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Mostly Mental
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Добавлен 12 апр 2020
Fibonacci Nim - Play Zeckendorf First?
Fibonacci Nim is a two-player strategy game where players take turns removing ducks from a pile. But how can we win? What does Fibonacci have to do with it? And what other beautiful math can we uncover along the way?
Other game theory videos: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV
Sections:
0:00 Intro
1:25 P and N positions
4:15 First move
5:45 Fibonacci?
6:25 Zeckendorf representation
9:22 Zeckendorf triangle
12:09 Strategy
15:24 Proof
19:13 Recap
20:33 Rematch
Other game theory videos: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV
Sections:
0:00 Intro
1:25 P and N positions
4:15 First move
5:45 Fibonacci?
6:25 Zeckendorf representation
9:22 Zeckendorf triangle
12:09 Strategy
15:24 Proof
19:13 Recap
20:33 Rematch
Просмотров: 413
Видео
Wythoff's Nim - Going for the Gold
Просмотров 1,3 тыс.Год назад
Wythoff's Nim is simple game with some deep math lurking behind it. Other game theory videos: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV Sections 0:00 Wythoff's Nim 1:37 P and N positions 3:58 Drawing a picture 5:32 Finding patterns 7:53 Beatty sequences 10:23 Differences 11:48 Putting it together 14:25 Rematch
Chiral Aperiodic Monotile - Spectre in the Machine
Просмотров 5 тыс.Год назад
Hot on the heels of the discovery of the aperiodic monotile known as the hat, the same group of authors have discovered an even better monotile: the spectre. This one doesn't need any mirror images. Huge thanks to Craig Kaplan for helping me make the animations! Video about the hat: ruclips.net/video/vtpswcAfWiI/видео.html The paper: arxiv.org/pdf/2305.17743.pdf Additional resources: cs.uwaterl...
Prime Knots - Knot Ready for Prime Time
Просмотров 692Год назад
Prime knots are the fundamental building blocks of knots in much the same way the prime numbers are the fundamental building blocks of numbers. This is part of a series on knot theory. Check out the rest here: ruclips.net/p/PLH5zdqQODdBhe2Kp5as7U7uudhNlEJk6H Sections 0:00 Intro 1:15 Connected sum 2:41 Prime and composite 4:09 Identifying prime knots 5:40 Primes and invariants 9:10 Counterexample?
Aperiodic Monotile - Mad as a Hat
Просмотров 26 тыс.Год назад
An aperiodic monotile (also known as an einstein) is a tile that can cover the plane, but never repeats. After decades of searching, mathematicians have finally found one. Sections 0:00 Intro 0:44 Tilings 1:33 Aperiodic tilings 3:45 Aperiodic sets of tiles 5:00 The hat 5:21 (Proof 1) Metatiles 7:12 (Proof 1) Meta-metatiles 8:54 (Proof 2) Family 10:59 (Proof 2) Common period 13:02 Open questions...
Knot Groups - A Walk in the Park
Просмотров 680Год назад
Fundamental groups and knot groups are powerful tools bridging the gap between topology and algebra. This is part of a series on knot theory. Check out the rest here: ruclips.net/p/PLH5zdqQODdBhe2Kp5as7U7uudhNlEJk6H Sections 0:00 Intro 0:46 Duck pond 2:51 Group fundamentals 5:27 Fundamental groups 8:07 Knot groups 11:05 Wirtinger presentation 12:59 Homomorphisms 14:51 Dihedral group 17:16 Prime...
Knot Coloring - Knot Just Black and White
Просмотров 502Год назад
Invariants are a powerful tool for distinguishing knots. Perhaps the most natural (or at least the most aesthetically pleasing) invariants are the colorings and quandles. This is part of a series on knot theory. Check out the rest here: ruclips.net/p/PLH5zdqQODdBhe2Kp5as7U7uudhNlEJk6H Resources: asclab.org/asc/sites/default/files/knot_quandle.pdf arxiv.org/pdf/2207.09257.pdf shell.cas.usf.edu/~...
Knot Theory - What's Knot to Love?
Просмотров 853Год назад
Knot theory is, as the name suggests, the study of knots. It's easy to get into (just play around with a string), but it leads to some deep topological structure. This is the first video in a series on knot theory. Check out the rest here: ruclips.net/p/PLH5zdqQODdBhe2Kp5as7U7uudhNlEJk6H Proof of Reidemeister moves: mathweb.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf (Page 19) Sections: 0:00 In...
Transfer Functions - Of Sound Mind
Просмотров 9402 года назад
Transfer functions are a powerful tool for modeling signal response. Join me and special guest Julian as we explore the theory and practice. Sections: 0:00 Intro 0:54 Motivation 2:00 Laplace transform and transfer function 5:18 Attenuation 10:16 Reverb 14:38 Showcase 15:43 Other applications
Hyperbolic Geometry - No Exaggeration
Просмотров 6 тыс.2 года назад
Hyperbolic geometry is geometry that plays by a different set of rules. Lines curve, triangles have less than 180 degrees, and the parallel postulate doesn't apply. Hyperbolic plane crochet pattern: static1.squarespace.com/static/5754f47fcf80a16bffa02c45/t/60a2dd27e2d1e010d14216b6/1621286201908/Handout-Hyperbolic_Crochet.pdf Explanation of Mobius transformations: ruclips.net/video/JX3VmDgiFnY/в...
Surreal Numbers - Bowl of Surreal
Просмотров 7 тыс.2 года назад
The surreal numbers are a number system (in fact, the largest ordered field of numbers) found within the structure of partisan games. This is the fourth and final part in a mini-series on game theory. Watch the whole series here: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV Sections: 0:00 Hackenbush 1:15 Advantages 3:10 Mixing colors 4:43 Towers 6:23 Set notation 7:27 Evaluation 9:20 Real n...
Nimbers (Part 2) - Leveling the Playing Field
Просмотров 6972 года назад
The nimbers hold beautiful connections between the worlds of game theory and abstract algebra. This is the third part in a mini-series on game theory. Watch the whole series here: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV Reference: www.neverendingbooks.org/on2-conways-nim-arithmetics Sections: 0:00 Intro 1:30 Division 3:38 Multiplication (but faster) 5:49 Division revisited 7:56 Square ...
Nimbers (Part 1) - Tipping the Balance
Просмотров 1,9 тыс.2 года назад
The nimbers are a number system hidden within the structure of impartial games. This is the second part in a mini-series on game theory. Watch the whole series here: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV Sections: 0:00 Motivating example 2:12 Sprague Grundy 4:13 Adding games 6:13 Subtraction 6:51 Twins 9:17 Turning Corners 12:30 Multiplication 13:24 Putting it together
Nim - Ahead of the Game
Просмотров 3,8 тыс.2 года назад
Nim is a classic game of strategy with some deep mathematics lurking just below the surface. This is the first part in a mini-series on game theory. Watch the whole series here: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV Sections: 0:00 Intro 1:22 P and N positions 4:13 Nim positions 5:41 Bitwise XOR (Nim addition) 7:06 Nim strategy 8:59 Rematch 9:49 Further questions
Hairy Ball Theorem - Combing a Coconut
Просмотров 4,4 тыс.3 года назад
The Hairy Ball Theorem says that a vector field on a sphere has a vector sticking straight out. Full paper: web.cs.elte.hu/~karatson/hairy-ball.pdf Sections 0:00 Problem statement 1:15 Plan of attack 2:19 Tightening constraints 3:33 Correspondence with larger sphere 5:12 Comparing volumes 7:15 Generalization
Visual Calculus - Tangentially Related
Просмотров 3,7 тыс.3 года назад
Visual Calculus - Tangentially Related
Square Wheels - Reinventing the Wheel
Просмотров 6 тыс.3 года назад
Square Wheels - Reinventing the Wheel
Nine-Point Circle - Finding Your Center
Просмотров 2,5 тыс.3 года назад
Nine-Point Circle - Finding Your Center
Young Tableaux - Think Inside the Box
Просмотров 4,1 тыс.3 года назад
Young Tableaux - Think Inside the Box
Stirling's Approximation - Close Encounters
Просмотров 2,8 тыс.3 года назад
Stirling's Approximation - Close Encounters
Catalan Numbers - On the Right Path
Просмотров 1,3 тыс.3 года назад
Catalan Numbers - On the Right Path
Recurrence Relations - Terms and Conditions
Просмотров 1603 года назад
Recurrence Relations - Terms and Conditions
Generating Functions - Powerful Counting
Просмотров 4004 года назад
Generating Functions - Powerful Counting
Pure Gold!
Glad you liked it. Thanks for watching!
beautiful ❤
Glad you liked it. Thanks for watching!
If monotiles cannot exist in 3d (which we do not know yet), I don't think the same would apply to 4d or any other nD where n=x^2, but I have no reason to believe this other than intuition
It's certainly possible, but I'd be a bit surprised. My guess is that either there are monotiles in every dimension (with possible exceptions of 4D, 8D, or 24D, since those are always weird), or there's some cutoff where it's possible for nD or lower but not (n+1)D or higher.
Great video
Glad you liked it. Thanks for watching!
if zero is the empty set, {}, and the constructor for Surreals starts off by dividing two empty sets, {|}, then the construction of the Surreals in fact only ever generates different names for zero: {} = 0 {|} = {{}|{}} = 0 {0|} = 1 = {{}|{}} = 0 {1|} = 2 = {{}|{}} = 0 {0|1} = 1/2 = {{}|{}} = 0 and no matter what you do, this is how every 'surreal' is constructed. which means you're an idiot.
add to this the fact that using the binomial theorem to expand the definition e reveals that e is the value of a non-Real Levi-Cevita series, despite the fact that e is supposed to be Real, and... oops. you're kinda wrong about everything, aren't you? I mean, if the Reals are continuous, as per Dedekind Completeness, then not only should you ask yourself where the hell the Surreals are supposed to exist, but now we also need to ask how e can be non-Real despite existing on the Real axis. but you're too dumb to have even noticed these problems in the first place, so... why are you attempting to educate others about things you do not remotely understand? watching your video is like trying to learn how to read from a cockroach running across a newspaper.
Who is the guy on the Mostly Mental videos on RUclips? I learned great gobs of stuff from his video on the Spectre tile. I would love to be able to email him, and bounce some ideas around with him. He introduces himself on his videos as Foxy or Doxy or something like that.
Hello, Foxy here. I'm just some guy on the internet who likes talking about math. My contact info is on my channel page, though I can't promise I'm very responsive.
Ces formes je les ai decouvert en 2013
It's certainly possible. The authors weren't the first people ever to look at these shapes. I went to a talk with Craig Kaplan, and he mentioned that in the research process he stumbled across a list of shapes for a completely unrelated problem which happened to also include the hat. This group just happened to be the first to prove and publish the aperiodic property.
You should write a book. I’d buy so many copies so I could have one at all times
I'm glad you like it. Thanks for watching!
can you provide the proof that the sprague grundy value of a position in the game of twins is the nim sum of the sizes of piles
Let's (for example) say we have a game of nim with piles 1, 3, and 4. Those have nim sum 6, which means there's a move which reaches a position with each value from 0 to 5. Now look at the corresponding game of twins, with red coins in spots 1, 3, and 4. Take the move corresponding to each of those nim moves, and you'll reach positions where the nim sums of the red coins are each value from 0 to 5. Also note there's no way to reach a position with nim sum 6, since the corresponding move in nim would also have Sprague-Grundy value 6, which isn't allowed. So we can always reach positions with each smaller nim value, but not the same value. And the end position (with all blue coins) has nim sum 0. That's exactly the relation we used to define our Sprague-Grundy values. So the Sprague-Grundy value of the twins game is the same as nim sum as the piles.
@@mostly_mental Thanks a lot! Can you provide me with some resources that can help me understand how to formally prove stuff like these (game theory, number theory stuff)? Love your content btw
@@vedantsingh5786 Glad you like it. If you're looking for a proper textbook, I first learned about the nimbers from "A Course in Game Theory" by Thomas Fergesen. I also highly recommend "Winning Ways for Your Mathematical Plays" by Berlekamp, Conway, and Guy. There are also plenty of good free resources online (I usually just search for "Combinatorial Game Theory PDF"). I don't know much number theory, so I can't help you with resources there.
how did you define the length rad3?
The hat and turtle were both discovered on a grid of kites, which are really just two 30-60-90 right triangles glued together along the hypotenuse. So the ratio of sides is sin(60), or sqrt(3).
Great explanation. In our remodeling we were considering using Penrose tiles. Then the hat and soon after spectre were discovered, now we are looking for a tile maker who'd make this for us.
I found this very interesting and you explained it very clearly. Thanks for making this video.
Glad you like it. Thanks for watching!
Yay! You're back!! I'm always excited to see your videos, friend :) Nim and game-theoretic extensions of number systems (like what Conway and Knuth developed) are truly an underappreciated love of mine. Thank goodness you have Aurora as your assistant!!
It's good to be back. Game theory is always fun to talk about, and I'm glad there are people who want to hear about it. And yes, Aurora is the best, and I couldn't do it without her.
I would like to see a video about Chomp
Excellent suggestion. I'm not sure I know much more than you do (and I'm pretty sure it's still an open problem, so no one else does either), but I'll do some research and see if anyone's published some partial results that could make for an interesting video.
@@mostly_mental hey, even if you aren't an expert on Chomp, you'd still have the most thorough RUclips education video about it. I encourage you to do it!! (I dont know what Chomp is, either, I just know that I love game theory and especially your videos)
9:15 proof If f_n is the highest fib which goes into some number, then [f_(n-1)]+ [f_(n-3)] + [f_(n-5)] ... + [1] is the largest number we can construct without using f_n and following the rules. we can expand each term[f_(n-2) + f_(n-3)]+ [f_(n-4) + f_(n-5)] + [f_(n-6) + f_(n-7)]... + [1 + 0 ] . This sequence is the sum of the first n-2 terms and it sums to f_(n) - 1 . (easy to show via induction.. f_1 + f_2 + f_3 = f_5 - 1, therefore f_1 + f_2 + f_3 + f_4= f_5 + f_4 - 1 = f_6 - 1) So the biggest number we can create without f_n is less than f_n and therefore less than the number we are trying to sum to. This means f_n must be used in our representation. then we can use the same process to deduce that in order to reach (our target - f_n) we must add the largest fib which fits into (our target - f_n). So in order to get a valid representation, we're going to get a single list of numbers we must use, therefore unique.
Looks right to me. Nicely done.
Great explanation! Glad you're back!
Glad you like it. Thanks for watching!
Aurora seems like a good sport!
She really is the best. She's taught me so much cool math. I'm lucky to have her as my assistant.
You know another thing that's also mental, only completely so, it's the topological inversion of a sphere inside out...
Yeah, that one's absolutely wild. I watched that video ~15 years ago, and it was a big part of my falling in love with topology.
cool facts about hyperbolic geometry: almost every path is close to straight equidistants (paths that stay the same distance from lines) converge to a horocycle (circle of infinite radius) as the guiding line goes away
is there a link to that game ? what great visual add. thanks for the vid
I'm glad you liked it. HyperRogue can be downloaded at zenorogue.itch.io/hyperrogue (or it's available on Steam).
i love your channel!!!!!!!! please don't stop making videos, they're great and i love learning math without the pressure of exams!
Glad you like it. Thanks for watching!
Good explanation, simple approach. Nice!
Glad you liked it. Thanks for watching!
Just turned in. Find it strange to be ending on hackenbush so I’m curious where you started
I started with impartial games and the nimbers (which I find more interesting than the surreals). If you're curious, you can check out the rest of the series here: ruclips.net/p/PLH5zdqQODdBiGrWszPScMO2Dvp0Ix_vpV
Does one orbit refer to one shell/one layer of the sphere? This video is the perfect blend of maintaining the technicality of the concept along with making it easy for the viewers. Thank you so much for such a wonderful video.
I'm glad you liked it. The orbit of a point is the set of points you can reach from that point using any sequence of a's and b's and their inverses. Everything in this video is referring to the hollow outer shell of the sphere, but it's not too hard to extend it all to the solid sphere (excluding the center, which has to be handled separately).
Understood. Thanks for the quick reply 😊
I highlighted the need to see this video on the Wikiversity article "Surreal number" and will attempt to include a link to this video on the Wikipedia article with the same name. Nothing on any of these wikis come close to explaining surreal numbers this well.
I'm glad you liked it, and I appreciate the enthusiasm, but I'd rather not be directly linked in those articles. Maybe just add a section summarizing the connection to Hackenbush?
@@mostly_mental I will be happy to oblige. Just to be clear, you want me to remove the links from Wikipedia or Wikiversity out to this RUclips video? I will wait until you verify that this is you wish (because I have no idea why you don't want the link). I control the Wikiversity page, so deleting that link there will be no problem. I will also delete the link I made at Wikipedia, but I can't guarantee that a Wikipedia editor won't reinsert it. I think I will just quietly revert my Wikipedia edit without comment. If I say anything it will draw attention to what they may decide is a good link from Wikipedia to this RUclips video.
@@guyvan1000 Thank you. I'm not a primary source on the topic, so I'm a bit uncomfortable being listed as a reference. Besides, everything I know comes from the books I mentioned at the end, which are both already listed on the Wikipedia page.
@@mostly_mental There is a difference between an external link and a primary source on Wikipedia, but I am not well enough versed in Wikipedia rules to know if a link out of the Wikipedia page is proper. Wikiversity has entirely different standards (one might say no standards...) I would like to include a link from Wikiversity to your video, but won't do it without your permission.
@@guyvan1000 I think I'm okay with a link on Wikiversity, so long as the math is the primary focus and not the video.
Amazing video!
Glad you like it. Thanks for watching!
How to measure the curve of the road ?
If you want to measure the length, the easiest way is to notice that the length of each arc of the curve must be the same as the length of one side of the square (which is 2). If you want to know how curved it is, the best measure is the radius of curvature, which essentially finds the radius of circle that best approximates the curve at each point. There's a formula here en.wikipedia.org/wiki/Radius_of_curvature (which comes out to cosh(x + c)^2).
sick bikies
Nice
thanks for the explanation of crossing paths I was struggling with it.
Good series! Too bad it seems to have been discontinued(?).
I'm still here. There's more to come when life gets less complicated.
@@mostly_mental I understand. Farewell
Sir, will you provide the reference book please?
I'm largely working from "Knot Knotes" by Justin Roberts.
@@mostly_mental Thank you so much sir for your reply. Your presentation and explanations are really amazing. Actually i thought of taking this topic as my master's project.🙏😊
@@suhana.a.a7949 I'm glad you like it. Thanks for watching!
By the way, keep Going! You are incredibly gifted at explaining complex ideas 🫂🛐⚡
One thing that strikes me here if I am understanding it correctly is that you have this kind of spectrum in which the hat side consists of a continuum of hats between chevron and spectre and the turtle side consists of a continuum of turtles between spectre and comet with all these intermediary tiles having the property of tiling the plane only in an aperiodic fashion but involving a certain number of reflected versions (proportion phi to the fourth power) with the spectre in the middle which can be tiled periodically or periodically but which heads a family of spectre mutants having chiral figures taking the place of its non-chiral line segments. This infinite family of spectre mutants has the property of tiling the plane only periodically. Cool!
Also: that the hat in the strict sense, like the turtle in the strict sense, is constituted by an assemblage of parts that you get from the (periodic!) tiling of the plane into hexagons with their corresponding dual (tiling in triangles). So that the whole aperiodic tiling is there in your face amidst this simple (periodic) tiling into hexagons and triangles. If I am right about that. Weird!
@@carlkuss Yeah, you've summed it all up pretty well. It's wild that stitching triangles and hexagons together is all you need to get so much complexity.
I’m having to pick some of this up for research, great video!
Glad you like it. Thanks for watching!
hello
another question, do you know where can i find something more specific about the "ein stein" problem that the hat broke? an article talking about this problem or something? thanks :)
"Einstein" is just another name for "aperiodic monotile". "Ein stein" literally translates to "one stone", so it's bit of a play on words.
hey, i would really appreciate if you could share your references for the aperiodic tiling with trapezoide tile demonstration, starting at 1:34. thank you!
strauss.hosted.uark.edu/papers/AHT.pdf describes the whole process in a bit more depth.
thank you so much man! :)
I appreciated the series. Thank you kindly.
Glad you liked it. Thanks for watching!
Please keep doing videos about this topic, i was looking for so long!! Thanks🎉🎉🎉
I'm glad you liked it. I'm currently in the middle of a series on knot theory, but I plan to come back to game theory when I have the time.
I came here after I saw someone making out while explaining hairy ball theorm
It’s a T-shirt, I can’t believe they said hat over T-shirt
Such high quality and so few views. Keep Going.
I'm glad you like it. Thanks for watching!
Thanks for asking the questions at the end. Can´t wait to hear more about this topic. This can´t be the end. Or will there be a alrge time spane where nothing happens again, who knows
Well, there was a followup paper a few months later. The same team found an aperiodic monotile that doesn't need to be flipped over to tile the plane. Video here: ruclips.net/video/3CxF-GkkjiU/видео.htmlsi=PtLXR4nAujCJeCwS Now we have the new techniques from these papers, and there's been a huge surge of interest in the problem. I'd be shocked if there aren't more exciting results in short order.
@@mostly_mental So, one question still open would be a shape with less "sides" or more sides or "zero" sides. Seems to me with no knowlege whatsoever you´d have to "fuse" two or even more repeating repetative tilling shapes together. And the questions you have mentionend in the follow up video.
How about 3D Aperiodic Monotile ?
There are a few known almost aperiodic monotiles in 3D. The disconnected Socolar-Taylor tile has a connected 3D analog that can tile layers aperiodically, but the layers might repeat (en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile ). And there are some shapes like the Schmitt-Conway-Danzer tile (en.wikipedia.org/wiki/Gyrobifastigium#Schmitt%E2%80%93Conway%E2%80%93Danzer_biprism ) that aren't periodic, but can still have screw symmetry. So far, no one's found a true 3D aperiodic monotile.
That's a t shirt not a hat
0:24 and i’m already subbed because your hand motions are so graceful
Glad to see all those hours practicing magic have paid off. Thanks for watching!
Nice playthrough and description; I feel like I actually understand this better instead of just learning an algorithm. Is a game like Othello compatible with analyses like this, or does the way the game works make it a lot more complicated?
The analysis we did here relies on the fact that Wythoff's Nim is an "impartial" game. That is, both players have the same moves available to them from any given position. But Othello is a "partisan" game; one player can only place white pieces and one can only place black. Partisan games tend to be a bit harder to analyze. You can use a lot of the same logic. For instance, the value of a position still depends on the values of the positions you can move to. But now in addition to P and N positions, there are also L positions, where the left player wins regardless of who moves next, and R positions, where the right player wins. I talked a bit about partisan games in my video on the surreal numbers (ruclips.net/video/9-nerkgryZ8/видео.html ). And I'd definitely like to revisit them in a future series.
@@mostly_mental Wow, thank you -- more than I knew to ask :) I suspected "impartial" but "partisan" wasn't there with me, and it's a richer distiction to have those words. I appreciate your related reference too; I had skimmed your list and not realized / I'll plan to watch that tomorrow!
What if we used logarithm with different base?
Logarithms in different bases only differ by a constant factor, so not much would change. Say we did this with base 2. In the step where we integrate (around 5:30), we would pick up a factor of log_2(e). Then when we get rid of the logs (6:58), we would end up with 2^(-n log_2(e)) instead of e^-n, but by logarithm rules, those are really the same thing. So we'd arrive at the same formula.
I like this more than the p-adic numbers
Yeah, the surreals are great. But my favorite number system is still the nimbers.
@@mostly_mental I enjoy the dual complex quaternions the most