Finally, a true Aperiodic Monotile!

mathematicians have finally discovered an aperiodic monotail S without Reflections let me catch you up tiles shapes that fit together with no gaps or overlaps humans are obsessed and have been for thousands of years there are two types that you need to know about periodic where the same thing happens over and over again I can pick it up I can move it over it looks exactly the same that's called translational Symmetry and if it has that it's periodic they seem to be quite common then there are clever ways of fitting shapes together like this so that when you go to pick it up and move it around it doesn't ever find a new place to fit no translational symmetry this is an example of a non-periodic tiling but you could use this shape in the normal way too so it can tile periodically it can do both so for half a century mathematicians have been trying to find a single shape which can only tile a periodically and on the 21st of March 2023 David Smith and Friends deliver the Hat ah we've done it an aperiodic no tile a single shape that can tile infinitely without any repetition The Crowd Goes Wild except they hate it because some of the hearts have to be backwards so does that actually count as a single tile perfectly comfortable to say yeah we we have an aperiodic monitor which is not to say that the question of whether one exists that does not require reflection it's not to say that that's not an important and deep question yeah let's study that next and try to come up with a non-reflective aperiodic monotone I I sort of I was telling my co-authors like we should we should we should maybe call that a vampire tile because it's a tile that covers the plane with no reflection oh they in fact did not call it the vampire tile and now a mere 10 weeks later on the 30th of May 2023 the moment you've all been waiting for an aperiodic no tail without reflex let's look at the anatomy of the Hat it's a 14-sided shape which consists of only two different length edges they can either be one or square roots one two three four five six seven eight nine ten eleven so thirteen okay so that slightly longer length that is two edges together with a 180 degree angle between them but the wonderful thing is when this was discovered it unlocked a whole Continuum of shapes which have this aperiodic monotyl property so like you can make one of the lengths longer and the other lengths shorter and it still works at the far end of the spectrum if you squish these marked lengths down to zero you get a Chevron and it tiles periodically and at The Other Extreme end you have a comet where the other lengths have been squished down to zero again periodic time at the midpoint you have a shape where all of the lengths are one and again it doesn't work it creates a periodic tiling but every single shape in between these will be a periodic so we have the hat and then the opposite of the hat which is called the turtle so start middle and end are periodic but everything in between has this magical aperiodic property then maths artist and tiling Enthusiast yoshiaki arakai was like wait let's make a combo Turtles and hats and hats and turtles both of which create aperiodic tilings but with two different tiles remember the aim of the game here is to get one tail hang on though if we Crank That dial to the middle the hats become Turtles the turtles become hats It all becomes one and the shape doesn't need to be reflected add some squiggly edges to stop it from ever tiling periodically and the Specter was born the one true aperiodic monotail of course as we're getting used to now this does open up infinitely many related aperiodic monotails one of the main reasons that this discovery has come around so quickly after the last one is the fantastic job David Smith Joseph Myers Craig Kaplan and Heim Goodman Strauss did of communicating their work and creating a space for people to play with it along with the very serious but incredibly readable paper that they put out they also put out a bunch of interactive resources so people could like explore build their own tiles print their own this fed into arakai's work which they saw took it on board and now here we are with a real hands-down no objections a periodic mono tile I've always wondered if people during different points in history were aware of the time that they were in you know for like different Golden Ages or revolutions things like that but definitely for this one I feel like the internet has led to a real golden age of maths collaboration unlike ever before we are having a moment it's an honor to witness and I can't believe I got to see it in my lifetime and I'm just so proud that these current maths Champions are inviting people to get involved and really listening to the the contributions that regular maths hobbyists have to make moths [Music] foreign

Channel: Ayliean

Views: 71,209

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Id: IfVwelta1fE

Length: 5min 36sec (336 seconds)

Published: Fri Jun 02 2023