin]

I think he got a couple things wrong but at least he tried

[@greebfewatani]

Nice presentation but algebra is taken from algabr means completion not balancing

[@steve-real]

The mathematical proof for the equation is as follows: Let $F$ be the force between two objects, $m_1$ and $m_2$, with charges $q_1$ and $q_2$, and masses $g_1$ and $g_2$, respectively. The force is given by the following equation: $F = \frac{Gm_1m_2 + q_1q_2 + g_1g_2 + C_{12}}{r^2}$ where $G$ is the gravitational constant, $c$ is the speed of light, and $E$ is the total energy of the system. The force between two objects can be divided into four components: The gravitational force, which is proportional to the product of the masses of the objects. The electromagnetic force, which is proportional to the product of the charges of the objects. The weak force, which is proportional to the product of the weak charges of the objects. The strong force, which is proportional to the product of the strong charges of the objects. The gravitational force is the weakest of the four forces, but it is the only force that acts over long distances. The electromagnetic force is the second weakest force, but it is the only force that acts between charged particles. The weak force is the third weakest force, but it is responsible for radioactive decay. The strong force is the strongest force, but it only acts between quarks. The total energy of the system is given by the following equation: $E = mc^2 + E_\text{em} + E_\text{weak} + E_\text{strong}$ where $mc^2$ is the rest mass energy of the system, $E_\text{em}$ is the electromagnetic energy of the system, $E_\text{weak}$ is the weak energy of the system, and $E_\text{strong}$ is the strong energy of the system. The electromagnetic energy of the system is given by the following equation: $E_\text{em} = \frac{q_1q_2}{4\pi\epsilon_0r^2}$ where $\epsilon_0$ is the permittivity of free space. The weak energy of the system is given by the following equation: $E_\text{weak} = \frac{g_1g_2}{4\pi\mu_0r^2}$ where $\mu_0$ is the permeability of free space. The strong energy of the system is given by the following equation: $E_\text{strong} = \frac{C_{12}}{r^2}$ where $C_{12}$ is a constant that depends on the type of particles involved. Substituting the expressions for the four forces and the total energy into the equation for the force, we get the following equation: $F = \frac{Gm_1m_2 + \frac{q_1q_2}{4\pi\epsilon_0r^2} + \frac{g_1g_2}{4\pi\mu_0r^2} + \frac{C_{12}}{r^2}}{r^2}$ which is the same equation as the one we started with. Therefore, the equation is mathematically proven.

[@2550205]

at 1:01 would have put the words or to use the scientific lingo not to be surprised of the letters representing the ideas represented by the collection of ideas being represented by letters as the word Distance inside the red area another word worth wording about and then arranged along the diagonal not shown in the absence of explanation of what every geometrician can tell every entertainer looking to bring a few more eyebells into the you tube of youness there are a lot of hours of entertainment available here on you tube this adds more to the particles making up the wave as language likes to put this together after taking that apart

[@mariyammapapabathiniratnam7862]

3:48

[@jamesraymond1158]

"Nicolai Ivanovich Lobachevsky was his name!" the wonderful Tom Lehrer song.

[@MrChrisayre]

I hate to say it but... at some point in any mathematics lecture/talk, you will hear the words "Sorry, haha, I lied earlier, but the point is..."

[@SpotterVideo]

What if we describe subatomic particles as spatial curvature, instead of trying to describe General Relativity as being mediated by particles? Quantum Entangled Twisted Tubules: "A theory that you can't explain to a bartender is probably no damn good." Ernest Rutherford The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory. Does it agree with the “Twistor Theory” of Roger Penrose? During the early history of mankind, the twisting of fibers was used to produce thread, and this thread was used to produce fabrics. The twist of the thread is locked up within these fabrics. Is matter made up of twisted 3D-4D structures which store spatial curvature that we describe as “particles"? Are the twist cycles the "quanta" of Quantum Mechanics? When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases = more Energy ). What if gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks. (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Force" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry. Modern physicists say the Strong Force is mediated by a constant exchange of Mesons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other. Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons? Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension? Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons . Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process. Gamma photons are produced when a tube unwinds producing electromagnetic waves. >>>>>>>>>>>>>>>>>>>>>> Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms. We know there is an unequal distribution of electrical charge within each atom because the positive charge is concentrated within the nucleus, even though the overall electrical charge of the atom is balanced by equal positive and negative charge. >>>>>>>>>>>>>>>>>>>>>> In this model Alpha equals the compactification ratio within the twistor cone. 1/137 1= Hypertubule diameter at 4D interface 137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted. The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.) >>>>>>>>>>>>>>>>>>>>>>> How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter?

[@minecraftsaurus5765]

Very interesting presentation. You missed crediting the Indians for some ideas on geometry (sine, cosine, pythogrean) and calculus (infinite series) that went to Arabs and then to Europeans. But credit does not matter. I am happy knowlege is not lost. (But who knows, new mathematics is still undiscovered, analog mathematics, quantum mathematics)

[@zimnesuhvezdie9027]

12:52

[@komolkovathana8568]

Good/simple presentation may satisfy Daddysitter,... Simple geometry satisfied Middle schoolers... Higher Postulation, (such as mine) maybe Falsified, or without proof... But rather give insight the implications of math inside Atomic geometry.. In terms of Fibonacci Ratio and TRIBONACCI Ratio Functions...for atomic mass, neutron mass, proton mass (1838.xx) compared to Electrons' (1.00).. Hope it useful, can kick-start the higher Scholars in high math/atomic physics..(!?!)

[@jamesraymond1158]

James Clerk Maxwell, not Clark, but you got the pronunciation right.

[@FreestyleSport101]

Little birdie told me that the artist discovered infinity not the mathematician?

[@zahid1909]

Without geometry, I would still remain as a piece of shapeless vacuum!!

[@scottbussler4041]

And then tragedy happens'...Ad starts playing

[@johncollins211]

Geometry and measurement were divine to the greeks. The human body was made by a divine plan of sacred ratios. For example the length from waist to knee is the same from the foot to the knee. From the top of the head to chin is 1/2 the length from shoulders to waist. We are all made with this design plan and they found symmetry to be beautiful. Some temples we're built integrating the same ratios as the human body.

[@yourguard4]

I don't think, that they based it on the human body...this "golden" ratio has its own basis: The ratio, which divides a distance in a way, that the ratio of the larger part to the smaller part, is the same as the ratio of the whole distance and the larger part. A/B = (A+B)/A

[@tarekazzam389]

Without Geometry, Algebra would have been monopolistically dominant. Ying/Yang - Balance or - Duality would have been disrupted and People's Brains too. Euclid, Carl Gauss, Lobachevski, George Bolyai, Bernhard Riemann, David Hilbert, Chen Chern, Shing Tung Yau, would have been either partly job-less or filling the Void of their Geometry - related Brain Areas & Geometry - related Neuronal Patterns & the Time that the Absence of Geometry would have created with some other Mathematical or Non - Mathematical Topic.

[@mahmoudbaraka987]

al-gabra means completion not balancing and al-mogabalah means balancing

[@mouse2078]

The first proof is not a proof. For example, it uses the non-axiom that the opposite angles are equal. First you must prove this from the axioms. It is an outline of a proof that you must work hard to make into an actual proof.

[@synapticaxon9303]

I'm not sure which was more breathtaking, bringing up the Declaration of Independence at the Ri, or butchering the recitation of it. That was like the New International Version, not the King George III edition! ;] Well you know Benjamin Franklin would have been smirking that smirk of his. :))

[@konradcomrade4845]

Calculating in Base 10 is hard enough, how could Babylonians endeavor to do it in Base 60. That is why their empire broke because they were staring too much and counting too intense the "sun-days" (and still they erred about 5.25 dys/yr) . I am going to try Base 12. Doing it in base 12 with new single-digit symbols and very regular one-syllable short words from 0 - 11 . It is building a nice numbers system (no Roman-derived quatrevingtdis!). Large numbers are grouped in 4 digits, readable from right to left and readable from left to right. Example: _dz stands for dozenSystem: 72_dec = 06*10_dz = 60_dz ; 180_dec = 130_dz ; 360_dec = 260_dz = 05_dz * 60_dz = 5_dec * 72_dec 3*72_dec = 216_dec = 03*60_dz = 160_dz. But in the conventional notations of Exponents and the Log there is an inconvenient entanglement, it is twisted.

[@kevinhanley3023]

Over and over I heard that you must be able to derive the theorems from 5 axioms. Why?

[@mariyammapapabathiniratnam7862]

2:35

[@babarizam4525]

He is an overexcited historian who is good at giving headache.

[@zimnesuhvezdie9027]

1:11

[@mathewmunro3770]

LOL, the video glitched just as he said you don't suddenly see me appear here then here.

[@gregpope4652]

If you like it then you should’ve put a grid on it

[@cinemaipswich4636]

There is a simple concept, that: An infinite circle is a straight line! Think about it, that you are a god, and you live outside of the Universe. You draw a circle of INFINITE radius. Then you ZOOM into the arc of that circle. What do you observe? But how near or far should I zoom? It does not matter, for scale is infinite. What do you observe? A straight line.

[@hugob8180]

so good, but the end was rushed, too bad the talk could not be 15 min more lengthy to explore einstein's works