Spiny Norman said:
The Dome Paradox: A Loophole in Newton’s Laws.
Now had a read of that.
In my opinion the initial presumption that Newtonian Physics is deterministic is not valid, so no paradox.
But I’d be interested to hear what others think.
For those who don’t have 20 minutes to spare, read here:
https://www.youtube.com/watch?v=EjZB81jCGj4
0:00:00 - I’ve got a question for you.
0:02:00 What will happen if I place this ball
0:03:00 right at the apex of this dome
0:07:00 and let go?
0:09:00 (ball clacking)
0:11:00 Well, you probably guessed that.
0:12:00 But now imagine we’re in a theoretical world
0:15:00 where this dome is perfectly smooth, no friction or bumps.
0:19:00 There are no disturbances
0:20:00 or forces acting other than gravity.
0:22:00 And I can balance this ball
0:24:00 perfectly, perfectly, perfectly at the apex.
0:31:00 Now, what do you think will happen when I let go?
0:37:00 If that was your guess, you’re right.
0:40:00 Newton’s first law says that an object at rest
0:42:00 will remain at rest unless acted upon by an external force.
0:46:00 (ball clacking)
0:48:00 Or so we thought.
0:51:00 This is a paper by an Australian philosopher of physics
0:53:00 named John Norton.
0:55:00 It claims that Newton’s laws
0:57:00 also say that at some random time,
0:59:00 the ball will spontaneously start rolling off the dome
1:02:00 without a cause.
1:04:00 No quantum mechanics involved.
1:06:00 Just good old Newtonian physics.
1:08:00 Why am I making a video about this? You ask.
1:11:00 There’s obviously some mistake.
1:13:00 A ball can’t just start rolling on its own without a cause.
1:17:00 But here’s the thing.
1:19:00 This paper was published in 2008.
1:22:00 It’s been 16 years since Norton published this result,
1:26:00 and no one can point out what’s wrong with it.
1:29:00 And it’s not the lack of trying.
1:31:00 This is a map that shows other research papers
1:33:00 that have talked about the dome.
1:35:00 There are rebuttals, rebuttals of those rebuttals,
1:38:00 outright denials, and a lot of heated arguments.
1:42:00 Some scientists think that this result
1:44:00 might even reveal a fourth law of motion,
1:46:00 and others think it’s a proof that humans have free will.
1:50:00 - I’m responsible for lots of provocative ideas
1:52:00 and philosophy that gets me into trouble.
1:54:00 - (laughs) Now, when I first read this paper, I thought,
1:58:00 surely this is just some kind of sleight of hand math trick.
2:01:00 But after weeks of investigating,
2:04:00 I realized just how deep this rabbit hole goes.
2:07:00 It goes beyond, hey, a ball is moving weirdly
2:10:00 in this one specific scenario.
2:12:00 It challenges the entire mathematical foundation
2:15:00 of Newtonian mechanics, the most widely used
2:18:00 and well understood framework in modern physics.
2:21:00 Humanity would not be where it was if it wasn’t
2:23:00 for Newton’s laws and the math derived from them.
2:26:00 While they can’t quite describe the inner workings
2:28:00 of an atom or when things are moving at near light speeds,
2:31:00 they describe pretty much everything in between.
2:34:00 So, how does this little dome challenge all of that?
2:38:00 And what does it mean for physics?
Why should we care?
2:41:00 Before we dive in, this is an extremely subtle topic
2:44:00 and I want you to understand
2:45:00 what this video is and is not about.
2:48:00 It’s not about whether a physical ball
2:50:00 will really move off the top of a physical dome
2:53:00 in the real world.
2:54:00 In reality, you can never place a ball right at the apex
2:57:00 without it rolling down
2:58:00 as you can’t create the ideal conditions.
3:02:00 And even if you could,
3:03:00 quantum mechanics would interfere at some point.
3:05:00 (soft music)
3:07:00 (screen whooshing)
3:10:00 This video is about what the mathematical theories
3:12:00 we use to understand reality tell us.
3:16:00 Now, I know some of you are thinking,
3:17:00 “What’s the point of that?
3:19
If this is purely theoretical
3:21
and doesn’t apply to the real world, who cares?”
3:23:00 Well, there are two reasons.
3:26:00 One, sure, we might be working in an ideal world
3:29:00 without friction or disturbances,
3:31:00 but anyone who’s done even high school physics
3:34:00 would know that that’s basically how we deal
3:36:00 with the real world in physics.
3:38:00 To make problems simple enough to solve,
3:40:00 we make assumptions like there’s no air friction,
3:43:00 objects are point masses,
3:44:00 and no energy is lost in collisions.
3:47:00 So, this idealized framework is pretty standard in physics.
3:51:00 And two, imagine two gigantic numbers.
3:55:00 Numbers so big that that many things
3:57:00 couldn’t physically fit into our universe.
4:00:00 Now imagine that if we found out
4:02:00 that our rules of addition didn’t work for them.
4:04:00 When we add them together,
4:06:00 we get something weird and unexplainable.
4:09:00 It wouldn’t physically affect anything in the real world,
4:12:00 but it would mean that our theory of addition
4:14:00 doesn’t work all the time,
4:16:00 and that’s something worth knowing.
4:19:00 That’s exactly the point of the dome.
4:22:00 You’ll see limits in Newton’s theory
4:24:00 that we weren’t aware of.
4:25:00 And how even the most well established theories
4:29:00 can still surprise us.
4:31:00 (notification trilling)
Hand-wavey explanation
4:33:00 Now, we’re actually going to ease into this
4:35:00 with a not too technical, hand wavy explanation
4:39:00 to give you some intuition for why this works.
4:41:00 In Newtonian mechanics, the laws of physics
4:43:00 work the same way forward and backwards in time.
4:47:00 If you record a physical process
4:48:00 and then play the recording backwards,
4:50:00 the reversed sequence
4:51:00 should still obey the same physical laws.
4:54:00 This is called time reversal symmetry.
4:57:00 Now, if you accept that it’s possible to nudge the ball
5:00:00 with just the right amount of force
5:02:00 so that it rolls up to the apex and stops there,
5:06:00 by time reversal symmetry, it’s a valid solution
5:09:00 for the ball to rest at the top of the apex for a while
5:12:00 and then roll down.
5:14:00 - The starting point of this is John Earman’s book on,
Newtonian Determinism
5:17:00 called “A Primer on Determinism.”
5:19:00 That came out, I think, sometime in the 1980s.
5:23:00 And John just alerted us to the fact
5:26:00 that there were all sorts of cases of indeterminism
5:28:00 all the way through physics.
5:32:00 - This is super important. Let’s break it down.
5:36:00 Newton’s laws are awesome at making predictions.
5:38:00 They were even used to predict the existence of Neptune
5:41:00 before it was ever observed.
5:44:00 If we are given the position and velocity
5:46:00 of something at the present time,
5:48:00 we can calculate its position and velocity
5:50:00 for any future time.
5:54:00 Said simply, the future state of things
5:56:00 depends on their present state.
5:58:00 And importantly, there is only one possible future state
6:02:00 for any present state.
6:04:00 If there were more than one possible future state,
6:06:00 well, that would throw our powers of prediction
6:09:00 out the window.
6:11:00 The idea that any present state has only one future state
6:13:00 is called determinism.
6:16:00 Determinism is a pretty huge part of the philosophy
6:18:00 of Newtonian mechanics.
6:20:00 Pretty much since its conception,
6:21:00 physicists have viewed Newtonian mechanics
6:23:00 as a deterministic theory.
6:26:00 The way determinism manifests itself
6:28:00 in the math of Newton’s theory is like this.
6:30:00 You might have heard that in maths
6:32:00 is the language of the universe.
6:33:00 In that case, the dialect of Newtonian mechanics
6:36:00 is differential equations.
6:38:00 The solutions to differential equations
6:40:00 tell you how things change over time.
Uniqueness Theorem
6:43:00 You can imagine that would be pretty useful
6:45:00 as all of motion is pretty much
6:47:00 how an object’s position changes over time.
6:50:00 The way determinism is expressed mathematically
6:52:00 is that each differential equation has only one solution,
6:56:00 one possible trajectory.
6:58:00 This is called the uniqueness theorem.
7:01:00 If a Newtonian equation had more than one solution
7:04:00 for the same initial conditions,
7:06:00 that would mean that a present state
7:07:00 had more than one possible future state,
7:10:00 and that would break determinism.
7:13:00 And that is exactly the problem with this dome.
Multiple solutions
7:17:00 The equation of motion that describes how the ball moves
7:19:00 when it’s placed at the apex has more than one solution.
7:23:00 It’s like one of those choose your own adventure books.
7:26:00 If you start at the beginning, the initial conditions,
7:28:00 you can end up reading entirely different stories.
7:32:00 But determinism says that Newtonian mechanics
7:34:00 is like a regular book.
7:36:00 When you start at the beginning
7:37:00 or when you’re given one set of initial conditions,
7:40:00 there should only be one story.
7:43:00 So, how did Norton come up with a scenario
7:46:00 that defies centuries of classical physics?
7:48:00 It must have taken years of hard work and toil.
7:51:00 - I mean this is really The work of just a short afternoon.
7:53:00 There wasn’t terribly much to do it.
7:57:00 - Oh.
7:58:00 So, here are the solutions to the equation of motion.
8:00:00 See? More than one.
8:02:00 This solution describes the ball
8:03:00 sitting at the apex forever, exactly what we would expect.
8:06:00 And this solution says that at some random time
8:09:00 called the excitation time,
8:12:00 the ball just starts rolling down the dome on its own.
8:15:00 So my first thought when I read this was,
8:18:00 “I do not understand. Where did this come from?
8:22
Why did this come from? How does it, what?”
8:26:00 Sure, we can plug the solutions into the equation of motion
8:28:00 and see that they both work, but that’s just like,
8:31:00 yeah, okay, it works because math.
8:34:00 Obviously those thoughts aren’t what you want
8:35:00 in an educational video.
Breaking determinism
8:37:00 So I thought, you know what?
8:39:00 The only way we’re gonna understand this
8:40:00 is if we just figure it out ourselves
8:42:00 from scratch right now.
8:45:00 So that’s exactly what we’re gonna do.
8:47:00 Imagine you hate Newtonian determinism
8:49:00 and you want to come up with some situation that breaks it.
8:53:00 What’s the first thing you would do?
8:56:00 Well, you might first wanna figure out
8:58:00 how to break determinism.
9:00:00 In other words, how to come up with a differential equation
9:02:00 that has more than one solution.
9:04:00 - There are conditions that tell you
Lipschitz Continuity
9:06:00 when a differential equation will have a unique solution,
9:09:00 and I knew one of those conditions
9:11:00 was the Lipschitz Condition.
9:13:00 And it’s just standard in mathematics books,
9:16:00 that they’ll say,
9:17:00 “Well, if you wanna have a unique solution,
9:19
you need special conditions.
9:21
The Lipschitz Condition
9:22
is one that will give you a unique solution.
9:24
If you don’t have it, here’s how uniqueness might fail.”
9:28:00 - Thanks, Norton.
9:29:00 Okay, Lipschitz Condition.
9:31:00 (bright playful music)
9:34:00 In the theory of differential equations, Lipschitz continuity is the central condition of the Picard-Lindelof theorem,
which guarantees the existence and uniqueness of the solution to an initial value problem.
9:47:00 Okay, that seems promising.
9:48:00 (bright rhythmic music)
9:52:00 Hmm.
9:53:00 So, it turns out there’s this really important theorem
9:56:00 in math called the Picard-Lindelof theorem.
9:58:00 It gives a set of conditions for an equation
10:00:00 to have a unique solution.
10:02:00 If all these conditions are met, this guarantees uniqueness.
10:06:00 But if we don’t meet all these conditions,
10:09:00 an equation can have multiple solutions.
10:12:00 That’s good. That’s exactly what we want.
10:15:00 An equation that has more than one solution,
10:17:00 so we can break determinism, which we hate.
10:21:00 The central condition of the Picard-Lindelof theorem
10:24:00 is the Lipschitz Condition or Lipschitz continuity.
10:27:00 If we don’t have Lipschitz continuity,
10:29:00 this can lead to splitting and branching of a solution.
10:34:00 So guys, we just need to break Lipschitz continuity.
10:37:00 How do we do that?
10:39:00 Well, the first step might be finding out what it is.
10:41:00 (bright rhythmic music)
10:43:00 Hmm, hmm. Yes.
10:46:00 Interesting. What could it be?
10:50:00 Hmm. Basically the lip shifts condition makes sure
10:53:00 a function doesn’t change too abruptly,
10:56:00 that a slope doesn’t get too big too quickly
10:58:00 or explode, if you will.
11:01:00 It definitely should not be vertical.
11:03:00 In math talk, it needs to be increasing by a finite amount,
11:07:00 not an infinite amount.
11:09:00 There are actually a lot of functions
11:11:00 whose slopes explode like this at some point.
11:13:00 A simple one is Y equals the square root of X.
11:17:00 It doesn’t look too crazy,
11:19:00 but let’s zoom in right around X equals zero.
11:22:00 (gentle playful music)
11:24:00 You can see the slope becoming more and more vertical.
11:28:00 And if we zoom in far enough,
11:30:00 we can see the slope blow up to infinity,
11:33:00 breaking Lipschitz continuity.
11:35:00 (gentle playful music)
11:38:00 Guys, we’ve made good progress.
11:40:00 We’ve found a function that breaks Lipschitz continuity,
11:43:00 which leads to multiple solutions
11:45:00 which breaks determinism, which we hate.
Okay, now what?
Breaking NEWTONIAN determinism
11:51:00 This isn’t a Newtonian system.
11:53:00 This is just a function,
11:54:00 an abstract relationship between X and Y.
11:57:00 We are trying to break Newtonian determinism,
12:00:00 which means we need to break determinism
12:01:00 within a Newtonian system.
12:04:00 Okay, so next question. What’s a Newtonian system?
12:08:00 Basically, any physical scenario
12:10:00 that’s described by Newton’s laws.
12:13:00 Objects moving under gravity, tension, friction,
12:16:00 or any other force that can be described by Newton’s laws.
12:20:00 So how can we use our function and get it to describe
12:23:00 some kind of Newtonian system?
12:27:00 Well, imagine you’ve been given a job
12:28:00 to build a slide at your local playground.
12:31:00 You can choose how steep and curvy it is,
12:34:00 which directly affects how fast the kids
12:36:00 will accelerate at each point.
12:39:00 But you’ve built it too steep and too curvy.
12:43:00 (body squelching)
12:45:00 Now you’ve been asked to make sure the slope and curves
12:47:00 are such that the kids stay on the slide,
12:50:00 and aren’t going so fast, they get scared.
12:53:00 So, you figure out the right acceleration
12:55:00 and you work backwards to build the shape of the slide
12:59:00 that ensures this acceleration.
13:01:00 (gentle rhythmic music)
13:02:00 So, just as a shape influences
13:05:00 the motion of an object on it,
13:06:00 if you start with what kind of motion you want,
13:09:00 you can reverse engineer the right shape.
13:12:00 Similarly, let’s say we want an object like a ball
13:15:00 to accelerate according to this relationship.
13:19:00 The simplest way to get a ball to accelerate
13:21:00 is to put it on a slope and add gravity.
13:24:00 A ball accelerating under gravity.
13:26:00 That sounds like a Newtonian system to me.
13:28:00 Let X be the position of the ball
13:30:00 and Y be the acceleration of the ball.
13:33:00 When the ball is at position zero,
13:35:00 its acceleration is also zero.
13:37:00 What could that physically correspond to?
13:40:00 Well, it could be a ball sitting still on a flat surface.
13:43:00 Now as the position of the ball increases,
13:46:00 so does its acceleration.
13:48:00 What can we do to our surface
13:49:00 to make the ball move like this?
13:52:00 Well, we could make it get steeper as position increases.
13:56:00 We end up with a kind of ramp.
13:58:00 And if we make it symmetrical about the origin,
14:01:00 we get a dome.
14:03:00 In fact, if we do the math,
14:06:00 we end up with exactly this dome.
14:11:00 So that’s how the dome breaks determinism.
14:14:00 As I mentioned, it sparked a lot of controversy.
14:17:00 A lot of people outright hated the dome
14:19:00 and tried to invalidate it.
14:21:00 Scientists and philosophers had plenty of problems with it,
14:24:00 but no one could agree on a specific definitive flaw
14:28:00 that completely invalidates it.
14:30:00 Nothing concrete enough to point at and say,
14:33:00 This is the reason the dome doesn’t work.
14:35:00 One of the biggest points of contention,
14:38:00 and what was my biggest problem with it
14:40:00 is this whole ball moving by itself thing.
14:43:00 Surely this is just wrong.
14:45:00 So, why are we even entertaining it
14:47:00 as a valid solution even theoretically?
We discard solutions all the time in physics.
14:14.Hate for the dome
14:52:00 So, why not just chuck it out?
14:54:00 Physics cares about math,
14:56:00 but math doesn’t care about physics.
14:58:00 It’s up to us to determine what makes sense
15:00:00 and discard what doesn’t.
15:03:00 But this raises the question.
15:05:00 How do we decide what makes sense?
15:08:00 How do we know when to discard a solution?
15:12:00 Well, a good indication
15:13:00 is when it doesn’t make physical sense
15:16:00 or if it violates a known law of physics.
15:19:00 Take the example of a ball being rolled against a wall.
15:22:00 Viewing this situation
15:24:00 through the conservation of kinetic energy,
15:26:00 we can say that the square of the initial speed, U squared,
15:29:00 is equal to the square of the final speed
15:32:00 after it’s hit the wall, V squared.
15:34:00 Solving this equation, we get two solutions.
15:38:00 Instinctively, this is the correct one
15:41:00 as it corresponds to the ball rebounding off the wall.
15:44:00 This one corresponds to the ball going through the wall,
15:48:00 which from experience doesn’t make any physical sense.
15:51:00 We know balls can’t travel through walls.
15:53:00 This solution both doesn’t make physical sense
15:56:00 and it violates Newton’s third law of motion.
15:58:00 When two objects collide,
16:00:00 they exert an equal and opposite force on each other.
16:03:00 So, we can safely chuck it out.
16:07:00 So can we chuck out this problematic dome solution?
16:11:00 Well, it seems pretty unphysical to me that a ball
16:13:00 can just start suddenly moving by itself.
16:17:00 But here, Norton’s got an answer for us.
16:19:00 He says that the whole unphysical argument
16:22:00 doesn’t apply here because the dome
16:24:00 is a mathematical creation within Newtonian theory.
16:27:00 Therefore, we have no prior knowledge
16:30:00 about whether the ball should stay there forever
16:33:00 or whether it could spontaneously move off.
16:36:00 The fact that we get an indeterminate solution
16:38:00 is not impossible, it’s just unexpected.
16:42:00 Okay, but what about the fact that this result
16:44:00 clearly breaks Newton’s first law of motion?
Which stated in its entirety is,
Does the dome break Newton’s 1st Law?
16:49:00 “In the absence of a net external force,
16:51
a body remains at rest or in a state of uniform motion
16:54
in a straight line.”
16:56:00 But here, Norton’s got an answer too.
16:59:00 He argues that we’re used to thinking of uniform motion
17:02:00 in a straight line over some time interval.
17:06:00 But in this context,
17:07:00 we need to apply the law to a single instant of time.
17:12:00 The law then becomes,
17:13:00 “In the absence of a net external force,
17:16
a body should have zero acceleration.”
17:19:00 In other words, zero net force equals zero acceleration.
17:24:00 So, is there a moment where this instantaneous version
17:26:00 of Newton’s first law breaks?
17:29:00 Well, before the ball moves,
17:30:00 or as Norton calls it, “before the excitation time”,
17:34:00 the mass is sitting still at the apex.
17:37:00 There’s no net force acting on it as gravity
17:39:00 is balanced out perfectly by the normal force
17:42:00 and it’s not accelerating.
17:44:00 This meets our conditions.
17:46:00 After the excitation time,
17:48:00 the mass is accelerating down the dome,
17:50:00 but it’s also got a net force acting on it.
17:52:00 Gravity.
17:53:00 It accelerates in accord with F equals an A,
17:56:00 so this situation checks out too.
17:59:00 And at the exact moment of the excitation time,
18:02:00 well, this is obviously the point of interest.
18:04:00 Now, I don’t generally like doing this,
18:07:00 but the only way I could think to explain this part
18:09:00 is through the math.
18:10:00 I don’t like doing that
18:11:00 because I don’t think it gives great intuition,
18:14:00 but I honestly don’t think
18:15:00 there’s an intuitive explanation for this.
18:17:00 If you take the solution
18:18:00 where the ball spontaneously rolls off the dome,
18:21:00 you can get its acceleration by differentiating twice
18:24:00 or just looking at the paper.
18:27:00 The exact moment the ball moves is when time, small t,
18:30:00 is equal to the excitation time, big T.
18:34:00 So when we plug that into our equation,
18:36:00 this term becomes zero.
18:38:00 So, the entire thing is zero.
18:41:00 No acceleration along with no net force.
18:44:00 Just as Newton’s first law says.
18:47:00 So, there is no actual moment we can pinpoint
18:50:00 where this instantaneous version of Newton’s law is broken.
18:54:00 So, why does the mass move?
18:57:00 Well, this is exactly Norton’s claim.
18:59:00 Newtonian physics is indeterministic.
19:02:00 There doesn’t need to be a why.
19:04:00 We are just so used to thinking about this in a causal way.
19:08:00 He says, “Our natural causal instinct is to seek
19:11:00 the first instant at which the mass moves
19:13:00 and then look for the cause of the motion at that instant.
19:16:00 We are tempted to think of the excitation time
19:18:00 as the first instant at which the mass moves.
19:21:00 But that is not so.
19:23:00 It is the last instant at which the mass does not move.
19:27:00 There is no first instant at which the mass moves.
19:29:00 The mass moves during the interval,
19:31:00 after the excitation time,
19:33:00 and this interval has no first instant.
19:36:00 So there is no first instant of motion
19:38:00 and thus no first instant
19:39:00 at which to seek the initiating cause.”
19:43:00 But I’m curious to know what you guys think.
19:45:00 Are you convinced by Norton’s arguments?
Leave your opinions in the comments below.
What does the dome mean for physics?
19:51:00 So now the burning question.
19:53:00 What does this all mean for physics?
19:55:00 Well, in the most extreme case, it could mean
19:58:00 that Newtonian mechanics is not a deterministic theory.
20:01:00 But it feels a bit strange to say
20:03:00 that an entire theory is not deterministic
20:06:00 because of a few edge cases that break it.
20:09:00 More likely is that Newtonian theory
20:11:00 is just not as clear cut as we thought,
20:13:00 and we don’t understand it, as well as we thought we did.
20:16:00 Some philosophers go so far as to say
20:19:00 there isn’t a single conception of Newtonian physics.
20:22:00 There is not one true version, but many versions
20:25:00 which are all correct
20:27:00 even if they’re all at odds with each other.
20:30:00 This journey into the dome has been so cool
20:32:00 because it revealed that something that we thought
20:34:00 was kind of pure and unified and whole and complete
20:38:00 could actually be this fragmented thing
20:40:00 that has many versions.
20:43:00 It’s surprising to think that after all this time,
20:45:00 we’re still not sure how to interpret
20:47:00 one of the oldest models of physics.
20:50:00 Is it that surprising though?
20:52:00 You might be shocked at what kind of questions
20:54:00 you’d think are pretty basic, that still puzzle scientists,
20:58:00 especially to do with something you’d think
21:00:00 we’d have figured out by now, us humans.
21:03:00 Scientists still can’t agree
21:05:00 on things like how we first started speaking.
21:07:00 Was it a slow, gradual process
21:10:00 or was it like a linguistic big bang?
21:12:00 Why do we sweat while other animals don’t?
21:15:00 What were the moments that made us human?
21:18:00 It’s a really fascinating topic,
21:20:00 and my friends over at Real Science
21:21:00 made an entire series about it called “Becoming Human”,
21:25:00 which I highly recommend.
21:26:00 It’s a deep dive into the defining moments of our evolution.
21:30:00 The different theories put forward by scientists.
21:32:00 And it’s one of my favorite things on Nebula.
21:35:00 A platform that’s quickly becoming the go-to space
21:37:00 for the best educational content on the internet.
21:40:00 Nebula is built and run by creators with the mission
21:44:00 to be the best place for us to make the work
21:45:00 that we couldn’t make anywhere else.
21:48:00 I’ve always wanted to take my videos to the next level,
21:51:00 and Nebula is enabling that for our entire list of creators.
21:55:00 Better stories, better production value.
21:57:00 Taking creators to the next level
21:59:00 is what Nebula is all about.
22:00:00 It’s completely ad free.
22:02:00 And most creators post their YouTube videos there
22:05:00 a week early.
22:06:00 To check out “Becoming Human”,
22:08:00 as well as hundreds of other top-notch, thoughtful videos,
22:11:00 completely ad free, sign up with my link to get 40% off.
22:16:00 That’s just $3 a month, or $36 for the entire year.
22:20:00 Seriously for the price of a latte,
22:22:00 you’re getting access to some of the most interesting
22:24:00 and creative content out there.
22:26:00 Go to nebulatv/upandatom
22:30:00 or click the link in the description.
22:32:00 You’ll be supporting this channel,
22:33:00 as well as the entire educational community.
22:36:00 Thank you and goodbye.
22:38:00 (lively techno music)