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  Simplex Noise in N (!) Dimensions - (Not) Having fun...  (Read 13581 times)
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Offline Grunnt

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Complex != complicated

« Reply #30 - Posted 2012-11-06 10:52:57 »

I'm not so sure about the statement that we "see" in 2D.

Us computer-monitor-dwellers "see" in 2D allright. So much, even, that apparently some of us think that human eyes perceive reality as a 2D rendering of a 3D model:

we really can only see in 2D dimensions.  And in the same manner for higher dimensions we can "see" a 2D projection of surfaces in that space

Normal people see in environments where they have depth perception and can see along x,y and z (depth) axis so to speak :-)

I highly doubt anyone can really imagine what a 4D or higher-D thingy would look like. It makes me think of Plato's Allegory of the Cave. Only then a cave with people with 3D glasses on, and the outside being 4D.

Offline matheus23

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« Reply #31 - Posted 2012-11-06 15:10:51 »

How the...

Propably I somehow clicked my own thread away, or somehow cleared all my "unread messages"...

What I want to say to imagining more than 3 Dimensions:

Actually you can't... It'd be a very big suprise if you'd be able to.
Imagine a 4D world (yeah... not imagine, but think of Smiley ). This world has a sun. The shadows the sun casts of 4D objects are 3D. How would you think of 4D objects, a sun and a 3D shadow being cast.

Also we "live in" a 3D world. For us it is 3-dimensional, but for scientists, there are theories where the world actually has 11 Dimensions! (M-Theory on en.wikipedia)
They based this theory on the fact, that gravity somehow behaves strange, when you assumed that the world is only 3-dimensional.

Also, one could say, we ARE able to think of a 4D world, when you see the time as one dimension of them.

I actually agree, because you ARE able to think of an animation, which changes it's object's position over time, which is 3-dimensional.

But what I PERSONALLY think about that is: saying time is one of [insert number]-Dimensions, is like projecting a [inserted number-1]-dimensional projection of the room.
It's what I did with my 3-dimensional noise. Our monitor is only capable of showing 2 dimensions: x and y. But how did I manage to show you a 3-dimensional noise? I projected it in time, and now it animates smoothly through all the "layers" of the 3-dimensional world.
You are able to animate through all the layers of "our 4D world" (in case you assume time is one dimension of them), since you only have to "deal with" 3 Dimesions "at a time".

What the good thing about assuming time is one of those 4 Dimensions we can imagine is: Just like one is able to bend the 3 Dimensions of room, one is also able to bend the 4th dimension: the time.
This allows Einstein's theory of relativity, because you need to bend the time, since light is not allowed to move faster or slower than c, the speed of light...

Thinking of 3 Dimensions is hardcoded in our brains. There is a guy, who is able to draw projectional, even given the fact that he is blind since he was born.

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Offline Riven

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Hand over your head.

« Reply #32 - Posted 2012-11-06 16:26:27 »

I consistently said '4 spatial dimensionals', imagining time as the 4th is bloody easy Smiley

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Offline Roquen

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« Reply #33 - Posted 2012-11-07 16:44:50 »

We only see in 2D.  Locally our environment is in 3 spatial dimensions, which mathematically becomes (minimally) a 4D projective space when we consider how light reaches each of our eyes.  The fact that we have two of these sensors aids us in decoding spatial relations, but it's easily tricked.

Charlie:  Look grandpa!  He's getting bigger!
Grandpa Joe:  No! The room is getting smaller!

Science museums, fun-fairs and amusement parks commonly have contrived 3D scenes which trick our perceptions.  Is that a cube or a hexahedron? In 2D Escher and Penrose, for instance, have images which give false "cues" to create physically impossible images.  As an aside there's some interesting work by an English artist who does 3D paintings, such that the image subtlety changes as you move parallel to it.

Consider a child's ball and a golf ball.  Two similar objects, although one has dimples and the other is smooth.  Why?  Because from any angle they have a similar 2D projected profile.  Now consider a solid cube, cone, plate and a cup without a handle.  We don't consider these to be very similar because they very different 2D projections...but they are all locally the same.  Likewise for a torus and a coffee cup, etc. etc.  Not that I'm claiming that seeing in 3D would be to see objects with the same topology as being very similar...I can't know, nor am I clever enough to conceptualize what it might mean to see in 3 dimensions.

WRT: dimensions.  Talking about dimensions is always tricky.  Especially when it's sometime useful to consider the same "thing" differently.  Simple example: It can be interesting to think of a quaternion as being a point in 4 spatial dimensions at times, and at others as 3D bivector in conjunction with an associated scalar.  The "standard" model of the 2D Eulicean plane is a 4D vector space.  Standard model of 3D is an 8D vector space.  So, like I say, dimensions are tricky to talk about without being very specific.

What does this have to do with anything?  Don't easily fall into the trap of thinking something is impossible untell you've really thought it though.  Otherwise you're just creating potentially false barriers for yourself.
Offline philfrei
« Reply #34 - Posted 2012-11-07 17:50:38 »

Well, it is an interesting topic (seeing in 2D vs 3D) and it is somewhat off topic. And many of the disagreements boil down to one's perspective. Are we looking at this from a reductionist point of view, or something more phenomenological?

Curious, without intending it, I used a figure of speech based upon 3D experience.

i will persist in saying that our brains are built to process visual info as 3D, and that the wiring is there from a very low level to bring out useful 3D cues from the flat projections on the inner wall of the eye, and that our experience of vision and seeing is that the world is 3D. And Darwinian evolution favors the ability to do it well, though the ability is not perfect.

The inherent tendency to create 3D is why techniques like using shaders are able to contribute to creating the illusion that a flat monitor is showing something with depth.

Thinking about the world as 2D takes a lot of imagination as well. I'm not sure that degree of flatness is truly comprehensible.

But in whatever D, we use our imaginations to get a foothold, and the degree to which the image or construct behaves in a way that is consistent with this space will vary in quality from person to person. Any mental image anyone has of the world is going to have some degree of incompleteness.

Whoo-hoo land!  Smiley

If anyone wishes to tussle further on this topic, perhaps we should take it to a back alley? (Another 3D figure of speech. Dang.)

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Offline Roquen

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« Reply #35 - Posted 2012-11-07 20:29:08 »

"Flatland: A Romance of Many Dimensions", Edwin Abbott Abbott, 1884
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