Simplex Noise in N (!) Dimensions - (Not) Having fun...

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#21

Er…

Ony your blog you said we can’t even imagine more than five dimensions.
I disagree.

I can imagine any amount of dimensions by simply stacking them into 3 dimensions.

Example:

2D: A square
3D: A cube
4D: A ‘row’ of cubes both separate and together simultaneously
5D: Same as 4D except a ‘stack’ of rows’
6D: A ‘cube’ of cubes that are simultaneously in the space of one cube and infinitely apart at the same time

Etc.

I know it’s not perfect as it still uses 3 dimensions, yet it is understandable as by adding another dimension, everything can still be in the same place as the already existing dimensions, just in a different place for the bew dimension.

In conclusion:

It is possible to think in any number of dimensions as long as your mind works logically and I am so going to make a game with 8 dimensions just to show I can.

If anyone needs further help I will be happy to start a new topic

#22

Imagining spatial dimentions above 3D is not as simple as you make it seem.

You can indeed say that adding 1 dimension duplicates the all previous dimensions infinite times:

[x] A line is an infinite series of stacked points
[x] A plane is an infinite series of stacked lines
[x] A volume is an infinite series of stacked planes
[x] A 4D-thingy is an infinite series of stacked volumes
[x] A 5D-thingy is an infinite series of stacked 4D-thingies
[x] A 6D-thingy is an infinite series of stacked 5D-thingies

Wait what? Thingies…?

Then you can say that by adding a dimension, you can transform a lower dimensional coordinate system without affecting it, like you can draw a curved line on paper, or you can wrap paper over a cilinder, but how do you wrap a volume over that 4D-thingy ? I’m not talking about relativity (time) here, we’re just dealing with spatial dimensions.

The opposite way of looking at it, is projecting dimensions into others (losing one dimension), like we can paint a picture (2D) of a landscape (3D), or look at the shadow (2D) of a structure (3D), or slice through a sphere to see a circle with the radius depending on how we sliced it - it might even be an oval, if we slice it with a curve, but that’s not quite projection anymore. How would we imagine that some 4D shape can be projected into 3D?

And how do you imagine this: for each added dimension you add an axis that is perpendicular to all other axes. I’m lost at imagining 4 spatial dimensions. :emo:

I like to imagine a world with more than 1 time dimension though, but life is too short… or is it?

#23

It is impossible to ‘see’ more than three dimensions, as that is how we are, but you can imagine without ‘seeing’

Time is something we cannot see, but we feel it and we move through it constantly as there is (for now) no friction that affects movement through time like in the normal three dimensions.

You can only ever be in one position per dimension, so if we can only see three dimensions, then anything in a different coordinate to us in another dimension is invisible to us. So imagine if you could be in the same place at the same time, but move through different positions in a fifth dimension. If movement through a fifth dimension was possible, you could eventually, find a ‘world’ no-one else lives, yet everything else is potentially the same.

Just like with time. You can be in the same place as someone else, but be in a different time and you could not see each other

#24

I fully grasp that, I just cannot imagine it.

I’d argue that we can see only two dimensions (the retina is a surface, not a volume). But that’s nitpicking.

0.1mm further into the next spatial dimension, there might be a bunch of cute fluffy bunnies bouncing around in aircastles eating soft candy supplied by a pink drain pipe. We don’t have to travel far, to get entirely change our surroundings, just like in a 2D world, one would exist on 1 piece of paper featuring a bunch of boring words, and 0.1mm further one would exist on a totally different piece of paper showcasing a photo of a rainforest. With no means to travel through that higher dimension, you’re just stuck in these lower dimensions.

By the way, there is no inherent friction in our three spatial dimensions.

#25

I meant friction from other objects.

Time for a new topic: http://www.java-gaming.org/topics/n-dimensional-theory/27709/msg/249113/view.html#msg249113

#26

I’d say that isn’t not nitpicking…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. Just a little more tricky to “understand” the surface :slight_smile:

#27

I’m not so sure about the statement that we “see” in 2D.

The retina may be flat, but there are two of them, and there is a lot of binocular processing that is hardwired in the cortex. Also, our concepts and memories of the objects include 3D.

Even with one eye, people can experience a great degree of depth perception. Apparently there is more than just the binocular aspect that produces this cognitively, but other ‘clues’ as well are hard-wired in to be used spatially.

#28

It certainly ripped a hole in classical set theory, but current set theory says that the statement is simply meaningless. :slight_smile:

#29

woh…derailed me thinks. When did we get to set theory?

#30

Oh, it was just a random sig in the thread that caught my eye. Back to what we were talking about, visual perception, right? :wink:

#31

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:

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

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.

#32

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 :slight_smile: ). 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…

EDIT:
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.
EDIT2:
LINK! http://en.wikipedia.org/wiki/Eşref_Armağan

#33

I consistently said ‘4 spatial dimensionals’, imagining time as the 4th is bloody easy :slight_smile:

#34

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.

#35

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! :slight_smile:

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

#36

“Flatland: A Romance of Many Dimensions”, Edwin Abbott Abbott, 1884