Cosmos - Carl Sagan - 4th Dimension

> how weird large-dimensional space is It's rather crazy that we humans can't really even intuit about a single extra dimension. Or even a single fewer! There's a lot of people who will say that they can visualize things in the 4th dimension but I've yet to find someone who can actually do this. This includes a large number of mathematicians (it's never the mathematicians that claim this...) I really like the animation in this Math Overflow post[0], because it has a lot of hidden complexity that most people don't think about. The animation is actually an illusion, and you are "hallucinating". That top image projecting a cube down onto a plane? Well... that isn't a cube. We've already projected the cube into 2D! Technically this is 3D. But the 3rd dimension isn't a spacial dimension, it is a time dimension. Which itself is a helpful lesson in learning about the abstraction of dimensions! So we hallucinate a cube, rotating, and then see the projected image on a plane, which we hallucinate as a square that isn't skewed but instead has depth. This is all rather wild in of itself. The truth is that we struggle to imagine 2D! And most people will claim to be able to visualize 2D and the claim will go uncontested. If you haven't read Flatland[1], I'd encourage everyone to do so. A lot of people get it wrong. They read it as an analogy 1 dimension down. Where we 3 dimensional creatures are analogous to the 2D creatures and a 4D creature would be as baffling as a 3D creature is to the

"This video might help. ▶ this video " Sagan's video proves that it's always helpful to investigate abstract concepts with an experiment. But the video also raises an important question: Can we derive true conclusions from wrong assumptions? Here the assumption that the "flatlanders" are really flat is wrong. Sagan notices that too and he says that his objects are not really flat, but they have thickness. Now these little cutouts have some little height but let’s ignore that, let’s imagine that these are absolutely flat. So the thickness of his objects are never zero. His flatlanders has a notion of third dimension because they have the third dimension. Their height is not zero. This is the typical rhetorical sophistry widely used in physics. We may also call it equivocation on the word zero. Here, Sagan defined the word zero both as nothing and something and he wants us to go along with this equivocation. You cannot have non-zero height and zero height at the same time. If you have zero height you do not exist. A door is either open or closed. Instead of flatlanders, Sagan may have used shadowlanders. Shadow is closer to two dimensional objects but even shadow has thickness. So, Sagan's assumption that his objects have no height is definitely wrong. Can we arrive at a correct conclusion from this wrong assumption? I guess not. Also it is clear that when we try to transform a 3-D object into a 4-D object, the object's morphology changes. A cube becomes something else. A cube does

In 3 dimensional space, you can’t. Draw a square on paper. The lines will be orthogonal. Draw a cube. The third dimension won’t be orthogonal. It can’t be. But, that doesn’t mean a third dimension doesn’t exist or can’t exist. The same thing happens with a hypercube. The fourth dimension won’t be orthogonal on paper. It won’t be orthogonal in three dimensions (you can build a 3D projection of a hypercube). This doesn’t mean it isn’t real or can’t be real. Whether you think of 4D space as something real we don’t have access to or something imaginary isn’t really too important. It’s just very helpful to realize it won’t have the concreteness of 3D objects in 3D space for you because you don’t have direct access to it. This video might help. ▶ this video

A fun way think about it is pressing on a filled balloon from the outside. Inside the balloon is our whole reality/dimension. Everyone is poking inwards and bending the rubber is what we see and how we interact with each other. But we're completely locked into our current senses. In some ways, it sounds awfully... "spiritual", right? The challenge is there's no point in trying to imagine what existence or thought outside of our reality looks like. Carl Sagan's FlatLand helps illustrate how stuck we really might be. ▶ this video

Good answer, a 4th dimension and the shadow projection analogy is brilliantly exemplified by Carl Sagan in this popular video: ▶ this video

I've been watching this vid on repeat trying to fully comprehend what's happening. This is another vid trying to explain the same thing - ▶ this video I really want to believe this is possible and this is how we'll be able to travel far into space. Like how we have to spend time walking around a very long wall when a bird can just fly over it in a fraction of the time, we wouldn't have to think about traveling light-years because there would something else we can do that's faster than a straight line.

For those trying to wrap their heads around this ... I find this segment by Carl Sagan from the original Cosmos to be pretty well presented on the topic: ▶ this video