I wasn't entirely sure where to put this, but I need a concept check.

I've been bending my mind around the concept of 4 spatial dimensions and what it would mean in a gaming environment, after some thought I managed to conceive of 4 spatial dimensions, my friend got lost when I was trying to explain it so I'll try and keep it clean so you get the concept before I pose my question.

A 1D line is a series of possible states of a 0D point from the lines lower limit / start to its upper limit / end

A 2D shape is a series of possible states of a 1D line

A 3D shape a series of possible states of a 2D shape

So by extension, a 4D area is a series of possible states of a 3D shape

so if I were to take a 4D area with the 4th dimension ranging from 0 to 1. am I right in saying that if W is my 4th dimension I may accurately represent the concept of 4 spatial dimensions via Lerping 3D points between a position regarded as W of 0 and a position regarded as W of 1?

if so am I also right in saying a 4th dimension with a range exceeding 1 the 3D points would follow a curve defined by the points positions per W unit?

to my understanding these points would actually represent the shape of 3D space as opposed to the objects within 3D space, meaning an object moving at for example 10mph in W0 may in W1 be moving at 100mph and then in another location 10mph in W1 could be 100mph in W0 via relative stretching and squeezing.

I hope I didn't just confuse myself just to sound like an idiot XD

I figured the idea of 4 dimensions in which to interact would be an interesting mechanic specifically for puzzlers, but I suppose the concept in other genres would allow for interesting 4 Dimensional level design, literately adding a new depth to the game.

regarding the title about interactivity, the idea is just like WASD have you moving in X and Y, I'd imagine Q and E moving you through W, only being able to see the world in 3 dimensions it would just look like things being warped when if my concept is right it would be moving through a 4th dimension.

Thanks for reading,

Any and all informations, examples, documentation on this concept or similar concepts would be greatly appriceated.

Bombshell

**0**

# interactive within 4 spatial dimensions

###
#1
Members - Reputation: **225**

Posted 01 May 2013 - 03:35 PM

###
#2
Moderators - Reputation: **39601**

Posted 01 May 2013 - 04:21 PM

Mathematically it isn't too hard.

We already do it to a lesser extent. The screen is a 2D surface. All 3D games simply project from 3D to 2D.

It is not too difficult to extend that to higher dimensions. I've seen many videos of higher dimension projections, up to 12D, which really just looks like nose to me.

A quick search found this video on youtube that demonstrates it. There are thousands more.

Most people are able to handle a 3D world easily because it mirrors what we are used to. Even so, working with it on computers is a learned skill. Most people can master it pretty easily but some people (especially those with only one function eye) have a difficult time.

You certainly can walk through a 4D world as you described, just be prepared for a difficult learning curve that few will master.

**Edited by frob, 01 May 2013 - 04:23 PM.**

Check out my book, Game Development with Unity, aimed at beginners who want to build fun games fast.

Also check out my personal website at bryanwagstaff.com, where I occasionally write about assorted stuff.

###
#3
Crossbones+ - Reputation: **7006**

Posted 01 May 2013 - 07:52 PM

The difficulty in visualizing a hyperspatial object is partially due to the fact that the increase in dimensional complexity is exponential. A point is simple enough, and a line is not much more complex, but a 2D object can be a circle, a square, a triangle, a scribble, a letter, etc, etc, etc. Likewise, an object in three dimensions has exponentially more complexity than an object in only two. A 2D person couldn't live the way that we live. How does he breathe? There's nowhere for the air to go! If you make a cross-section of him so that he has lungs and such then his digestive system will essentially become a split that goes right through him and he'll fall to pieces. You might potentially make a jellyfish-like creature, but that's about as complex as a flatlander could get unless we consider some serious changes to biological functions. For 3D people these things aren't concerns. Our digestive systems don't cut us in half because the third dimension holds the sides together, and we can cover all the bits up by wrapping them with skin, which is quite convenient.

Now here's a real brain-bender for you.

Time

*is*a fourth spatial dimension. Because of the way we perceive time it's very difficult to reconcile this, but consider the following:

If I draw a box on the ground around a flatlander they're trapped. If I draw a box on the ground around you then you can just step over it. Your third dimension of freedom allows you to bypass the barrier. Now what if I lock you in a room? If you were able to move freely through time then you could simply step into a time when the walls were passable (door is open, or even before the building was built) and just step out of the room.

We consider time differently because we can't move freely in time and because our perception does not allow us to see through time. We have an advantage in that we can remember certain things about the direction we call 'past', and we can make some inferences about the direction that we call 'future', but our inability to see directly into either means that we have to rely on these primitive tools that exist in the current point in time with us.

Now I know you're looking for a different take. Really what you're asking about is an additional dimension that we could interact with in a way more like the way we interact with width, depth, and height. Understand that doing so contiguously would be so disorienting for a 3D being that they would have no means of navigating. The 'laziest' means of doing this would be to have a small number of 3D planes that the person can switch between. A more dedicated means of simulating it would be allowing a person to 'rotate' into a different dimension. This would be obscenely intense in terms of environment creation, and it would extremely disorienting for the player. For instance, we can look at the 'lazy' implementation as being like Zelda's Link to the Past. A contiguous implementation would be one where Link could control how far he is between the light world and dark world with the same amount of granularity as he has in his normal motion. The problem there is that the 'in-between' space between the two places would be a gibberishy fuster-cluck that would probably not even really be navigable.

There's an indie game in production right now that does something a little between. It's called Miegakure (mee-A-gaw-koo-ray). That game works by having several planes similar to the light and dark world, and the player can rotate along an axis that makes the ana/kata dimension replace the dimension of width, so the player can rotate dimensions, then step onto another plane, then rotate back into the 'normal' 3D space of a different plane. The planes end up coming in slices of a few feet across while the player is rotated. This is closer to what it would be like, and probably the limit of what a human can really deal with, but it's not really a 'correct' simulation. In a 'correct' simulation the 'planes' would be infinite just like the width or height of the 3D universe is infinite (theoretically), and the width of each plane would be the Planck length. In short, the world would look entirely like gibberish, and even reliably picking a plane to rotate back into would be impossible.

**Edited by Khatharr, 01 May 2013 - 08:00 PM.**

There are ten kinds of people in this world: those who understand binary and those who don't.

###
#4
Moderators - Reputation: **39601**

Posted 01 May 2013 - 08:27 PM

Not quite.Now here's a real brain-bender for you. Time is a fourth spatial dimension.

Time is orthogonal to space, and it is a valid dimension for representation purposes.

Time can be traveled. Normally we just go in a single dimension, but numerically and mentally nothing prevents us from examining past and current events or considering future events.

A game is certainly able to use in-game time as a dimension that can be traveled.

This can be an incredibly fun mechanic, as seen in games like Braid.

However, that does not make it a

**spatial**dimension.

Time is not a spatial dimension. You cannot move an object in the spatial directions "forward, up, right, and future". That would break most of physics.

Check out my book, Game Development with Unity, aimed at beginners who want to build fun games fast.

Also check out my personal website at bryanwagstaff.com, where I occasionally write about assorted stuff.

###
#5
Crossbones+ - Reputation: **7006**

Posted 01 May 2013 - 10:09 PM

I'd argue that it wouldn't break the math. Only our understanding of it. Consider time for flatlanders. We can make a strip of 8mm film where each cell represents a single instance. If we clip them all out from the strip and lay them on top of one another in order then there's a spatial and even geometrical relationship within flatland. Their time dimension expressed in our 'extra' spatial dimension reveals this.

Not quite.Now here's a real brain-bender for you. Time is a fourth spatial dimension.

Time is orthogonal to space, and it is a valid dimension for representation purposes.

Time can be traveled. Normally we just go in a single dimension, but numerically and mentally nothing prevents us from examining past and current events or considering future events.

A game is certainly able to use in-game time as a dimension that can be traveled.

This can be an incredibly fun mechanic, as seen in games like Braid.

However, that does not make it aspatialdimension.

Time is not a spatial dimension. You cannot move an object in the spatial directions "forward, up, right, and future". That would break most of physics.

The only evidence we have of time being different from space is the way in which we experience time compared to the way in which we experience space. This most likely says more about us than it does about time.

Remember the twin paradox. Theoretically I can move an object forward, up, right, and future. I'd have to be really, really fast in order for it to be even slightly noticable, but it's within the realm of physics. Forward, up, right, and past would be significantly more difficult.

**Edited by Khatharr, 01 May 2013 - 10:12 PM.**

There are ten kinds of people in this world: those who understand binary and those who don't.

###
#6
Members - Reputation: **496**

Posted 02 May 2013 - 10:24 AM

Time is not a spatial dimension. You cannot move an object in the spatial directions "forward, up, right, and future". That would break most of physics.

*A priori*, it makes just as much sense to measure movement with respect to a spatial axis - for example, to say that a particle 'moves' because we happen to observe its x-coordinate changing as we change its y-coordinate.

###
#7
Crossbones+ - Reputation: **7006**

Posted 02 May 2013 - 03:02 PM

Isn't it that we tend to measure time in delta and we tend to measure space in absolute distances? T minus 5 seconds is 5 seconds ago, etc.When you're doing special relativity, however, in certain contexts time gets a negative sign and spatial position doesn't, which suggests to me a fundamental distinction of some sort between time and space.

Ah, you have to give in on one or the other of those. If we're talking about real life then it's not possible to retain a spatial dimension because space itself is moving (expanding). If we're talking theoretical then we have to allow that an object can exist in two places at the same time because of the possibility of time travel.

Note that it's possible to move through time without changing spatial position, but it's not possible to move through space without changing temporal position

There are ten kinds of people in this world: those who understand binary and those who don't.

###
#8
Members - Reputation: **496**

Posted 02 May 2013 - 06:03 PM

Isn't it that we tend to measure time in delta and we tend to measure space in absolute distances? T minus 5 seconds is 5 seconds ago, etc.

We measure space in deltas too - 5 km west of New York, etc. I'm getting at a different distinction, though. Mathematically, special relativity treats time and space slightly differently. For example, the 'pythagorean theorem' in special relativity is delta-s^2 = delta-x^2 + delta-y^2 + delta-z^2 - delta-t^2; note the sign difference between the spatial and temporal terms.

If we're talking about real life then it's not possible to retain a spatial dimension because space itself is moving (expanding).

I don't know much about general relativity, but my understanding is that points in space are preserved, but the distance between them increases. In that case, there's no problem with talking about a spatial position.

If we're talking theoretical then we have to allow that an object can exist in two places at the same time because of the possibility of time travel.

Leaving aside the issue of how this time travel works, I'd say that's not really movement. You have two instances of the same object, but they're not connected by a smooth transition - or rather, they are connected, but their smooth transition goes forward in time, then back in time through a wormhole or similar, not straight from position to position.

###
#9
Crossbones+ - Reputation: **7006**

Posted 02 May 2013 - 09:05 PM

We measure space in deltas too - 5 km west of New York, etc. I'm getting at a different distinction, though. Mathematically, special relativity treats time and space slightly differently. For example, the 'pythagorean theorem' in special relativity is delta-s^2 = delta-x^2 + delta-y^2 + delta-z^2 - delta-t^2; note the sign difference between the spatial and temporal terms.Isn't it that we tend to measure time in delta and we tend to measure space in absolute distances? T minus 5 seconds is 5 seconds ago, etc.

This equation does not emerge from a difference in time as a dimension, but as a difference in the way we treat time. The signage is different there *because* space is being measured in deltas and thus requires an offset. This is not requiring a negation of time, it's relating space to time through a negation. We could just as easily write an equation that determines the volume of a cube with a cut-out section 's' by L*W*H - sL*sW*sH. This is not a measurement of negative length, it's just a length being used in a negative mathematical relation.

I don't know much about general relativity, but my understanding is that points in space are preserved, but the distance between them increases. In that case, there's no problem with talking about a spatial position.If we're talking about real life then it's not possible to retain a spatial dimension because space itself is moving (expanding).

But a point is not a real object. Even subatomic particles occupy more than a Planck length and are thus subject to movement through spatial expansion - at a bare minimum.

Leaving aside the issue of how this time travel works, I'd say that's not really movement. You have two instances of the same object, but they're not connected by a smooth transition - or rather, they are connected, but their smooth transition goes forward in time, then back in time through a wormhole or similar, not straight from position to position.If we're talking theoretical then we have to allow that an object can exist in two places at the same time because of the possibility of time travel.

If you consider the dimension of time as being spatial in nature then the movement is contiguous. That's what I'm saying. If you're talking about moving backwards in time through a wormhole then the movement is still contiguous, but requires at least one additional dimension for the wormhole to function.

There are ten kinds of people in this world: those who understand binary and those who don't.

###
#10
Members - Reputation: **496**

Posted 03 May 2013 - 03:16 AM

This equation does not emerge from a difference in time as a dimension, but as a difference in the way we treat time. The signage is different there because space is being measured in deltas and thus requires an offset. This is not requiring a negation of time, it's relating space to time through a negation. We could just as easily write an equation that determines the volume of a cube with a cut-out section 's' by L*W*H - sL*sW*sH.

But a point is not a real object. Even subatomic particles occupy more than a Planck length and are thus subject to movement through spatial expansion - at a bare minimum.

If you consider the dimension of time as being spatial in nature then the movement is contiguous. That's what I'm saying. If you're talking about moving backwards in time through a wormhole then the movement is still contiguous, but requires at least one additional dimension for the wormhole to function.

###
#11
Members - Reputation: **487**

Posted 04 May 2013 - 03:59 AM

A 1D line is a series of possible states of a 0D point from the lines lower limit / start to its upper limit / end

A 2D shape is a series of possible states of a 1D line

A 3D shape a series of possible states of a 2D shape

So by extension, a 4D area is a series of possible states of a 3D shape

A 1D line is a series of possible states of a 0D point from the lines lower limit / start to its upper limit / end

A 2D shape is a series of possible states of a 1D line

A 3D shape a series of possible states of a 2D shape

So by extension, a 4D area is a series of possible states of a 3D shape

Swept "shapes" perhaps?

(Sphere with no velocity == sphere. Sphere with velocity == capsule? Continuous collision detection?)

###
#12
Crossbones+ - Reputation: **7006**

Posted 04 May 2013 - 04:00 PM

The sign difference is necessary, but what I'm saying is that it's not a negative quantity being added. It's a positive quantity being subtracted. It's like saying that 3 feet to the right is the same as minus 3 feet to the left. There's no 'minus 3 feet', it's just 3 feet in a different direction. Consider what delta_t^2 means there and why it's present. (it's c^2 * delta_t^2, btw) The comparison of dr^2 and c^2dt^2 indicates whether the spacetime interval indicated is spacelike, timelike or lightlike. In other words, is the spacetime interval vector longer in space, longer in time or roughly equal.

That's not really true. The point of delta-s as I defined it above is that it is a 'distance' metric between two points that does not change when you move to a different reference frame. If you try to define the metric without the sign difference, you'll get a 'distance' that changes when you transform coordinates to a reference frame with a different velocity. That sign difference is necessary and fundamental, and shows up throughout special relativity.This equation does not emerge from a difference in time as a dimension, but as a difference in the way we treat time. The signage is different there because space is being measured in deltas and thus requires an offset. This is not requiring a negation of time, it's relating space to time through a negation. We could just as easily write an equation that determines the volume of a cube with a cut-out section 's' by L*W*H - sL*sW*sH.

In theoretical space, perhaps, but if expansion wasn't motion in real life then we'd never have realized that the universe is expanding. Remember that all positions are relative - iow imaginary. Until you can demonstrate a reference frame that is fixed (thus breaking relativity) you can't really say that anything is holding still.

Expansion doesn't imply movement. Even if you have an object that's not a point particle, every point on that object can remain at a fixed position as space expands, so the object gets larger but experiences no change in spatial position.But a point is not a real object. Even subatomic particles occupy more than a Planck length and are thus subject to movement through spatial expansion - at a bare minimum.

"Motion" through x without "motion" through t would appear as length in x. (a spacelike interval because dt = 0)

This is easier to discuss with graphs:If you consider the dimension of time as being spatial in nature then the movement is contiguous. That's what I'm saying. If you're talking about moving backwards in time through a wormhole then the movement is still contiguous, but requires at least one additional dimension for the wormhole to function.

In case A, an object moves through both space and time at less than the speed of light, as normal. In case B, an object moves through space without moving through time; this what I'm saying is impossible. Case C is the wormhole. I agree with you that movement along the red line is continuous. The red line is always moving through time as well, though, so at no point is the object moving 'horizontally', or even necessarily faster than the speed of light.

In retrospect, I was imprecise when I said 'that's not movement' - sorry about that. What I meant was that it's not movement directly through space from instance to instance like the horizontal movement in the diagram.

The problem here is that the common concept of motion depends on a change in position over time. When considering time spatially it's necessary to rotate such that an additional dimension is implied in order to observe the delta. I think we're on the same page with that, though.

Considering this, let's go back to the origin of dimension theory (ostensibly) and talk about the fly in the room. We can describe the fly's position at any point in time as x, y, z. If we elevate the observer by one dimension we can see the whole journey of the fly through the room as a contiguous spacetime object. From that perspective we could describe any point in the journey as x,y,z,t. If we move back down to a three dimensional space, but this time use x as a delta instead of t we still get sane data, but from a different perspective. A computer simulation of this would be interesting. In t,y,z space the fly can occupy more than one point at the same x, or can be a line along the t dimension.

**Edited by Khatharr, 04 May 2013 - 04:16 PM.**

There are ten kinds of people in this world: those who understand binary and those who don't.

###
#13
Members - Reputation: **496**

Posted 08 May 2013 - 11:36 PM

Consider what delta_t^2 means there and why it's present. (it's c^2 * delta_t^2, btw) The comparison of dr^2 and c^2dt^2 indicates whether the spacetime interval indicated is spacelike, timelike or lightlike. In other words, is the spacetime interval vector longer in space, longer in time or roughly equal.

In theoretical space, perhaps, but if expansion wasn't motion in real life then we'd never have realized that the universe is expanding. Remember that all positions are relative - iow imaginary. Until you can demonstrate a reference frame that is fixed (thus breaking relativity) you can't really say that anything is holding still.

Considering this, let's go back to the origin of dimension theory (ostensibly) and talk about the fly in the room. We can describe the fly's position at any point in time as x, y, z. If we elevate the observer by one dimension we can see the whole journey of the fly through the room as a contiguous spacetime object. From that perspective we could describe any point in the journey as x,y,z,t. If we move back down to a three dimensional space, but this time use x as a delta instead of t we still get sane data, but from a different perspective. A computer simulation of this would be interesting. In t,y,z space the fly can occupy more than one point at the same x, or can be a line along the t dimension.

###
#14
Crossbones+ - Reputation: **7006**

Posted 09 May 2013 - 03:34 AM

This is an equation designed to compare space with time, though. If we were comparing xyt against z (using whatever terminology) then I see no reason why it wouldn't work. (In the 'rotated' fly-space, for instance.)It is a comparison, but it's interesting to me that using that comparison is the way you construct an invariant. My argument is that if time and space were truly interchangeable, you'd expect an invariant quantity to have dt show up interchangeably with dx, dy, and dz. If you try to construct the invariant interval that way, though, it doesn't work. (I know about the c^2; it's easier to use natural units, and in this instance it's conceptually simpler for people who are unfamiliar with relativity.)

If there were no difference between theory and reality then there would be no need for theory. This is a tool that we use to understand things, yes. But this indicates that our understanding is not perfect. In this case we're imagining a universe that holds still for us to examine. You can create a reference frame, but what's actually holding still there? You've asserted that something can hold still in space while moving through time, but can you actually show evidence of this ever happening? In order for something to be shown as not moving we'd first have to demonstrate a reference frame thatI can't call anything absolutely stationary, but my argument holds true in any inertial reference frame. I can certanly find an inertial reference frame and determine the position of objects within it without breaking relativity. I don't quite understand what you mean by theory, by the way; theory attempts to describe real life as well as it can. The expansion of the universe is extremely theoretical.

*itself*is not moving. This is against relativity. We can say that two objects are moving in parallel at the same speed, but that only means that they're holding still in reference to one another. It doesn't mean that either of them is not moving in space.

It suggests to me that the fly interacts with the dimension of time the same way that we do. Just like a flatlander can't freely interact with the third dimension, we can't freely interact with the fourth. This does not effect the nature of the third dimension, though. It's simply a trait of the flatlander. However, it's worth considering that we regularly observe objects which are larger than a point. This suggests that 'motion' through space independent of time may simply be so common that we don't notice it. If we consider the fly as a point then it's not apparent, but if we remember that the fly is a collection of particles then even when t is the delta the fly is occupying more than one 'point'.I agree more or less with what you're saying here. Using your setup, my point is that if you use t as a delta, the fly can only be at one location at once, but if you use x, y, or z, the fly can at multiple locations simultaneously - that suggests to me that there's something different about t.

There are ten kinds of people in this world: those who understand binary and those who don't.

###
#15
Members - Reputation: **496**

Posted 11 May 2013 - 12:37 AM

This is an equation designed to compare space with time, though. If we were comparing xyt against z (using whatever terminology) then I see no reason why it wouldn't work. (In the 'rotated' fly-space, for instance.)

You've asserted that something can hold still in space while moving through time, but can you actually show evidence of this ever happening? In order for something to be shown as not moving we'd first have to demonstrate a reference frame that itself is not moving. This is against relativity. We can say that two objects are moving in parallel at the same speed, but that only means that they're holding still in reference to one another. It doesn't mean that either of them is not moving in space.

It suggests to me that the fly interacts with the dimension of time the same way that we do. Just like a flatlander can't freely interact with the third dimension, we can't freely interact with the fourth. This does not effect the nature of the third dimension, though. It's simply a trait of the flatlander. However, it's worth considering that we regularly observe objects which are larger than a point. This suggests that 'motion' through space independent of time may simply be so common that we don't notice it. If we consider the fly as a point then it's not apparent, but if we remember that the fly is a collection of particles then even when t is the delta the fly is occupying more than one 'point'.

**Edited by before-it-was-popular, 11 May 2013 - 12:40 AM.**

###
#16
Crossbones+ - Reputation: **7006**

Posted 13 May 2013 - 07:54 PM

Language fails me at this point, but I can visualize this, both with the fly and with an entity that can move freely in t. I honestly believe that this is a limitation in the ways that this has been examined. The dr2 - dt2 equation is used to determine the spaciness or timeyness of the interval. If we're rotating ourselves in the way described we'd have to start talking about depth versus fruit-saladness, or something. In order to give that a fair shake we'd have to use reference frames that are oriented in the same dimensionality.I'm pretty sure it wouldn't work. dr^2 - dt^2 is an invariant because of the equations for coordinate transformations between different reference frames, which should be the same in rotated fly-space. If you construct a similar equation to compare z-distance with xyt-distance, you'll get a quantity that can change from reference frame to reference frame, regardless of how you view the space it's measured in.

Don't we observe this behavior in QFT, though? Paticles that pop in and out of 3-space? An object that is 'stationary' in time would be encountered thuswise, wouldn't it?I'm not trying to claim that anything is holding still in the absolute sense that you mean - as you seem to agree, movement isn't meaningful in any absolute sense. I am claiming that something can hold still in a particular reference frame - that is, that its spatial coordinates in that reference frame could be constant. That much seems clear to me; in a reference frame attached to my desk, my monitor is stationary.

If I restrict my attention to inertial reference frames - that is, reference frames that aren't rotating, accelerating, or breaking the speed of light - I can then make my claim that objects can be stationary in space but not in time. In short, it doesn't matter that I have no absolute coordinates, because my argument holds true in any set of relative coordinates I choose, as long as those coordinates correspond to an inertial reference frame.

I'm trying clumsily to use the fly as a volumetric particle. I'm mixing up the analogy a bit. Essentially I'm just pointing out that even elementary particles have volume, which could be viewed as a non-orthogonal intersection with 3-space. In other words, if you consider spacetime and draw a line that runs along the t-axis then you'd get a stationary point in 3-space. Since a point has no volume, this is a kind of persistent virtual particle, failing to meet the Planck length, it holds no locality. If you tilt the line so that it's not t-aligned then the point gains volume in 3-space since its intersection begins to become linear and can gain locality. Taking this to its logical extreme, we'd end up with a volumetric object which occupies less than a Planck time, another virtual particle.Talking about the fly being in two spots at once was a little sloppy of me; the more precise thing to say would have been that the fly's movement could satisfy things like dx/dt = 0 or dx/dy = 0, but never things like dt/dx = 0, with dt in the numerator. According to special relativity, that much is true regardless of how the fly interacts with the fourth dimension.

As you say, you can view the fly as a collection of particles; your suggestion, then, is that there's one 'fly' particle, and at any particular time-step it moves through the positions of all the elementary particles in the fly? Even disregarding the issue of multiple elementary particle types, that's not movement in the sense that I was discussing - it's a jump, not a smooth progression.

**Edited by Khatharr, 13 May 2013 - 07:56 PM.**

There are ten kinds of people in this world: those who understand binary and those who don't.