Roger Penrose - Forbidden crystal symmetry in mathematics and architecture

Sir Roger Penrose provides a unique insight into the "forbidden symmetry" of his famous penrose tiles and the use of non-repeating patterns in design and architecture.
It is a rigorous mathematical theorem that the only crystallographic symmetries are 2-fold, 3-fold, 4-fold, and 6-fold symmetries.
Yet, since the 1970s 5-fold, 8-fold, 10-fold and 12-fold "almost" symmetric patterns have been exhibited, showing that such crystallographically "forbidden symmetries" are mathematically possible and deviate from exact symmetry by an arbitrarily small amount. Such patterns are often beautiful to behold and designs based on these arrangements have now been used in many buildings throughout the world.
In this Ri event Sir Roger Penrose reveals the mathematical underpinnings and origins of these "forbidden symmetries" and other related patterns. His talk is illustrated with numerous examples of their use in architectural design including a novel version of "Penrose tiling" that appears in the approach to the main entrance of the new Mathematics Institute in Oxford, officially opened in late 2013 (www.maths.ox.ac.uk/new-building).
The tiling is constructed from several thousand diamond-shaped granite tiles of just two different shapes, decorated simply with circular arcs of stainless steel. The matching of the tiles forces them into an overall pattern which never repeats itself and exhibits remarkable aspects of 5-fold and 10-fold symmetry.
Similar features have been found also in the atomic structures of quasi-crystalline materials. The initial discovery of such material earned Dan Shectman the 2011 Nobel Prize for chemistry, his work having launched a completely novel area of crystallography.
Images of the completed Mathematics Institute in Oxford courtesy of Vanesa Penrose.
The filming and production of this event was supported by the Science and Technology Facilities Council: www.stfc.ac.uk. Production by Edward Prosser. Additional camera operation by Mark Billy Svensson.
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[@M.-.D]

So incredible to see Professor Penrose win the Nobel Prize. One of the greatest minds.

[@StaP876]

It's really nice to listen to him. I love his universe.

[@Ungtartog]

The dude is pushing 90 and still razor sharp!

[@vhawk1951kl]

you understand what "incredible" means? apparently not

[@M.-.D]

@@vhawk1951kl unbelievable as in no murmurs seemed to be hinting at his selection.

[@debszta2]

Roger Penrose: The man that got into science because he liked to light things on fire when he was a kid. Love this guy.

[@jacobjorgenson9285]

And his father and two brothers are scientist and another is a chess grand master.

[@truthneverchangeswakeup2762]

Don't most guys🤭 He's so serious trying to show this "phenomenon"? Just look at Snowflakes, not one alike I've been told. " If you only knew the magnificence of the 3,6,9, then you would have the Key to the Universe. " Nikola Tesla 🕊️

[@jakeford9073]

Yer my dad was a leading virologist and he got into science because he like blowing stuff up. He made a hydrogen bomb and blew up the shed and caught his room on fire a few times... I think it's quite common among scientists lol

[@truthneverchangeswakeup2762]

@@jakeford9073 Reminds of my brother, Mike making lye bombs that required aluminum foil. And our Great Auntie's 1962 Comet with the paint burnt off the hood.. Big patches right down to the bare metal.. P. S. Our Great Grandmother made soap hence the lye availability 🤣

[@life42theuniverse]

Combustion is fascinating. I enjoyed restarting the campfire in the morning from the coals just with air and kindling. Later I learned it was oxidation and latent heat.

[@petebop]

Just wow, such a brilliant display of a truly intellectual human, proving their intuition with math! The passion and humility! Thousands of years of thoughts collected and dissected to find the beautiful "chaotic" symmetry around us, bravo Sir!

[@gwahli9620]

On videos like this, even the ads are more classy. I got an ad from a lab equipment company.

[@mokopa]

Did you buy?

[@gwahli9620]

@@mokopa No. We're just a software company and have no need for that kind of equipment.

[@gordonrussell9110]

why you not using ad blocker?

[@gwahli9620]

@@gordonrussell9110 For ethical reasons.

[@Sam-xg3mr]

I got an ad for the Reno 9-1-1 reboot. So much for that theory.

[@jeanqnguyen4542]

Love his hand drawn and written overhead slides, no one does it anymore

[@mwilson14]

I'm almost 15 minutes in and I'm totally engrossed into the lecture. This is tremendously fascinating and informative. This fully explains the observations I've made under the microscope of my synthetic corundum (ruby and sapphire) as well as the formations observed in natural corundum. I'll probably watch this several more times to take it all in. We're lucky as a society to have resources like this available for free.

[@unvergebeneid]

Somehow, when something famous is named after a person, I always assume that this person is dead. It always amazes me when I find out that not only is the person still alive, I actually get to hear a talk by them!

[@MrBollocks10]

Still going!! Like a Duracell bunny.

[@prototype8137]

What fascinates you about him? Ive heard a few lectures and hes apart of humanities problem. Pushing bs black hole theory, dark matter, and other occult symbolism as many of these "famous" do. Science is intentionally attacking empathy and we see it having effect in the world. There is no denying this. The impact that these people have on the next gen is real and kids dont know better. Apparently some adults are still indoctrinated into our upside system and cant see the problems. Its complicated but this man is no hero and no genius. Living a life dictated by memorization and numbers does things to people too.

[@iamjimgroth]

@@prototype8137 Are you quite alright? You sound like you might have hit your head on something. You should get that looked at by a professional.

[@dionysianapollomarx]

@@prototype8137 tool, that isn't how to refute his entire body of work.

[@jackcascione6137]

)

[@DIMentiaMinecraft]

Whatever you do, don't take the stairs to his office.

[@SongbirdAlom]

I laughed really hard. Thank you.

[@ddaro_ss]

I initially wanted to stop the video at around 8 mins in but - contrary to some comments complaining about how boring the talk was - i just got so engrossed that I finished the whole thing. The slides may have limited the talk in some ways (IMO mainly just time-wise) but without the slides we wouldn't have the awesomeness that was 26:00 to 28:00. A powerpoint simply wouldn't have the same oomph factor. In any case, it's always inspiring to see someone so passionate about their work.

[@xrisku]

Greg Queen of Everything schizophrenia is believed to be a defect in the brain in regards to timing. unlike your comment which is just ugly.

[@ElMauber]

Imagine seeing these patterns implemented in modern architecture. Absolutely mind blowing Mr. Penrose, bravo! 👏🏼

[@jach8047]

Sir Roger never fails to amaze me. Thank you.

[@jacderida]

I've watched this twice now. It's inspired me to take on a wee project to do some programming to generate some of these tiling patterns. Fascinating lecture!

[@TheParadoxOfParadox]

Brilliant to hear! Did you ever end up with any code? Is any of it available on github or anywhere else? :)

[@mark11967AD]

I always had trouble understanding math teachers with the exception of a few. Mr. Penrose would definitely fall in the latter category. Wonderful almost magical quality of his connection to mathematics. Universal or infinite patterns of symmetries or near perfect are indeed fascinating and probably very relevant to future discoveries of great import.

[@brendawilliams8062]

You have to be able in your minds eye to paint a Rembrandt before you put your sights on modern art. He is genius in that he relates it to math.

[@TheRoyalInstitution]

If you want to share the incredible moiré pattern demonstration, we've posted it here: Incredible Moiré Patterns!

[@Beach_comber]

+MrAaronvee "The Royal Institution" is short for "The Royal Institution of Great Britain". It's a proper name. The RI has different aims from the Royal Society. Plenty of FRSs also belong to the RI. Eric Lathwaite was never director of the RI, he just gave a few children's lectures there.

[@uusees7907]

+MrAaronvee childrens lectures!! my oh my! the pseudo effort just as such effort. interesting wáy to worthy it attention. not wronging none nor is it credible ya ...consern..? ... aaaw!

[@neurophilosophers994]

The thing I love about Penrose is that he loves mathematics and he has no clue that most of the people that he is explaining it to have no idea what he's talking about. It reminds me of myself explaining anything to anyone.

[@TechNed]

What a wonderful and enlightening presentation. One feels that there are more applications as yet unknown, for this.

[@mcee823]

Just love the Royal Institute! Penrose, Dawkins, going to sleep with geniuses! From Faraday and keeping the standard high. Makes you proud. For everyone, even dummies like me can enjoy it. Love it, keep it up!

[@ivanjelenic5627]

Haha, yeah, same for me!

[@glutinousmaximus]

Roger is perhaps one of the unsung heroes of the 20th Century. His work is brilliant and his book "The emperor's new brain" is a triumph. I corresponded with him a few years ago, and he is a very down-to-earth guy. Good stuff.

[@shuddupeyaface]

To be is to encompass a space. To encompass a space, you have to equally displace. The two realities are fundamentally intertwined. Thank you Mr Newton.

[@Ungtartog]

I really want to use this pattern in some construction project. One of the cool things about it is that you do not *have* to do a layout first, or think too deeply about how you are meeting the edges of your field. You can't exactly buy penrose tiles from the flooring section at Home Depot.. so it would mean cutting and wasting a lot of tile.. and then you have to figure a way to score the alignment patterns on them in an attractive way. Almost easier to buy a kiln and make the tile from scratch yourself. But then I had another thought! To put this in concrete.. all you need is two stamps! Much faster way to create a penrose floor.

[@Max_Le_Groom]

I'm so grateful for this. Thank you so much. 😊👏

[@tushkoli]

This is so simple yet fascinating!

[@nextblain]

Is there any way you can put where is Roger actually pointing at while he is showing his slides? I just seem to see a picture and I get confused where on the picture should I look when he is talking about it.

[@JohnDlugosz]

Some years ago, I wanted to have a small bathroom floor tiled with Penrose rombic style tiles, but every contractor ran away screaming. It's interesting to see that people have actually done that now.

[@kennyd6738]

John Długosz kept you from running scared at the price

[@brianmackey7682]

What prevents you from doing your own tiles?

[@scottleft3672]

@@brianmackey7682 Usually the pitch of the bedding.

[@poorfa4s]

I am a flooring contractor as a side job--you would have made my day asking me to do that.

[@A_Man_In_His_Van]

Easy job, unless you have to hand cut the tiles. If you can buy them as a working kit no problem, it lays itself out.

[@KozmykJ]

I particularly like the Rhomboid pattern with it's suggestions of 3D that jump out as one focuses on different loci.

[@warner631]

i found myself short of breath a number of times. This is absolutely something of great importance.

[@moonzestate]

The ancient Greek biographer Plutarch quoted Plato as writing, “God geometrizes continually.”

[@whoeveriam0iam14222]

"this point" "that point" doesn't really work without a pointer for the camera =[

[@TheNefari]

It's really impressive if you move the the top pattern and come up with stripes. And i kinda got the impression, that there is more to it than that. If you rotate the top shape, it almost looks like it is zooming in on the pattern.

[@AGamesCorner]

that is the exact feeling i got. and when you are zoomed in at maximum the strips appear.

[@ivanjelenic5627]

he did say you can make it as big as you want it, i guess you missed that

[@drcovell]

Interesting intersection of disciplines. Mathematics as art and vice-versa.

[@ebrelus7687]

That's the essence of art. Showing the mathematical harmony of world, thats the utility of art. Random shapes without meaning or natural pattern is not art. The thing that pleases eye in art is the harmony, the proportions, even in the human forms, facial shapes. Art is reproduction of natural order, laws, patterns. Other part of it is visualization of ideas but still meeting the overall laws. And this is the essence of civilization - building on natural foundation - not destroying things, doing opaque, rebelling against the essence of world or to prove loss of own basic roots or human reflexes (and most of modern "art" is exactly that, meaningless ugliness, betrayal of world creation laws to prove own childish ungreatfullness to the world which brought us to be, to build on ruins of the old/real world, neverending progress for sake of progress, everlooking for a different, new, extreme in selfhate, sadomasohism, insane objective of disproving basic intuition that art for art is not art; formlesness, antiform is the modern cult of soulless aberrant nihilism). The best crossovers between math and nature brought us the best human inventions. The looking for unreal, utopias, desacralisation of nature brought us communism, cultural destruction & unnatural magic-pill-based, not treating causes of diseases caricature of medicine. Good day to you :-) Take your daily dose of sun or fishoil/vitamin D please. Take care.

[@MadderMel]

I love his quaint hand drawn diagrams !

[@yellowpanda04]

incredible stuff, very interesting thanks for posting this

[@zzasdfwas]

"Something doesn't look quite right. That tile on the end doesn't work with the one on the other end." I'm not convinced that Penrose is human.

[@NathanOkun]

What you are seeing here is TRUE GENIUS -- someone to whom problems that stop others in their tracks are merely somewhat tedious, not barriers to solutions. "The improbable we do at once, the impossible takes a little longer" in a person...

[@xutamm]

No rulers used in the construction of images? Knowledge and intuition always present in his modus operandi. Nice stuff to show to young people involved with science. Not everything has to be computable...Why pen and paper still have their power? Hands on experience adds something...

[@biokode]

Awesome! I love this.

[@emasolie4135]

Love geometry, and especially love Sir Roger, precious man.

[@Spiller333]

He deserves the Nobel prize simply for tileing the plain asymmetrically.

[@Spiller333]

Wait what....

[@peacefuldownpour581]

i wanted to ask,... for the square, triangle, parallelogram and hexagon tiling, i can find many points as my axis of rotation so as the whole tiling can overlap itself, but how about penrose tiling? does it have only a single point as its rotational axis or are there an infinite number of points for me to choose as the axis of rotation for the penrose tiling to have 5 fold symmetry...

[@zackcovell]

When you subdivide the pattern is it a halving or another ratio of sizing?

[@Baobei666]

Amazing. Transcript or reference of the articles mentionned would be great!

[@zagyex]

What I like the most about this is that its consequences has lead to thoughts on consciousness and then on to physics and so 'everything'

[@hawk0485]

stunning!

[@amallee8825]

Very enjoyable... Would love to understand this more..

[@realcygnus]

superb content !

[@fretlord4201]

This really makes me wonder about the difference between dimension and perspective. Is a cube really a cube? How many more sides might be out of sight, or merely an illusion caused by relative movement? At certain points during the rotation, it looks a lot like the prime number spiral. Neat stuff!

[@asparagusmelontoast]

A cube could be a hexagon or

[@wayneyadams]

I studied crystal symmetries and point groups as they relate to electron diffraction spectroscopy and still found this lecture incredibly difficult to follow. I think the problem is that he moves along so quickly without explanation of so many of the things he says are "obvious." They are not "obvious" to the average people watching this video who have never had an advanced Chemistry (or maybe Physics) class. PhD does not necessarily mean good teacher.

[@jonathanwalther]

The case about "obvious" seems to be a maths thing. I remember a math professor who used the word "trivial" a lot, when most of the students didn't find the topic/proof at hand trivial at all. I learned: there are us mere mortals and there are some mathematical minds, capable of highly abstract thinking.

[@mrautistic2580]

If I could spend five years of my life on this kind of math, I would love it. For all those questioning where this math takes you...it can form the basis for some pretty sweet future tech.

[@bluesewage980]

five years later! how do you view this subject now? I'm intrigued and glad that i stumbled upon this at the right time in space!

[@nandakumarcheiro]

Crystal condensing symmetry breaking during lava condensation produces differential crystal veins as a function of temperature as applied in metal retrieved at different temperatures.

[@eamonnsiocain6454]

The Moiré effect is particularly interesting.

[@Sebentheyargimachine]

Does somebody have any information on the type of granite used that made the pattern look different in dry and wet surfaces? 53:29

[@Kurtlane]

Does Mr. Penrose or anyone else have a book that describes all this in detail? This is great stuff, and I'd love to slowly read and savor it. Thanks.

[@TarisRedwing]

Those granite and stainless steel tiles at the end are beautiful

[@23Fibonacci]

6:00 Roger: "Straightforward..." Me: < rewinds to 4:00 >

[@Apeskinny]

Steve Smith I get it....! Duh! Back to the beginning 8

[@greencoder1594]

20:54 Tiling the plane without repetiton: «I knew you can do it with five tiles, then I started fiddling around for a bit and realized you can do it with two» Some people are paid to fiddle around and make creative progress.

[@TripcussionShorts]

Hahaha yes specially when they have a knighthood not us mere mortals

[@daieast6305]

ah yes, those people are called fiddlers or maybe more often violinists

[@SoccerQuiz]

Fall in love with structure & shapes, make u doing things like that

[@catonthemoonrecords]

Amazing!

[@robertm8246]

I was inspired to use this on the job - Penrose Tiling on a concrete floor

[@mirusvet]

This many is simply amazing.

[@OrangeJackson]

26:30 blew my mind.

[@camrodam]

Truly a WTF moment

[@asymptoticsymphony6101]

whoa. this is amazing

[@TripcussionShorts]

Thank You Sir Roger Pentagon Rose

[@arborsamurai]

Marvelous!

[@oracleatdeptford2970]

He is on to the something That leads to everything. Jolly good show . What what.

[@heldlightning7118]

Yep yep

[@0xde57]

beautiful!

[@nandakumarcheiro]

Roger really wanted to show that the wobbling crystal symmetry is broken to have differential boundary of crystals for example as opal black crystal or rubies as differential veins of crystal condensing boundaries of lava as a function of temperature condensing differential.

[@FXTripazha]

Came here just for this guy. He is sooo cooool!! 😍

[@robdev89]

Fascinating.

[@AdastraRecordings]

Seeing these as atomic arrangements and not just tiles, it begs the question, is this a possible solution for the neutron packing problem? The cubic solution bothers me immensely, but this has me wondering now...

[@fsommen]

This video shows how brilliant Professor Penrose really is.

[@dianamari_lv]

Is it possible that this pattern could be achieved by placing only triangles (a combination of equilateral, acute and obtuse) given that the triangles are different sizes so as to differentiate the patterns?

[@lenglel]

Yes. Any two-dimensional shape can be represented by triangles. In the case of a circle, however, this would require an infinite number of triangles.

[@dan_rust_maths]

If you google 'Robinson triangle' you'll find a version of the Penrose tiling using just triangles (essentially what you get by cutting the kites and darts into two types of triangle).

[@daieast6305]

can not wait for the 'forbidden' part

[@holdmybeer]

I love his tie.

[@robertevans6218]

I noticed that at time frame 27:59 when intersecting lines began to form that a fractal "cardioid" pattern may have been forming which implies that the "Phi" value may be at the "heart" of the crystalline structure.

[@jahnvikashyap230]

Good Catch. It indeed is.

[@syamsularifinmohamadyunus8144]

Great sharing

[@Reynoldsrobert]

Reminds me of the multiple universe theory, which I think of in this way where one really fits inside of another, perhaps more complex one, ad infinitum.

[@aibrainlet8041]

is this math the same causal reason we have 12 notes in a musical octave even though the intervals between them are not exactly precise? The patterns that emerge when plotting frequency ratios seems to be the same type of patterns here

[@kakarotlifted7302]

Interesting.

[@Alexagrigorieff]

@@kakarotlifted7302 The reason for 12 notes in an octave is that the circle of fifth has 12 points. 1.5^12 ~ 130, which is very close (1/3 halftone) to 128 - 7 octaves. Intervals in well tempered 12 halftone chromatic scale are reasonably close to natural intervals.

[@Palifiox]

I read about this 35+ years ago, but he lost me at 5 - 6 minutes. "This one", "that one." Which one is he talking about? Pointing at them might have helped.

[@s7a2g65]

Codenwarra Cove You have to experience the overlays around the 26min mark.

[@richtourist]

@The Royal Institution Am i right in thinking that that bench is actually the same one that has always been there? I mean, are we looking at the same bench from which Michael Faraday enunciated his field theory of electromagnetism?

[@TheRoyalInstitution]

It's actually the fourth Faraday desk. When the first one broke from all the wear and tear, we commissioned another one built to the same specifications, with a few added improvements. And the same happens whenever it breaks down. The current one can be unlocked, elevated and moved on castors, and also has inbuilt charging and display cables. Considering the desk is used almost daily for demonstrations and lectures, we feel that a roughly 70 year lifespan is pretty decent for a desk.

[@richtourist]

I wonder how long it would take Faraday to understand the concepts behind an HDMI cable. Quite a lot has happened in 200 years. Faraday could sit in the audience and lead the way by asking questions of a panel of experts in the relevant fields. That would be a fascinating journey to watch. You can't get Faraday, but you could get a celebrity in a room full of boffins and record the process of them asking questions until they felt able to present a lecture live on TV from the RI. That would broaden the public understanding of science. A reality TV show covering the celebrity exploration, followed by a real lecture at the RI. That could be a horrible load of fake nonsense, or it could be something wonderful and inspiring. Jodie Whittaker (Dr Who) explores and lectures on...

[@dan8535]

brilliant

[@salimhuerta2699]

Roger is an absolutely wonderful human being top quality that one haha

[@FeliciteThosz]

"No matter how big the region is, you can always find it in the other one." This threw me in relation to "there's more than one pattern possible" until I realized that identifying a region necessarily means reduction to a FINITE section (no matter how large) outside of which you could always create variation. Only if infinite patterns perfectly overlapped could you say that there is only one pattern. But you could have the same pattern for light years and light years of space, and then--bam!--variation.

[@pris0nergaming641]

When he draws the green circles over the green pattern I see the seed of life. and in the end the fruit of life. Nice TY

[@sambeard4428]

Are there currently any applications of this tiling pattern in cryptografie? Looking at how these patterns are arranged and what interesting properties they pertain i imagine very smart algorithms could be constructed using this knowledge

[@KaliFissure]

5.and 12 are the combination for the dodecahedron which is the only polyhedron which allows vertex to same vertex uninterrupted surface path. Geodesic.

[@DavidAKZ]

Are Penrose tiles scale independent ?

[@peterandrew2097]

Please look carefully at the surface patterns of the Maclura Pomifera, called osage orange, spherical seed pod. Hypothesize that the five-fold Penrose pattern may form the basis of its seemingly confusing design. Requires careful analysis. What do you think?

[@SpiritBladeFox]

Reminds me of fractals or the Turing pattern

[@asadalbra]

that was so cool,

[@pris0nergaming641]

Is this the stacking of 8d and 24d oranges?

[@LaurentLaborde]

i don't understand but i love it :)

[@Dina_tankar_mina_ord]

what does this have in common with angle graphene layer

[@jh1859]

Kepler, Escher, and that Japanese bloke got so close to entering this garden path of delights.

[@marcos9204]

So intelligently humble

[@TheHimmus]

thumbs up from me this was very informative

[@jfreeman2927]

all my life I've searched for an asymmetric yet infinite grid. it's a basic unity that allows individuality by being infinite. a universe that is designed for both free will and determined fate-- a paradox. I understand why this kind of symmetry is forbidden. it undermines the orderly conformity that people find so comforting these days.

[@mohscorpion2]

oh my , the man is a true genius

[@alexwilson8034]

Honestly I find it incredibly charming that in 2015+ he STILL uses slide projectors instead of PowerPoint

[@friendlydragon8999]

Sir Roger Penrose is awesome as a brilient thinker

[@frankrwalsh]

What about the dodecahedral Iron pyrite crystals?

[@keleighwolf]

I wish he was more Escher in thinking & less an inelegant Chompsky simulacrum. Fractals exist as representations of sound as a commutable entity. More Parallax & Tycho Brahe. Then du Sautoy & black hole firewalls. Have a lovely day, Dr Cliff x

[@hansolo2121]

This man's comb over makes him a 'legend' indeed. Wow. Just WOW. And it is MC Escher who made him famous (not the other way around).