Let's say, hypothetically, I have a four-dimensional gun. Don't ask me how I have it, (or, if applicable, how I'm holding it), and I shoot something, maybe you, an animal, etc. What happens to the target of the four-dimensional bullet?
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Let's say, hypothetically, I have a four-dimensional gun. Don't ask me how I have it, (or, if applicable, how I'm holding it), and I shoot something, maybe you, an animal, etc. What happens to the target of the four-dimensional bullet?
It gets shot?
Its like asking what happens when you cut a 2d piece of paper with 3d scissors. The scissors occupy other spaces than the paper, but the fact that they occupy the same space as the paper *as well* means they can cut it.
The harder question is what happens if you shoot a 4d object with a 3d bullet? Because there's different ways for it to be 3d in a 4d space - just having the matter be confined to 3d doesn't mean that its interactions would be confined to 3d necessarily, so it might have an effective thickness in that 4d dimension due to the interaction scale. Or it might just pass through completely and not interact at all, if the interactions are also bounded to only exist in the same 3d subspace (or if the interaction thickness is significantly smaller than the length scales of the things making up whatever gets shot). E.g. if you have an edge thinner than an electron, can it actually 'cut' anymore, or does it just sail through?
4d in what sense? We exist in 4d spacetime as it is, so every gun is 4d. If the gun and bullet occupy 4 spatial dimensions, you can miss in exciting new ways which will seem utterly baffling to your 3 spatial dimension perceiving eyes and mind. Accuracy becomes nearly impossible if we assume that only the gun is 4d, and you and the target exist only in 3d space. Being unable to perceive the extra dimension, you will find it essentially impossible to figure out the 4d angle you are shooting at, so can easily miss the target even with the muzzle pressed to their head. In this scenario, on the off chance you hit, the effects will be similar to getting shot normally.
If the fourth dimension is temporal, and the gun/bullet can move freely in time, then you can shoot the target last Wednesday. Good luck aiming though, now you have to compute the gun's aim point in time, and while we are fairly proficient at guessing where something will be, I don't think we ate very good at calculating the 4d angle necessarily to aim into the past.
I’m pretty sure you’d be able to make a wound inside the target without effecting the outside of it if you aimed correctly. I think it’s analogous to using a 3D gun to shoot a 2D target printed on a piece of paper.
Very observant. I think you're right. This concept works that way both in 1D and in 2D, that usually means it's true for 3D. But I do think you can only do this from outside of the target's 3D "plane". As in: I can't shoot a hole through only the center of a 2D target if I'm standing in the same 2D plane as it, if I fire from its plane I'm hitting the edges. So if you can see the 4D gunman coming he can't shoot a hole through only your innards. Not without taking a few steps to the 4D side and becoming completely undetectable to 3D people in the process that is.Quote:
Originally Posted by Maat Mons
Not enough data.
If you wanted the target to be "shot" that would be sub-optimal as it would leave a hole the size of the bullet rather than a line the width of the bullet. Maybe for something behind heavy armor or earthworks you'd want an exploding projectile.Quote:
Originally Posted by Maat Mons
Or maybe you're treating cancer and you want to shoot only some tumors.
Same thing that happens when you shoot a 2-dimensional piece of paper with a 3-dimensional bullet. You put a hole in it.Quote:
Originally Posted by TheHalfAasimar
Now, the hole might have an unusual orientation. A 2d paper might view the hole as wholly internal, intersecting none of its edges. In the same way, a 4d bullet, depending on orientation it is shot from, might simply leave a hole through your center, while leaving the skin untouched.
I'm looking to prank this one coroner friend of mine.Quote:
Originally Posted by Quizatzhaderac
Reminds me of a very real technique called gamma knife - it is pretty cool and very useful.Quote:
Originally Posted by Quizatzhaderac
A 3d slice of it's 4d hypervolume would intersect with the target and do what bullets do.
The 4d bullet would also weigh 6 times more than its 3d equivalent, since its hypervolume is composed of 6 volumes. So depending on its momentum and angle it might just knock the target into the 4th dimension. From the targets POV the world would shift weirdly since it only sees a 3d slice of the 4d world. And from an onlooker the target might just disappear.
Another thing to consider is whether the 4d space has some medium in it, like air, you can put regular air in the 6 volumes, they would be stuck in their 3d slices. If that happens then an onlooker would also be able to hear the gunshot.
Just a small nitpick: hypervolume is a completely different quantity than volume (think of the difference between volume and its surface), so there is no direct way to tell the mass of such a bullet (assuming a known 4d shape) if a regular definition of mass would make sense at all here.Quote:
Originally Posted by Mastikator
On the other hand... If we assume mass is a function of volume, it should be 8 times as heavy as a 3D bullet of the same dimensions in the same way that a cube has 6 times as much surface area than a square of the same dimensions. (Right? 8 times, not 6, because the 3D build plan for a hypercube contains 8 cubes?)Quote:
Originally Posted by Radar
It's "hypermass", a function of its hypervolume, that's a bit trickier. We can (presumably) calculate how much it would be* based on extrapolations from lower dimensional properties, we just have no references for what that number would mean.
*If the length of a line is n the surface of a square of that size is n^2, the volume of a cube of that size is n^3, and so the hypervolume of a hypercube of that size should be n^4. The hypermass is the hypervolume times the hyperdensity. Okay, I have no idea how to calculate that.
That would just be the surface volume the same way as a cube has a surface area consisting of 6 squares. Most of the object is in the inside, so that mass would be most likely defined by the hypervolume.Quote:
Originally Posted by Lvl 2 Expert
Calculations are not a problem once the whole situation is well established. There are bigger unknowns here like for example how in general this hyperdimensional bullet interacts with our reality. I'd guess there are many possibilities to set this up with one of the main things would be how to embed our reality in some hyperdimensional model.Quote:
Originally Posted by Lvl 2 Expert
But what makes you think hypervolume has mass? Mass as far as I know is a property of volume. 3D volume. And we know what a 4D object's volume is the same way we know what a 3D object's surface is.Quote:
Originally Posted by Radar
That's what I find one of the most fun things about speculating about more dimensions, the calculations. Because it really is quite simple. We have pretty good ideas about how to make a reasonable model: we extract logical predictions from the patterns appearing in 1D, 2D and 3D objects.Quote:
Calculations are not a problem once the whole situation is well established. There are bigger unknowns here like for example how in general this hyperdimensional bullet interacts with our reality. I'd guess there are many possibilities to set this up with one of the main things would be how to embed our reality in some hyperdimensional model.
A 1D "square-analog" (a line, I mean a line) contains 1 line. If that line is 2 cm long the total length of line is 2 cm.
A 2D square contains 4 lines and one square surface, for a total line length of 8 cm and a total area of 4 cm^2.
A 3D cube contains 12 lines, 6 square surfaces and 1 cube, for a total of 24 cm, 24 cm^2 and 8 cm^3.
A 4d hypercube or tesseract contains 8 cubes, because the pattern for how many things make up the next step is 4, 6, ... It contains 24 faces and 32 lines (there are logical reasons to figure this out, but I blanked and just checked one of those 3D hypercube drawings, people much smarter than me have thought about this). It also contains 1 hypercube, for a total of 64 cm, 96 cm^2, 64 cm^3 and 16 cm^4.
(The Wikipedia article on hypercubes has numbers for up to 10-cubes, for anyone looking to puzzle around with what the pattern is.)
From there we can figure out rules for non-cube objects. The square cube law say that if any 3D object becomes 2 times as long without changing its porperties is gets 4 times as much surface and 8 times as much volume. n, n^2, n^3. You can derive these formulas from a cube but they're true in any 3D object. So that's true for 4D objects too. So we can find out what happens when we make a 4D object twice as large in all directions by just plugging in those numbers we found for a tesseract. It's surface still becomes 4 times as large, it's volume 8 times as large and its hypervolume 16 times as large. It just adds an n^4. Mass is a function on volume, so it gets n^3 times as large when the bullet gets n times larger, as an example of the logic we can use.
Now, of course, getting back to your actual point, the physics are a lot harder, because our physics aren't 4 dimensional. There is a huge amount of articles out there on how time travel or faster than light travel of whatever is possible if we just assume this weird setup with 6 or 9 or whatever spatial dimensions. But the basic logic in this thread looks sound. A 4D gun could create a 3D hole in a person without an entry or exit wound. But it could only do so if fired from outside of our 3D "plane", and the trigger mechanism probably doesn't work with 3D hands, requiring at least some hypervolme or hypermass to operate, the same way a truly 2D object with no volume at all would probably just pass between the matter of one of our guns. That is, if truly 2D objects existed, which they don't because this is a 3D world.
Sorry, ever since me and a classmate figured out how to play 4D, 5D and further Battleship (the idea works with any number of dimensions, but the rules of the game quickly make it less fun to play if you go above 3D) I've been a bit of a rambler on the subject.
I was going to say this tooQuote:
Originally Posted by Maat Mons
What I assume is that the mass is a property of the whole object and not just its surface. Keep in mind that you can also (for example) get many different 3D slices of the 4D object. Whatever the mass or its hyperdimensional equivalent would be, it should be a property of the whole thing and not some slice thereof. Still, a definite answer is not possible before a physical model is established.Quote:
Originally Posted by Lvl 2 Expert
Another argument against using the surface for calculating any properties like mass is the following: take a given volume of some material. You can form it in many different ways with almost arbitrary surface area (larger than a ball of the same volume). Would a change of the surface area change the mass? No, so the surface is not a good indicator for mass. My intuition says that it would also not work well in higher dimensions.
Sounds pretty awesome! Have you ever heard of Miegakure (sadly not available yet as far as I know)? Or 4D Toys from the same developer, which is actually finished and obtainable? There are also some games that are 3D but use non-Euclidean geometry (for example Hyperbolica).Quote:
Originally Posted by Lvl 2 Expert
I think it can be helpful when evaluating a 4D argument to replace each term with its lower dimensional equivalent to see if the analogous situation holds.
“That would just be thesurface volumesurface area the same way as acubesquare has asurface areaperimeter consisting of6 squares4 lines. Most of the object is in the inside, so thatmassheat transfer coefficient would be most likely defined by thehypervolumevolume.
What I assume is that themassheat transfer coefficient is a property of the whole object and not just itssurfaceexterior. Keep in mind that you can also (for example) get many different3D2D slices of the4D3D object. Whatever themassheat transfer coefficient or itshyperdimensionalvolumetric equivalent would be, it should be a property of the whole thing and not some slice thereof. Still, a definite answer is not possible before a physical model is established.
Another argument against using thesurfaceexterior for calculating any properties likemassheat transfer coefficient is the following: take a givenvolumearea of some material. You can form it in many different ways with almost arbitrarysurface areaperimeter (larger than aballcircle of the samevolumearea). Would a change of thesurface areaperimeter change themassheat transfer coefficient? No, so thesurfaceexterior is not a good indicator formassheat transfer coefficient. My intuition says that it would also not work well in higher dimensions.”
At least, I think some of those uses of "surface" were meant to refer to an object's exterior, not a 2D space.
Hah. Was thinking the same exact thing. Very useful for zapping specific things (like tumors or other masses) inside something you dont want to harm (like say a brain).Quote:
Originally Posted by Radar
The snarky answer to the 4d gun/bullet questiion is that the gun, bullet, and target area already 4d objects, if we take time into account. What do you think "leading the target" is about (There's a great Full Metal Jacket quote one could put in here if they wanted).
If we're actually thinking in terms of physical shape (like a hypercube), I think there is something to consider. The 4d gun/bullet would still only interact with the 3d target where they intersect. I guess it depends on whether we're assuming that the target is actually a 4d object, but is only aware of 3 dimensions (or I suppose a third party is only able to percieve it via one 3d perspective maybe?), which is different than the object actually only having 3 dimensions.
It's relevant when considering the effective mass for causing damage though. A higher dimension object can't actually cause damage in a dimension that a "smaller" object does not have. Imagine dropping a box (cube) onto a piece of paper (plane). Only by dropping it across one of the two dimensions that the paper has do we cause any damge to the paper. If the paper is lying flat on a surface, and we drop the box on it (also with its leading side flat to the surface/paper, we do no damage, because the paper has no substance across that dimension to be damaged/compressed/broken/bent/whatever.
The same should happen with a 4d object striking a 3d one. The parts of the 4d object that don't intersect don't cause any damage.
Each cross-section of the hypervolume is a volume with massQuote:
Originally Posted by Lvl 2 Expert
And so the total mass would be the sum of all cross-sections taken from the same angle
There are an infinite number of such cross-sections. And this isn't the sort of infinite sum that converges to a finite number.Quote:
Originally Posted by Bohandas
Uh, its exactly the kind of infinite sum that converges to a finite number. It's the exact same kind of calculation you do in 3d, integrating say over all 2d cross-sections of thickness dr.Quote:
Originally Posted by Maat Mons
Check the dimensional analysis. Mass does not contain length units, density does. So a 3d density and a 4d density are not the same thing, but a 3d mass and 4d mass are.
The summation that underlies the integral to find a volume, as you noted, assigns a thickness. Area times arbitrarily small thickness dx is an arbitrarily small volume. So, you’re adding an infinite number of infinitely small volumes to get a finite volume. That works. The example I was responding to was adding an infinite number of masses that aren’t arbitrarily small. Hence the problem.
I’m well aware that a 4D density is not the same as a 3D density. That is, in fact, the whole point of the post I made. I order for the post I was responding to to make any sense, the each 3D mass would need to be multiplied by a one dimensional density and an arbitrarily small length. This would have made each summand an arbitrarily small 4D hypermass, and the result would have been a finite 4D hypermass.
Basically, an integral needs a dx, a dr, or a dsomething, and the post I was responding to had none.
Surface in the same sense as a circle is a "surface" of a disc. In mathematical terms,Quote:
Originally Posted by Maat Mons
a disc: x^2 + y^2 <= r^2
a circle: x^2 + y^2 = r^2
The mass of an infinitessimal 4d volume element will also be an infinitessimal. The math, at least, works, even if the poster was not explicit about the dx (but in talking about adding up cross-sections rather than going and writing down an integral or sum explicitly, I find that to be implied taking the message of that post in good faith). It's like complaining that someone wrote down an indefinite integral when really what they mean is 'pick the limits appropriately to what you're trying to model'.Quote:
Originally Posted by Maat Mons
As for the physical intuition justifying the math, next paragraph...
To connect this to a physical picture, we have to identify what's the physically real thing when we change the dimensionality. Is the 3d density the physically real thing, so now all 4d objects have an infinite mass? That doesn't make sense even in 3d, because ultimately the density is a summary statistic over a finite volume and infinitessimal mass densities aren't physically real in a literal sense (infinitessimal probability amplitudes on the other hand, maybe).Quote:
I’m well aware that a 4D density is not the same as a 3D density. That is, in fact, the whole point of the post I made. I order for the post I was responding to to make any sense, the each 3D mass would need to be multiplied by a one dimensional density and an arbitrarily small length. This would have made each summand an arbitrarily small 4D hypermass, and the result would have been a finite 4D hypermass.
The natural assumption is to extend the rotational invariance of physics in 3d to 4d, meaning that the distance between particles comprising a material (which is summarized as a density when considering a sufficiently large volume) is the invariant. So that justifies saying that the 1d, 2d, 3d, and 4d densities of materials would have a particular relationship due to the forces between their component pieces only caring about distance at some small enough level, rather than explicitly caring about direction.
Or in terms of dimensional analysis, the thing that gets you from a 3d density to a 4d density is a characteristic length-scale associated with spacing between particles in a material. That is the thinnest a '3d' volume of that material in a 4d space can be in the 4th direction.
I dunno about all these theoretical concepts, but 4d guns implies the existence of 4d roses, which, of course, results in an Appetite 4d Struction, which totally rocks.Quote:
Originally Posted by TheHalfAasimar
I wanted to come up with some witty reply, but I am simply in awe of your pun. *slow clap of utmost respect*Quote:
Originally Posted by Imbalance
Note that all 3d guns fire bullets in approximately a straight 1d line, or more accurately, in a 2d parabola. The 4d gun will do the same.
If you want to hit a 3-dimensional object's interior without penetrating its surface, you don't need a special gun; you just need to take any old gun into the fourth dimension so that the bullet's path only intersects our 3d world at one spot in the object's interior.
This is no different from shooting a hole in a piece of paper without hitting the paper's edge. Just shoot it from outside its plane -- as we do every time we aim at a paper target.