I think you are looking for Lebesque measure, wikipage.
Quote: “For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume.”
Wonderful answers all around, but this seems to be the succinct, specific one-word answer: it’s a Lebesgué!
You’d just continue saying ‘volume’, alternatively ‘k-dimensional volume’ or ‘volume of the n-dimensional object’. Like for spheres: https://en.m.wikipedia.org/wiki/Volume_of_an_n-ball
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[1]
I’m going to start calling area “2-dimensional volume”
Only if you also call length “1-dimensional volume”.
What happens if I turn the dimensional volume up to 11?
Well if it’s in 1-dimensional space, then you have a line the length of 11 units.
Well, you could just make 10 higher and make that the highest
“But- but this one goes to 11.”
A popular example of a four-dimensional polytope is the Tesseract, which is just a 4D cube. Four dimensional and beyond polytopes have what is called a hypervolume. This can be calculated by using Lebesgue measure, which is beyond my understanding of mathematics.
Fun fact: four-dimensional analysis is common in the development of modern parallel supercomputing!
Also try the math comms this is more of a math question and also I really really wanna know the answer.
4D ounces
To freedom
is the only chance I have
4D also has duration
Only if time is your fourth dimension. OP is likely asking about a fourth spatial dimension, since that’s much more in keeping with the progession of 1D > 2D > 3D
Ah, I see
Do we not consider time a spatiall dimension? It’s the measure of change, how isn’t that spatial? No time, no change. The volume of 4D is duration, i don’t follow how it could be anything else. Is there another set of dimensions that scales up differently that I’m unaware of? Maybe that considers space-time as emergent or something? Idk. Could you point me a direction, porfavor?
If it was a spatial dimension we wouldn’t have to call it space-time because all of the dimensions would be space. 4D space is extremely difficult to imagine because our brains are built for a 3D world.
Think about a cube for a moment. That’s a 3D object. You could take a 2D slice out of it and imagine that, right? The slice you take out of it might be a square, or a triangle, or a trapezium. It depends on the angle and position of the slice.
Now let’s scale that up to taking a 3D “slice” out of a 4D object. You’re still doing the same thing, taking a lower-dimensional piece of a higher-dimensional object. So we know that a cube is an object in which every side is parallel to another side, all internal angles are 90 degrees, and all sides are the same length. It’s like the 3D equivalent to a 2D square, right? We can generalise that up to 4D even if we can’t picture it. This shape is called a hypercube or a tesseract. Taking a slice out of a tesseract that’s one dimension lower, like we did with taking a 2D slice out of a 3D cube, would give us a 3D object.
As it is, we have no direct evidence of spatial dimensions beyond the third existing. Some theories do propose the existence of them in order to find answers to very difficult problems in physics - string theory, for example - but as far as I know none of them are generally accepted to be likely to be true. It’s a case of “the maths works out if we assume this” but we do not have evidence upon which to assume it.
Temporal dimensions - time, that is - work quite differently to spatial ones. Things don’t seem to be able to move freely in time like they do in space. In fact they can seemingly only go in one direction in time. It’s sometimes useful to consider time as a 4th dimension in the overall space-time because it allows us to do useful maths with it that seems to line up wth real life (particularly with regard to relativity and how moving close to the speed of light in spatial dimensions affects things), but that doesn’t mean that time is space.
The volume of 4D is duration
Duration is only a measure of the scale of something in time. It’s like length, it’s a measure in one dimension. Say you have a 3D shape. You know that all of the internal angles are 90 degrees and all of the sides are parallel to another side. In other words, it’s a cuboid box, right? If you know the volume of this box and you also know two lengths (say, height and width), you can calculate the third length, right? And if your box is 4D and you know the 4D volume and three lengths, you can work out the fourth length. However, if you know the duration and two lengths you have no idea about what the third length is. It could be anything.
“Four-dimensional space” and “tesseract” are probably good wikipedia articles to start with. Not Tessearct the prog metal band though.
Thank you for writing all this out for me.
I’m aware of the hypercube/tesseract, notably that it’s shadow is our 3D object - which can be mindbending to try and wrap your head around.
Intuitively I’ve always understood it as the ‘moment’ is the slice of 4D. If you pause the universe with your magical TV remote, thats a 3D paused picture. Therefore a, let’s say a persons 4dimensional self would look more like a worm from their birth to their death and every slice would just be a moment of their life. Conceptualizing our 4D selves is admittedly beyond me and I don’t really find the drive in myself to attempt really. Maybe I just don’t have the stomach for conceptualizing myself as a worm.
Based of the perceptions that I exist at min in 3 + 1 and my mastery of the dimensions diminishes the high I go. Like throw some lines at me dawg, I got this shit all day. I can drop that linear algebra like a MOTHAFUCKA (💀 In my defense, I am a silly goose). We are incomprehensible to dot lives. Total mastery over it
2D? Graphing? I LOVE that shit! Let’s go! I can pop into and out of existence (explained easily enough in flatland). I am dancing in these parameters, not a god, but skilled enough to necessitate making it artsy to keep my interest, if that makes sense…? Like how Nietzsche couldn’t just write out Zarathustra, he had to make it a parable on parables, like the bible, and then in as much pentameter, cadence and alliteration he could agonize out of himself. Y’know: Prose.
We operate as engineers in 2D, designing reality
3D ok hold on a minute. In 3D we step back. We are field techs. Operators and witnesses. We don’t have the perspective to engineer the 3D, or even the language to relay the absolute knowledge that would be needed. Becoming an engineer of the 3rd dimension would mean we have cracked the teleporter, even if thats just as a very specific atomic 3D printer - the end product is the same. A teleporter (be it actual teleportation or CntlC+CntlNull+CntlV) is the same thing as a time machine. To teleport from where you are to a mile away, the speed of the earth around the sun, the sun’s speed around SagA*, the galaxies speed towards the Great Attractor all have to be quantified just so the coordinates are correct and you don’t materialize 55miles above Earth’s atmosphere with zero momentum - just enough time to see the planet speed away at 250,000mph before your brain shuts off from no oxygen. And coordinates like that necessitate 7 points. 6 to mark Height x length x depth and the 7th for the starting perspective. You know this. Fucking…Stargate.
We need 3 to accurately see 2D. We need for 7 to accurate see 3D. Can we extrapolate it would take 11 for 4D? Or would it be 15?
In specific applications where it is useful to consider time as a 4th spacial dimension.
So if you’re not talking about relativity, it’s probably not.
