The twist was extra insane to anyone who kept up with the dev logs. CodeParade went through way too many hoops just to make Hyperbolica, then does an even more brutal and tedious job making 4D Golf. Literally the vast majority of the work was on the physics engines, each one way harder to make than the last, and then he just makes 2 of them in one game?!
@@KacaBiru Well, Hyperbolica had 3 custom physics engines, kinda. The initial world with higher space curvature in the length and width dimensions but not height, that cowboy world with lower space curvature, and that final world with higher curvature in all 3 dimensions. That said, reusing Hyperbolica's engine to make another engine sounds like something CodeParade would do.
Cosmodeus ( 29:19 ): "I've seen unfathomable horrors fall upon those not ready" Samut (smug): Like what, a quadruple bogey? Samut (during the next world) ( 1:48:33 ): I am physically ill ... this game is taking years off my life
To resolve the great torus debate: He is describing a _surface_ called a Clifford torus, which requires 4 dimensions to not self-intersect. You are describing a _surface_ called a donut torus, which requires 3 dimensions. Neither of those are the cartesian product of anything because they are surfaces. The cartesian product creates _topologies,_ unless someone states otherwise, which no-one did. The cartesian product of two circles is the 2D topology called a torus, which does describe both of those.
I think what the original one was refering to is that you can identify pairs of pairs ((a, b), (c, d)) and quadruplets (a, b, c, d) this way if you have a subset in 2d and you have a cartesian produc with another 2d subset, you get a subset in 4d example: let C={(x, y) | x²+y²=1} (C stands for circle, T stands for torus) then T = C×C = {((x, y), (z, w)) | x²+y²=1 and z²+w²=1} which we identify to {(x, y, z, w) | x²+y²=1 and z²+w²=1}, which is a surface embedded in 4d you can also use different parameters and say that C={(sin a, cos a)} and T~{(sin a, cos a, sin b, cos b)} from a topological standpoint T is just a simple 3d torus since you can look how it's unfolded if you use (a, b) as coordinates in a square and glue together the edges in 4d you kinda first stretch the torus so that it's as high as its diameter, squish the inside and outside parts together so it looks like a cylinder and then unsquish those same two parts into the 4th dimention such T is identified to the Clifford torus (maybe rescaled) let me know if I got anything wrong, the only thing I utilized that might not be conventional is ((a, b), (c, d)) ~ (a, b, c, d) conclusion: topologically 3d torus and Clifford torus are the same it makes some sense to talk about cartesian product of curves in 2d forming surfaces in 4d
@@creativenametxt2960 The ((a, b), (c, d)) ~ (a, b, c, d) ~ (a, (b, (c (d, {})))) thing always happens with cartesian products, i think. It does in fact make sense that 2d curves form 4d surfaces. Without context it is unusual to intend that interpretation. Especially after referring to your circles with S1.
@@chri-k I am not sure that's the case because doing a product of two 5d spaces would result in a 10d shape if we still use ((a, b, c, d, e), (f, g, h, i, j))~(a, b, c, d, e, f, g, h, i, j) then we have (a, b, c, d, e) = A×(cos(u), sin(u), 0, 0, 0) where A is a constant rotation matrix and (f, g, h, i, j) = B×(cos(v), sin(v), 0, 0, 0) expanding the matrices with zeros and rearranging the rows/columns you can get (I am fairly sure) (a, b, c, d, e, f, g, h, i, j) = C×(cos(u), sin(u), cos(v), sin(v), 0, 0, 0, 0, 0, 0) where C is a constant rotation matrix so you would simply get a rotated Clifford torus same with transltion of the original sets of points, but I do agree that if you were to scale them the should be the same size I think it makes sense to talk about 2d shapes by default embedded in 2d space anyways, but specifying that it has to be a unit circle in 2d is probably still good becase that specifies to the reader that you think of a circle as a set of coordinate pairs and that the cartesian product is the set of coordinate quadruplets (again, assuming the identity thing) probably a question of exact definitions of each term and what the people arguing interpret it as?
For some reason i thought it would end up skewed. That's pretty clearly not how that works even without doing the math -- the only way for a rotation to skew it would be if they were rotated in the extra dimensions the cartesian product adds, but then the operation isn't a cartesian product anymore. The point is, especially after referring to your circles as S1, you can't use the by far less common interpretation (as well as the one which doesn't fit what the previous person said) without saying what you are doing. Topology is the default interpretation, as well as the one that already was implicitly taken immediately from the start, when Samet said that it makes a torus. My point with the top comment wasn't that there is an error in reasoning -- there are none -- it's that neither of them applied the #1 rule of communication in math: make sure you have the same point/framework of reference, although Samet did try once. Although i seem to have failed at that myself.
Hey, at least you only invited people to challenge you with FOUR D golf levels ! :D Thank you so very much for this series, I really hope to see more of it in the future, as you seem to by far be the best at understanding and explaining this game/concepts of those I have watched so far!
"The mini golf is still the hard part" NEVER have I felt more seen. This chatter has seen the 9th dimension and lived. What a god. By far the most confusing part of the controls is knowing the way the rotations work. in the "3D" view, moving the mouse rotates your view in the X-Z plane when moving sideways and X-Y plane when moving up&down. When left-clicking, it rotates your view in the X-W plane when moving sideways and Z-W plane when moving up&down. In "Volume" view, Y is "hidden", and there's no reason to let the player make the course look upside-down, so "up&down" is disabled and you can only rotate your view in the X-W direction when left-clicking, since in volume view, looking "up&down" is the ZW plane. That's why "aligning" yourself at a right angle in Volume view prevents the course "sliding" out from under you in 4D - lines you up with the slice in the ZW plane. The thing that's happening 2:13:30 is just like this. Right clicking in the "3D" view rotates your view in the XV and ZV planes. Middle click rotates your view in the VW plane (hence why it's only visible in volume view. The axes visible in the "3D" view are XYZ, so the VW plane is entirely invisible. It can look like you're perfectly aligned in Volume view, but you still slide off because even though you're aligned in the ZW plane, you're NOT aligned in the ZV or VW planes. It can look like you're facing the right way, but you're actually trying to put with your view entirely diagonal. When you "align the ghost" in Volume view you're aligning yourself with the VW plane, allowing you to find what you're looking for using just the XZV look controls. Which is why you have to ONLY use right click, since using left click slides the W axis in a weird direction. Because the player's "view" is directional, It's basically an arrow pointing in four (later five) directions at once. And you NEVER get to "see" what V direction you're pointed in. Unless you look at the compass. Heehee.
27:00 An example of a true statement not provable from the axioms of a well-known system is the Goodstein theorem („every Goodstein sequence terminates“), which can be formulated in Peano arithmetic (PA), but as proven by Laurie Kirby and Jeff Paris in 1982, the Goodstein theorem cannot be derived from the Peano axioms. The proof of the Goodstein theorem is thus not carried out in Peano arithmetic, but a stronger system, usually that is Zermelo-Fraenkel set theory (ZF) without Foundation and Power Set (ZF⁻−P), or a similar set theory system. What’s important is that the result of “true, but not derivable” must _also_ be carried out in a stronger system, that is, the Kirby-Paris theorem is likewise carried out in some version/extension of ZFC (ZF with the Axiom of Choice). if PA could prove that PA can’t prove some statement A and neither its negation ¬A, PA would be inconsistent as that result implies that PA proved it can’t prove everything which is the same as PA proved it’s consistent - which is what Gödel showed that PA cannot do, unless it’s inconsistent. So, yes you can add axioms. ZF is basically that to PA, not really by intention, but definitely in practice. It is noteworthy that both the Goodstein and the Kirby-Paris theorems are not PA theorems, but (for the sake of argument) ZF theorems. Now, there’s two reasons why ZF can prove the Goodstein theorem (G): Either ZF is consistent and the Goodstein result is really true in PA, or ZF is inconsistent (and can prove anything). However, aside Gödel sentences, adding two or more axioms can lead you astray: Consider Zermelo-Fraenkel set theory (ZF) without the Axiom of Choice (AC). You might be saying: Oh, we can’t prove that all sets can be well-ordered, but we also provably can’t construct a set that must not have a well-order, so add an axiom that lets us well-order all sets (that is, add AC). You might be also saying: Oh, we can’t Lebesgue-measure all real subsets, but we provably can’t construct a set that must not have a Lebesgue-measure, so add an axiom that lets us measure all real subsets, that is add the Axiom of Determinacy (AD). The issue is, ZF+AC proves that there is a non-measurable set (called a Vitali set), so under ZF, AC and AD are incommensurate: ZF+AC proves “there’s a Vitali set” and ZF+AD proves “there is no Vitali set”. AC and AD are not on the same level of daringness or basic-ness. AC is on the same level as ZF, that is if ZF+AC is inconsistent, already ZF was. ZF and ZF+AC are equiconsistent. On the other hand, ZF+AD is equiconsistent to ZF+AC+“there are infinitely many Woodin cardinals” which is strictly stronger than ZF: The existence of any large cardinal (a Woodin cardinal is large) is enough to make an extension of ZF strictly stronger than ZF, as a large cardinal is enough to produce a model of ZF.
Yeah, those 'corrugated iron' textures seem to ignore momentum when rolling off them and only preserve the relative motion of the ball to the floor. Not certain why but it's probably to make certain things easier.
But graphics cards are very optimized to draw and manipulate triangles (or their nth dimension equivalents) so in 3D a sphere is drawn as a mesh of triangles, in 4D it's a mesh of tetrahedrons and in 5D a mesh of penta(whatever they're called I don't remember). However there are very few symmetries in 5D so finding a sphere-ish 5D object was a challenge and the reason the "ball" looks so lumpy.
Having played Beyond, I was laughing when I saw the cluelessly fitting "It's actually only 4D" in the background.
The twist was extra insane to anyone who kept up with the dev logs. CodeParade went through way too many hoops just to make Hyperbolica, then does an even more brutal and tedious job making 4D Golf. Literally the vast majority of the work was on the physics engines, each one way harder to make than the last, and then he just makes 2 of them in one game?!
Are you just implying there will be a 4D Hyperbolica?
@@KacaBiru Well, Hyperbolica had 3 custom physics engines, kinda. The initial world with higher space curvature in the length and width dimensions but not height, that cowboy world with lower space curvature, and that final world with higher curvature in all 3 dimensions.
That said, reusing Hyperbolica's engine to make another engine sounds like something CodeParade would do.
@@KingMako30 apparently the hyperbolica physics engine was based on the marble marcher physics engine, so yeah, that is something he would do
Cosmodeus ( 29:19 ): "I've seen unfathomable horrors fall upon those not ready"
Samut (smug): Like what, a quadruple bogey?
Samut (during the next world) ( 1:48:33 ): I am physically ill ... this game is taking years off my life
The 6D directions are probably Beavis and Butthead. XD
1:59:13 “bada bing, bada boom, wtf wait what?“ is certainly a funny line
To resolve the great torus debate:
He is describing a _surface_ called a Clifford torus, which requires 4 dimensions to not self-intersect.
You are describing a _surface_ called a donut torus, which requires 3 dimensions.
Neither of those are the cartesian product of anything because they are surfaces.
The cartesian product creates _topologies,_ unless someone states otherwise, which no-one did.
The cartesian product of two circles is the 2D topology called a torus, which does describe both of those.
I think what the original one was refering to is that you can identify pairs of pairs ((a, b), (c, d)) and quadruplets (a, b, c, d)
this way if you have a subset in 2d and you have a cartesian produc with another 2d subset, you get a subset in 4d
example: let C={(x, y) | x²+y²=1} (C stands for circle, T stands for torus)
then T = C×C = {((x, y), (z, w)) | x²+y²=1 and z²+w²=1} which we identify to {(x, y, z, w) | x²+y²=1 and z²+w²=1}, which is a surface embedded in 4d
you can also use different parameters and say that C={(sin a, cos a)} and T~{(sin a, cos a, sin b, cos b)}
from a topological standpoint T is just a simple 3d torus since you can look how it's unfolded if you use (a, b) as coordinates in a square and glue together the edges
in 4d you kinda first stretch the torus so that it's as high as its diameter, squish the inside and outside parts together so it looks like a cylinder and then unsquish those same two parts into the 4th dimention
such T is identified to the Clifford torus (maybe rescaled)
let me know if I got anything wrong, the only thing I utilized that might not be conventional is ((a, b), (c, d)) ~ (a, b, c, d)
conclusion:
topologically 3d torus and Clifford torus are the same
it makes some sense to talk about cartesian product of curves in 2d forming surfaces in 4d
@@creativenametxt2960
The ((a, b), (c, d)) ~ (a, b, c, d) ~ (a, (b, (c (d, {})))) thing always happens with cartesian products, i think.
It does in fact make sense that 2d curves form 4d surfaces. Without context it is unusual to intend that interpretation.
Especially after referring to your circles with S1.
@@chri-k I am not sure that's the case
because doing a product of two 5d spaces would result in a 10d shape if we still use ((a, b, c, d, e), (f, g, h, i, j))~(a, b, c, d, e, f, g, h, i, j)
then we have (a, b, c, d, e) = A×(cos(u), sin(u), 0, 0, 0) where A is a constant rotation matrix and (f, g, h, i, j) = B×(cos(v), sin(v), 0, 0, 0)
expanding the matrices with zeros and rearranging the rows/columns you can get (I am fairly sure) (a, b, c, d, e, f, g, h, i, j) = C×(cos(u), sin(u), cos(v), sin(v), 0, 0, 0, 0, 0, 0) where C is a constant rotation matrix
so you would simply get a rotated Clifford torus
same with transltion of the original sets of points, but I do agree that if you were to scale them the should be the same size
I think it makes sense to talk about 2d shapes by default embedded in 2d space anyways, but specifying that it has to be a unit circle in 2d is probably still good becase that specifies to the reader that you think of a circle as a set of coordinate pairs and that the cartesian product is the set of coordinate quadruplets (again, assuming the identity thing)
probably a question of exact definitions of each term and what the people arguing interpret it as?
@@creativenametxt2960 uhh, i think my comment got nuked. I think that's either because I edited it too much or because UA-cam did a UA-cam
For some reason i thought it would end up skewed. That's pretty clearly not how that works even without doing the math -- the only way for a rotation to skew it would be if they were rotated in the extra dimensions the cartesian product adds, but then the operation isn't a cartesian product anymore.
The point is, especially after referring to your circles as S1, you can't use the by far less common interpretation (as well as the one which doesn't fit what the previous person said) without saying what you are doing.
Topology is the default interpretation, as well as the one that already was implicitly taken immediately from the start, when Samet said that it makes a torus.
My point with the top comment wasn't that there is an error in reasoning -- there are none -- it's that neither of them applied the #1 rule of communication in math: make sure you have the same point/framework of reference, although Samet did try once.
Although i seem to have failed at that myself.
Hey, at least you only invited people to challenge you with FOUR D golf levels ! :D
Thank you so very much for this series, I really hope to see more of it in the future, as you seem to by far be the best at understanding and explaining this game/concepts of those I have watched so far!
1:32:35 Can’t wait for 9D golf.
This is what ive been waiting for!!
Awesome playthrough, really enjoyed how much anger the beyond levels got out of you XD
"The mini golf is still the hard part" NEVER have I felt more seen. This chatter has seen the 9th dimension and lived. What a god.
By far the most confusing part of the controls is knowing the way the rotations work.
in the "3D" view, moving the mouse rotates your view in the X-Z plane when moving sideways and X-Y plane when moving up&down.
When left-clicking, it rotates your view in the X-W plane when moving sideways and Z-W plane when moving up&down.
In "Volume" view, Y is "hidden", and there's no reason to let the player make the course look upside-down, so "up&down" is disabled and you can only rotate your view in the X-W direction when left-clicking, since in volume view, looking "up&down" is the ZW plane. That's why "aligning" yourself at a right angle in Volume view prevents the course "sliding" out from under you in 4D - lines you up with the slice in the ZW plane.
The thing that's happening 2:13:30 is just like this. Right clicking in the "3D" view rotates your view in the XV and ZV planes. Middle click rotates your view in the VW plane (hence why it's only visible in volume view. The axes visible in the "3D" view are XYZ, so the VW plane is entirely invisible. It can look like you're perfectly aligned in Volume view, but you still slide off because even though you're aligned in the ZW plane, you're NOT aligned in the ZV or VW planes. It can look like you're facing the right way, but you're actually trying to put with your view entirely diagonal.
When you "align the ghost" in Volume view you're aligning yourself with the VW plane, allowing you to find what you're looking for using just the XZV look controls. Which is why you have to ONLY use right click, since using left click slides the W axis in a weird direction.
Because the player's "view" is directional, It's basically an arrow pointing in four (later five) directions at once. And you NEVER get to "see" what V direction you're pointed in. Unless you look at the compass. Heehee.
Matt Parker had a video about the $1 vs $20 meme, in which he also explained the misconceptions people had about this.
Don't forget about the Marble Mode that you unlocked
27:00 An example of a true statement not provable from the axioms of a well-known system is the Goodstein theorem („every Goodstein sequence terminates“), which can be formulated in Peano arithmetic (PA), but as proven by Laurie Kirby and Jeff Paris in 1982, the Goodstein theorem cannot be derived from the Peano axioms. The proof of the Goodstein theorem is thus not carried out in Peano arithmetic, but a stronger system, usually that is Zermelo-Fraenkel set theory (ZF) without Foundation and Power Set (ZF⁻−P), or a similar set theory system. What’s important is that the result of “true, but not derivable” must _also_ be carried out in a stronger system, that is, the Kirby-Paris theorem is likewise carried out in some version/extension of ZFC (ZF with the Axiom of Choice). if PA could prove that PA can’t prove some statement A and neither its negation ¬A, PA would be inconsistent as that result implies that PA proved it can’t prove everything which is the same as PA proved it’s consistent - which is what Gödel showed that PA cannot do, unless it’s inconsistent.
So, yes you can add axioms. ZF is basically that to PA, not really by intention, but definitely in practice. It is noteworthy that both the Goodstein and the Kirby-Paris theorems are not PA theorems, but (for the sake of argument) ZF theorems. Now, there’s two reasons why ZF can prove the Goodstein theorem (G): Either ZF is consistent and the Goodstein result is really true in PA, or ZF is inconsistent (and can prove anything).
However, aside Gödel sentences, adding two or more axioms can lead you astray: Consider Zermelo-Fraenkel set theory (ZF) without the Axiom of Choice (AC). You might be saying: Oh, we can’t prove that all sets can be well-ordered, but we also provably can’t construct a set that must not have a well-order, so add an axiom that lets us well-order all sets (that is, add AC). You might be also saying: Oh, we can’t Lebesgue-measure all real subsets, but we provably can’t construct a set that must not have a Lebesgue-measure, so add an axiom that lets us measure all real subsets, that is add the Axiom of Determinacy (AD).
The issue is, ZF+AC proves that there is a non-measurable set (called a Vitali set), so under ZF, AC and AD are incommensurate: ZF+AC proves “there’s a Vitali set” and ZF+AD proves “there is no Vitali set”. AC and AD are not on the same level of daringness or basic-ness. AC is on the same level as ZF, that is if ZF+AC is inconsistent, already ZF was. ZF and ZF+AC are equiconsistent. On the other hand, ZF+AD is equiconsistent to ZF+AC+“there are infinitely many Woodin cardinals” which is strictly stronger than ZF: The existence of any large cardinal (a Woodin cardinal is large) is enough to make an extension of ZF strictly stronger than ZF, as a large cardinal is enough to produce a model of ZF.
GENIUS plot twist
We have reached the dimension of recordable video-cassette boxes!
plot twist is such a troll
It just straight up pulls a middle finger at you right as you start to understand the game
1:02:15 is rather funny in retrospect
8:03 Could have tried to bounce off one of the walls at that hole
I got to 20:00 and was still confused so skipped to 1:38:00 and am still confused but at least it looks like a nice trip
21:56 momentum was not conserved?
I think rotational inertia would make it not super smooth, but certainly not that jarring.
Yeah, those 'corrugated iron' textures seem to ignore momentum when rolling off them and only preserve the relative motion of the ball to the floor. Not certain why but it's probably to make certain things easier.
Wow UA-cam compression did NOT like this one lol
sursum and deorsum are latin for up and down
THANK YOU SO MUCH BRO , can you plz finish tandis , you still have more Levels to complete, i realy enjoy ur videos plz , god pless you bro
well what happens in 5D if you just take all points that have a distance of 1 or less from a specific point? is that not a 5D sphere?
그 모양을 프로그래밍하려면 무한한 용량이 필요합니다.
But graphics cards are very optimized to draw and manipulate triangles (or their nth dimension equivalents) so in 3D a sphere is drawn as a mesh of triangles, in 4D it's a mesh of tetrahedrons and in 5D a mesh of penta(whatever they're called I don't remember). However there are very few symmetries in 5D so finding a sphere-ish 5D object was a challenge and the reason the "ball" looks so lumpy.
It's too damn late to fuck with 4d