Earlier quoted context omitted.
Wouldn't that be xray vision?
An xray is still a 2D projection, just with different wavelength light.
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4D Toys: a box of four-dimensional toys
131–140 of 148 posts
Re: 4D Toys: a box of four-dimensional toys
#132Earlier quoted context omitted.
That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise. In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little…
>The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise. You might be right, I don't know. Could you explain a bit more how you mean?
Complex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in exactly the same way as scaling & rotation operators on 2-dimensional vectors.)
If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector.
In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.
Re: 4D Toys: a box of four-dimensional toys
#133Re: 4D Toys: a box of four-dimensional toys
#134Re: 4D Toys: a box of four-dimensional toys
#135This is so cool, but its driving me crazy. I was wondering if someone could provide me with more resources that help intuit about 4d space. For example, in Miegekure, he walks through the 4th dimension to get to the other side of the wall, but thats assuming that no part of the wall extends into the forth dimension (aside from rubble). What would happen if he switched back to the normal 3 dimensions in the middle of…
I found this Matt Parker lecture provided some good tools for getting your head around the fact that extra dimensions really do let you go around things, not through: https://youtu.be/1wAaI_6b9JE?t=2299 (the whole thing is worth watching, but the bit about higher dimensional visualization starts there)
I'll save people the effort and point out that the only relevant things (to the given questions) the entertaining video shows are a proper klein bottle and the nature of projection into 3D. In 4D, klein bottles don't self-intersect, just as mobius strips don't self-intersect.
It does not cover the question of putting oneself inside a wall or how to go around 4d objects, and the projections show detail that a 4d person would not see.
Re: 4D Toys: a box of four-dimensional toys
#136This is so cool, but its driving me crazy. I was wondering if someone could provide me with more resources that help intuit about 4d space. For example, in Miegekure, he walks through the 4th dimension to get to the other side of the wall, but thats assuming that no part of the wall extends into the forth dimension (aside from rubble). What would happen if he switched back to the normal 3 dimensions in the middle of…
There are some simplifying assumptions that may help with some of your questions:
> What would happen if he switched back to the normal 3 dimensions in the middle of the wall?
The 4D cell the player is in is clear, no matter how you rotate the world around them. The cross-sectional representation of the world can misleadingly show you things you (and light) can't reach, though.
> In miegekure everything is kind of discretized (grassy area to desert area), but in reality that would be continuous.
It's as discretized as Minecraft, but with an extra dimension. In Minecraft, you can't simply go through things, just as the Miegekure won't let you.
> What would that actually look like, for example, the area right next to the wall?
All common representations of 4D space show you a cross section or x-ray projection, which, as with such things in life, show you internal details you can't see with visible light. Unfortunately, there seem to be no realistic representations available.
> if I were sitting in an easy chair and started looking down the 4th dimension what would happen?
There's no way to answer that without knowing what you're actually sitting in. An easy chair is a 3d object and a 4D you can't sit in it.
> How does gravity work in those 3 dimensions (2 of our spatial dimensions + 1 of the hidden dimension)?
Locally, remember that gravity is pretty much just a constant force in a particular direction. That works in 4D, too. If you're looking for physics answers, the answer is that most such forces would be incredibly weakened if they dissipated in an extra dimension.
> For example, since the three dimensional projection of a hypersphere changes diameter, does that mean the 4d dimensional analogues of earth are just different size earths?
This is answered in the Matt Parker video linked to in another reply. The shape doesn't change diameter - that's just a feature of the physically unrealistic but pretty representation that's most common. It could be called a shadow or projection.
> in one of the miegekure videos the windmill in the new 3d space is like a cross section of the windmill but extending for a distance
Remember that for it to exist in miegekure, it has to be a 4d object. It is one that has been chosen to look like a windmill in a particular cross-section. In other directions, the game designer simply chose how it behaved. It is, in essence, not a windmill, but something carefully made to look like one when looked at in one particular way.
Re: 4D Toys: a box of four-dimensional toys
#137This and Miegekure show cross-sections or projections of a 4d space that are common but that show details that could not possibly be seen by functional 4D eyes. Are there any attempts out there at showing a 3D representation of what 4D eyes could see? (For example, if you have a solid 4D cube, you can only see the outside of it, but the cross-section shows parts of the inside, as happens you take 2D cross-sections of a 3D cube)
Re: 4D Toys: a box of four-dimensional toys
#138I wonder if, after playing many hours with this game, the brain will suddenly "grasp" the concept of 4d.
I think it will. I'm aware of two somewhat related experiments: upside-down vision and magnetic field vision. In first experiment, participant wore vertical flipping glasses for a week. He was walking, eating, cycling and so on, never taking glasses off. At the end of experiment he was pretty good at it and it seemed natural. When glasses were finally off, he felt very discomfortable for some time. https://en.wikiped…
Check the following links for information on such a belt. It's a spin-off from research on "sensory substitution" at a german university (in Osnabrück).
http://www.feelspace.de/navibelt/ http://www.feelspace.de/research/
Re: 4D Toys: a box of four-dimensional toys
#139Earlier quoted context omitted.
Wouldn't that be xray vision?
An xray is still a 2D projection, just with different wavelength light.
It's somewhat like having the alpha on everything turned down to varying levels - you can see through the skin, bones and metal implants are still at 100% alpha - but fundamentally no different than what looking at someone gives you: a 2D plane.
Re: 4D Toys: a box of four-dimensional toys
#140Earlier quoted context omitted.
If you sliced a tesseract then presumably you'd encase it in the knife and a square, with no depth (the depth is in the 4th dimension that we're not experiencing), would appear? Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in…
It might help if you consider spheres as a surface . 3D spheres are a 2D surface wrapped into a 3D space, likewise, hyperspheres would be a 3D surface wrapped into a 4D space. There's no "infinite spheres on its surface", I think the "rotation" is a better analogy. Take a line rotated around an orthogonal axis and you have a circle, a circle rotated around an axis orthogonal to the other two is a sphere, a sphere rot…
Are all spheres hyperspheres? When you rotate in the 4th, or higher ordinal, dimensions don't you get the same shape?