The first thing with the sphere can be demonstrated with just your arms. It was trending on social media a bit ago. If you put your arm out straight palm facing down, pull your arm towards your chest so your arm makes a 90 degree angle with your elbow, then rotate your arm again 90 degrees so your finger point towards the sky, and the go back to the starting with your arm facing out straigh, your hand will have rotated 90 degrees. You can repeat that again and your hand will now have your palm facing upwards.
cubing is all about this concept-for 5x5 you could rotate the middle 9 squares of the top surface by first rotating the middle sections down, then to one side, then back up from the side to the top. in this way, you can see that puzzle cubes are functionally spheres in this way
Something I noticed: If you stop at three turns in on the G2 surface (so the Rook has just entered the inner ring) and _turn_ the whole figure so that the rook is ‘on top’ of the inner ring (like on the sphere), the Rook will have the same orientation as if it were on the sphere
Yes, I think this could give a hacky way to write the code to search for good mazes on the genus two surface. You could reuse the sphere code with some modifications for what happens when you go inside the holes.
This comes from the gauss-bonnet theorem. A a surface (without edge) of positive/negative euler characteristic must have, on average, positive/negative curvature. A sphere has euler characteristic 2 and this genus 2 surface has euler characteristic -2.
@@neopalm2050 I’m not knowledgeable enough to understand what you said but I do recognize gauss at least. What would a characteristic 1 or -1 look like? Would 0 be considered no curvature?
@@ionic7777 Simply put, curvature is what causes holonomy. If you enclose positive curvature in a path, you will find an equal amount of holonomy in it. Let's say you take a sphere with radius 1. This has curvature +1 (per square unit). If you enclose a region of area pi/2 (which covers an eighth of the total 4pi of the sphere's surface) like one of the octants shown in the video, you'll see holonomy of +pi/2 radians. You'd see the same holonomy even if you made the sphere bigger since you enclose more area with an eighth but the sphere has less curvature per unit of area when it's bigger. An eighth of a sphere will always have pi/2 units of curvature no matter the size. The gauss bonnet theorem relates the euler characteristic of the surface with the curvature. If we assume the surface has no edge, the theorem states "the total curvature of a surface is 2pi times the euler characteristic". This euler characteristic can be described as V-E+F for any connected graph with sufficiently simple faces. For example, on a sphere you can have a graph (the octahedral one) with 6 vertices, 12 edges, and 8 faces. 6-12+8 = 2. For the genus 2 surface shown here, you can see a graph with 10 vertices, 20 edges, and 8 faces. 10-20+8 = -2. All edgeless orientable surfaces either have euler characteristic 2 (genus 0 i.e. sphere), 0 (genus 1 i.e. torus), -2 (genus 2 i.e. this), -4 (genus 3), ... If you want a different euler characteristic (but still have no edge), you'd have to look at non-orientable surfaces, which can be somewhat complicated. For example, the "real projective plane" has euler characteristic 1. Non-orientable surfaces always either have an edge (like the mobius strip), or intersect themselves (like the klein bottle). You might look at the torus with its euler characteristic of 0 and remark that clearly this surface is curved. If you total all the curvature though, the positive and negative curvature would exactly cancel out.
The word holonomy refers to the rotation caused by moving through curved space or on a curved surface or through hyperbolic space or on a hyperbolic surface.
Each corner is a 90 degree turn. So In a sense it's unsurprising in one case there are 4-1 turns and in the other it's 4+1 turns. Each right turn looks like the figure did a CCW turn relative to the direction of motion.
There is a game called Hyperrogue, that plays on a tiling of the hyperbolic plane. It has holonomic rotation (since it has non-zero curvature), which can be disorientating when playing.
Idk if “bend and snap” was an intentional pun but that was a goof reference to legally blonde lol, but I always love these videos I hope we can see more in the future with some interesting characteristics
This is genius! Help elementary and junior high school students develop geometric intuition, which they could recall in college. It helps learning math!
Are some of the 3d models actually available for 3d printing on our own? i would love to print one for a mathematician friend, but shapeways is too expensive unfortunately :/ as always, great content :)
@Russell Phelan yeah. 170dollars are a little bit too much for a student. i guess that 170 is reasonable if he puts in this time. But i dont have that money, but access to 3d printers
Now I am wondering if it's possible to play chess on a different surface. Maybe a cylinder where the ranks revolved around but the files don't. A torus won't. But this surface... Perhaps
Thing is - pawn is always rotating when it changes direction. even on a flat surface. you can easily see this if you "unfold" its path into a straight line: at every point where it had a turn it will rotate on 90 degrees in opposite direction. But on a flat surface you have exactly 4x90 = 360 degrees rotation so you simply don't notice it, while on a sphere you have one "missing" 90deg turn and on this surface - you have one extra so it can be easily noticed
neat! it's interesting how this differs from a simple toroid, which is rotationally quasi-flat; the distinction seems to lie in the curvature where the two "component" toroids join?
Do the arms only rotate or can some surfaces cause them to mirror? And if so, wouldn't it be nice to have the rook's arms differently colored for better visualization? As always, really interesnting. Topology is a beautiful subject.
In principle, a non-orientable surface would have paths that mirror the arms. It doesn’t seem easy to do the mechanical design for a puzzle that would have this feature.
You mean have the paths for the rook go through the surface (leaving the panel out there)? Interesting… the paths would have to be double sided even if the surface is only one sided, otherwise I think the object would not be connected. And moving the rook through the holes in this model is already fiddly - it could only be worse with a complicated self-intersecting surface.
@@henryseg Without the panels it wouldn't self-intersect. Think of a Mobius strip. Locally it has 2 rails even though there's only one. I'm hoping that the track crossings would hold it together, but maybe it's guaranteed to fall apart? If so, then maybe you can keep the panels but cut holes in them where the rails pass through.
Having five 90 degree angles has nothing to do with genus 2 which in turn has nothing to do with negative curvature. You could have negative curvature on a different genus object and you could have designed the maze with a different number if 90 degree turns
@@henryseg Ah interesting! I see what you mean now. But i think i am able to construct a maze without curvature even when the paths have to go "all the way around" in a straight line. Both on a genus 2 and on a genus 1 and 0 object? And perhaps also the converse?
The Klein bottle has Euler characteristic zero, so it naturally has flat, euclidean geometry. So, just like the chessboard, it wouldn't have any interesting holonomy effects.
I'm sure a topologist *could* argue that's just one hole, somehow, lol Something irks me about calling those rotations. I get that it appears to rotate from our point of view, but it feels wrong. Is there not a more appropriate name than rotation? Plane shifting? The object is not rotating any more than a chess piece on a flat board it's just a consequence of the shape of the plane it travels on. We're cheating because we have the perspective of a god. If this is considered a rotation then it's not possible to walk a straight line because every surface is curved which means youre constantly rotating no matter what you do, unless youre slowly actually rotating in the opposite direction) If we consider those lines that travel across the surface to be straight, then we cannot consider straight line travel to be rotational. If we call those rotations, then we should also describe the straight lines as curved. Otherwise we're swapping perspectives mid-sentence
Sorry to be this guy, but I'm literally getting my PhD in this. I personally would not say that a genus 2 surface has negative curvature. While the genus 2 torus does have *points* that have negative curvature, this is not true for all points. So I think this is a bit of a misnomer. Further more this does not take into account the flat genus 2 torus which has curvature 0 everywhere.
Sure, no embedding of the genus two surface into R^3 has constant gaussian curvature. It is true however that by Gauss-Bonnet, any cellulation in which the faces all have the same number of sides will show a holonomy effect as you go around any of those faces. It's usually a good idea to avoid getting bogged down in the technical details for the RU-vid audience! The "flat genus 2 torus" - are you talking about a translation surface version of the genus 2 surface? I would say that there the curvature is concentrated at the cone points?
@@henryseg "It's usually a good idea to avoid getting bogged down in the technical details for the RU-vid audience!". Fair enough, and sorry about that, but this is the first time I have had this card so I wanted to use it! By Flate genus, I believe you can identify the edges of a hexagon to create a quotient space for the genus two torus.
@@georgebabus2030 If you identify opposite edges of a hexagon you get the usual genus one torus. I think you have to go to an octagon to make it genus two.
The word genus refers to a group of things of the same type or kind that can be demonstrated to be so by the possession of common characteristics. It in no way refers to holes, perforations, or hollows of any sort. When a group of people adopt a word and give it a wholly new definition, simply on their own whim, that is called co-optation. To do so is always the clear mark of a dishonest and loathsome scoundrel of the most disgusting "genus".🤣🤣
Language evolves - that's just how it is. Every word came from some other word usage, often with a slight (or not so slight) change in meaning. According to www.etymonline.com/word/genus, apparently, the logical meaning of the word "genus" as a "kind or class of things" predates the biological use.
@@henryseg A typically false and self-serving gloss, as expected. The word genus in biology ALSO clearly means a type with common characteristics, as a genus is a group of closely related species that would have been capable of reproducing in the recent past (in geological terms), but which have since diverged while retaining those similarities. I can only surmise that you work in academia, for only there are such linguistic gymnastics encouraged and condoned. And this is why honest scholars have abandoned such institutions in droves, to avoid association with the likes of you.🤣🤣
@@archenema6792 I don't understand why you'd come to a video made by math nerds for math nerds and criticize it for using well established technical language. I'm legitimately concerned for you, don't you have something better to do with your time?
Very cool! I don't know a whole lot about topology but I didn't know the genus 2 double-donut was tiled by right-angled pentagons. Pretty cool how a sphere gets tiled by 3 right angles, the torus by 4 (so it's flat) and the genus 2 by 5. Does this pattern keep going I wonder? Also I knew the torus was flat but never really knew that it was because it was sort of in the perfect spot in a series like that.
Since any Platonic solid can tile a genus-0 sphere and any Euclidean tiling can tile a torus, does this mean that any regular compact hyperbolic 2D tiling can tile a surface of some sufficient genus like with these pentagons, ignoring any geometrical distortion that may arise as a result?
I thought it would rotate clockwise, but not because it has negative curvature, but because it rotated 5 times, while in the sphere it rotated 3 times, I assumed that if 3 rotations is equal to counter clockwise, 4 would be neutral, and 5 would be clockwise By rotations I mean, different rail segments, or number of corners passed through
These are all very closely related ideas. EDIT: Assuming all the sides are geodesics (the equivalent of "straight lines" on a curved surface), the curvature contained inside an n-sided shape equals the sum of its angles minus the sum of the angles of a flat n-sided shape (which is 180°·(n−2))
I think the key here is that the shapes consist entirely of 90 degree angles. You can tile a sphere with regular squares or regular pentagons, but in that case the angles won't be 90 degrees any more.
@@FreeFireFull they don’t have to be 90° or triangles/pentagons, they can be anything so long as the number of edges of the polygon formed is not a multiple/divisor of the number of edges at each vertex On a plane this property is always false. But it is true for a pentagon tiling on a sphere, so there will be holonomy, however since there are 1 more edges per polygon than per vertex, as opposed to 1 less, the direction of rotation is reversed compared to the triangles.
This is negatively curved space (I mean, "Surface," not "Space") so if you move the rook arround the pentagon clockwise, it will be rotated anticlockwise. [Edit: He said that he would move it anticlockwise. So if it is a negatively curved surface then the rook will rotate clockwise.] But by how much? On the dodecahedron where you moved the rook arround the pentagon, it rotated by ¹/6th. That can't happen here. The direction of rotation is determined by the curvature of the space or surface. The degree of rotation is determined by the angles. If the sphere with the triangles had 20 triangles, then they wouldn't have right-angles, they would be 60° angles, so the rotation wouldn't be the same amount. Actually, IS the shape negatively curved? You've got a positive curve, then a negative curve... No, the inside of a sphere is not negatively curved. Moving the rook around the inside of a sphere would have the same effect as moving it around the outside. (A direction that is clockwise on the outside of the sphere would be anticlockwise viewed from the inside, so what about a path that takes the rook around a loop that moves between the inside and the outside of a Kline bottle?) Henry describes the path around the right angled pentagon as "clockwise" but arn't parts of it anticlockwise? [Edit he said that the path is anticlockwise, but arn't parts of it clockwise?] 5 * 90°= 450° 450° - 360°= 90° Even if some of the corners cancel other corners out, there's an odd number of corners 90°+90°+90°+90°+90°=90° 90°+90°+90°+90°-90°=270°=-90° 90°+90°+90°-90°-90°=90° 90°+90°-90°-90°-90°=-90° 90°-90°-90°-90°-90°=-270°=90° So, the answer is definitely 90° but is that clockwise or anticlockwise? The curves that the rook must move along, appear to be: Clockwise, anticlockwise, clockwise, clockwise, clockwise. 90°-90°+90°+90°+90° =270° clockwise = 90° anticlockwise. [Edit: I mean clockwise. Every clockwise move, rotates the rook anticlockwise. Every anticlockwise move rotates the rook clockwise] [Edit: No, the opposite of that. Every move clockwise rotates the rook clockwise. -90°+90°-90°-90°-90° =-270° clockwise =90° anticlockwise.] [Edit] No, the opposite of all that. Actually, one could imagine a distortion of this shape wherin all the curves were clockwise. 90.°-90°-90°-90°-90°=-450° =90° clockwise. [Edit] clockwise doesn't rotate the rook -90°, it rotates it 90°, so: 90°+90°+90°+90°+90° =450° clockwise =90° anticlockwise.
The way to figure out the movement is to track the change of motion at the corners. When ever you switch the face you press on to move the rook it has "turned" because the arrow pointing along the path is suddenly perpendicular to the path. Pushing on the face on the right side of the rook causes it to be traveling translated 90 degrees clockwise with respect to the new motion.
That is a nicely illustrated toy. I would have loved to be able to play with that stuff in college. Also loved that coffee mug you were showing next to the sphere. 😁
Why doesn't the rook rotate when it travels around certain paths on the genus two surface? Can't a two holed torus span hyperbolic space? What makes those paths special?
well, if all the angles in some pentagon shape on the genus-2 surface are right angles, that feels a lot like walking around a vertex on a hyperbolic plane where there's 5 squares around a vertex and ending up where you started again, but being rotated, but with opposite holonomy to the sphere with all-right-angle triangles, so maybe the same thing applies to the genus-2 surface edit: just watched more of the video, my hypothesis was true, yay
I wonder if there is a number system similar to the quaternions that works for this system I'm trying to work it out at the moment, but i don't know if it's anywhere close to self-consistency
the sphere used quaternions, and the flat sheet can be described with C x C. So I wonder of this can be represented with the split quaternions or something similar.
As I was typing my guess I realized that I was wrong. I was going to say that it would rotate the same way. Make a full rotation and then a little bit more. But then I realize that it would rotate the other direction. Because a triangle has less sides than a square and a pentagon has more. So it would be like an expanding hyperbolic grid thingy. Forget what it's called. I don't know how far it would rotate but without fully visualizing it I'm just going to guess that it makes a 180 about face.
Nope, just one quarter rotation. I realized my mistake. A pentagon only has one more side than a square. A triangle has one less. But I was thinking about the triangle and added two. whoops.
paused at 2:00 to guess: I believe if the rook goes around that pentagon counterclockwise (anticlockwise if you prefer!) it will come back having rotated 90 degrees clockwise. So, that's like a -90 degree holonomy? I don't know how it's measured. Maybe it would make more sense to say that it's 270 degrees.
Odd number of points allow for directional change and even doesn't? This actually makes sense. Shapes with even number of points are all at 90deg angles despite not appearing as so and odd number points have at least one connection between two points that's not at 90deg. That's where the rotations occurs.
I really don't know if this is criticism or praise for the video yet, but at some point I just had to pause and visualize if genus one is holonomic or not. Part of me wishes that bridge was included in the video, but another part is glad I had to pause and think about it
