
View Full Version : 4d object visualization ..
planaria 031404, 02:57 AM honestly i dont understand people and their inability to see 4d objects
.. a ball that moves through time can be described as a string ..
what is actually hard is visualizing COMPLEX 4dimensional shapes.. (try to visualize every detail of 2 galaxies colliding .. 5dimensional cubes are also easy it is simply a cube that move both ways in time at the same time.. so in a sense the cube can stretch through time as well as move through it ..
the best definiton of dimensionality that i have heard is simply giving something a new way to move.. although i would add that things like color , meaning , taste , conductivity, etc, can all add a form of "dimension" to an object.
the very definition of dimensions is whats really confusing to me .. because a dimension can be so many different things ..
what do you classify dimensions as ..
honestly i dont understand people and their inability to see 4d objects
.. a ball that moves through time can be described as a string ..
what is actually hard is visualizing COMPLEX 4dimensional shapes.. (try to visualize every detail of 2 galaxies colliding .. 5dimensional cubes are also easy it is simply a cube that move both ways in time at the same time.. so in a sense the cube can stretch through time as well as move through it ..
the best definiton of dimensionality that i have heard is simply giving something a new way to move.. although i would add that things like color , meaning , taste , conductivity, etc, can all add a form of "dimension" to an object.
the very definition of dimensions is whats really confusing to me .. because a dimension can be so many different things ..
what do you classify dimensions as ..
I neither do not understand the people that have have difficulty to visualize a 11 dimensionnal space. This is very easy, you visualize a space in N dimensions and take N = 11. :D
I try to explain multidimensional models to this scienceminded friend of mine and I am absolutely baffled at his inability to even visualize 3 dimensions from a 2D drawing let alone 4 or 5 dimensions. Is it difficult to visualize 11 dimensions? Yes, of course, we don't need 1100f's sarcasm to point that out. I cannot visualize 11 dimensions.
Come to think of it I cant even visualize 4 dimensions. The best I can do is visualize a 3D representation of a 4D object, or at best draw a 2D model of a 3D representation of a 4D object, but some people can't even do that!!!
A 4D cube can easily be visualized. Picture a cube. It has 8 corners so label them in your head A1, B1, C1, D1, E1, F1, G1, and H1. Then picture a smaller cube floating at the center of the larger cube. Label this smaller cube's corners A2, B2, ... H2 in correspondence with A1, ... H1. Now, finally, draw a line connecting A1 to A2, a line connecting B1 to B2 and so on. Once this is done you have successfully made a 3D model of a 4D cube. The only thing is that in 4 actual dimensions, ALL the angles of the object you just created are 90 degree angles (yes even the angles created by the lines connecting the two cubes' respective corners).
I once thought that the only explanation for existence (not just spacetime) must be an infinite dimensional system. Basically a dimension for every dichotomy in nature, be it physical or conceptual (up/down, sacred/profane, hard/soft, subjective/objective, etc) But then I figured that I was taking the notion of dimensions a bit too liberally.
You asked what a dimension actually is and I think the physical definition of it has alot to do with coordinates. Classical thinkers where sure that there were only 3 dimensions because you never needed more than 3 coordinates to precisely specify a location in space. Time was introduced as a fourth dimension cuz we needed a 4th coordinate to specify the location of an "event" (not just an object). I personally believe in 10ish dimensions, but not in the string theory sense. I think the extra dimensions are extended, not curled up. Why don't we see them? I am inclined to think that this is a result of our binocular vision and not some feature of the universe as a whole. Why can't we move through them? We are actually moving through them all the time (me thinks).
John Connellan 031404, 04:37 PM honestly i dont understand people and their inability to see 4d objects
Oh yeah, thats because it is impossible.
.. a ball that moves through time can be described as a string ..
A string is a 3D object
what is actually hard is visualizing COMPLEX 4dimensional shapes..
exactly
the best definiton of dimensionality that i have heard is simply giving something a new way to move.. although i would add that things like color , meaning , taste , conductivity, etc, can all add a form of "dimension" to an object.
what do you classify dimensions as ..
No, I've always thought of dimensions being numbers solely describing the position of an object in spacetime. It has nothing to do with properties of objects (like colour) but I might be wrong here.
planaria 031404, 07:12 PM The only thing is that in 4 actual dimensions, ALL the angles of the object you just created are 90 degree angles (yes even the angles created by the lines connecting the two cubes' respective corners).
mathematically yes.. but visually no .. for instance what i mean by visualizing a 4d object as a string i meant more as a visual tracer .. for instance when a bright light pases infront of your eyes you see the path that it took. you are in essence seeing a visualization of a 4 dimensional object.
percieving things mathematically and perceiving them visually do not always lead to the same results.
i wonder what it would be like feel a 4dimensional object .. or to smell a 4dimensional smell that would be interesting ... i wonder if we feel and smell these already and dont even realize it ?
Rappaccini 031404, 08:48 PM Smelling fourdimensional smells?
This doesn't seem like real physics or mathematics to me.
John Connellan 031504, 05:08 AM No as I have said above, smell is a property of an object not a dimension the object is in. Another way of differentiating between these two is that dimensions will apply to the universe as a whole whereas properties will not.
planaria 031504, 02:24 PM actually smell is more a form of measurment imo . but this is off topic ..
but to put this to rest i think it depends on what you concentrate on for visualization.. if you are a hardcore mathematician it will be impossible for you to visualize a 4d object .. because you do not think visually you think numerically ..
although im sure there are exceptions to this rule .. but by and large lots of mathematicians seem to have this.
I wonder what it would take to see in 4 diminsions?
Math helps plot out 4d objects in a 3d graph helping us view a 4d object.
http://research.microsoft.com/~hollasch/thesis/chapter4.html
A smell or sense of touch of a 4d object would be the similiar as a being in a flatland (2d plane) touching a cross section of a sphere (a circle). He gets an accurate discription of a sphere but he has to wait until the sphere crosses through his plane. while the sphere is in his plane he recognizes it only as a circle. In his case he senses an x and y axis but not a z axis. No matter what this 2d guy does he is unable to view the height of a sphere. Similiarly in a 3d view the only way we can see a 4d object is by 3d slices of the object as the objects moves through our 3 dimensions. We are limited to sensing only a 3d slice of the object at a time.
planaria 031504, 08:11 PM which is why tracers from light shining/moving at your eyes is a perfect example.. its not slices its a blur
as for smelling and touching ...i think smelling a 4dimensional object would be smelling a 'blur' of all the smells that it could emit through time.
touching is a similar idea..
imagine being able to feel a big rock while at the same time simultaneously being able to feel it in rubble dust reforming into another rock etc..
it would be very confusing to be sure ..
I wonder what it would take to see in 4 diminsions?
Math helps plot out 4d objects in a 3d graph helping us view a 4d object.
http://research.microsoft.com/~hollasch/thesis/chapter4.html
A smell or sense of touch of a 4d object would be the similiar as a being in a flatland (2d plane) touching a cross section of a sphere (a circle). He gets an accurate discription of a sphere but he has to wait until the sphere crosses through his plane. while the sphere is in his plane he recognizes it only as a circle. In his case he senses an x and y axis but not a z axis. No matter what this 2d guy does he is unable to view the height of a sphere. Similiarly in a 3d view the only way we can see a 4d object is by 3d slices of the object as the objects moves through our 3 dimensions. We are limited to sensing only a 3d slice of the object at a time.
No its a slice we dont get a blur of that information we get a slice of it.
If time is the 4d that slice is now.
John Connellan 031604, 05:18 AM actually smell is more a form of measurment imo . but this is off topic
I was talking about the other form of the word 'spell' which is the scent of an object. That is a property. U are talking about the verb 'to smell' i guess? That is indeed a measurement (or 'sense') of the surroundings. I can't see why a 4d object would 'smell' any different to a 3d object with the same scent?!
planaria 031604, 12:22 PM I was talking about the other form of the word 'spell' which is the scent of an object. That is a property. U are talking about the verb 'to smell' i guess? That is indeed a measurement (or 'sense') of the surroundings. I can't see why a 4d object would 'smell' any different to a 3d object with the same scent?!
ok lets say there is a 4d tomatoe.. the reason that it would smell different is because we run into 4d tomatoes all the time.. a tomatoe in 3 space dimensions and 1 time dimension grows, ripens , rots, so you would smell all these smells at the same time.. since the 4d object is a sum of a 3d tomatoe through time. i guess a 4d tomatoe can also be an interval of time of a tomatoe too .. so you would smell all the possible smells of that the tomatoe can emit through that interval.
i guess you are right about emitting smell and smelling.. but if there is no observer , then was there ever a smell :)
i am very interested in this subject, and spend countless hours trying to explain the fourth dimension to my friends and even people on here.
could someone try and explain to me how the 5th, 6th, and 7th ... etc dimensions work? A person once told me some mathematical explanation of like 10 to the negative 33 thousand or some odd number, but this made no sense to me really, i guess i am more of a visual person.
I can understand up to the 4th dimension, Time. and i have made several theories regarding Time, most of them based around my assumption that there are many "layers" of time that overlap one another, to create the "dimension" of time.
But my mind stops there, and i cannot figure out what they mean when they say theres up to 11 dimensions? Although i do not doubt it for a second, I just cannot comprehend it at this time... can someone shed some light on this?
on the 4d object visualization i think it boils down to this:
 you either CAN or CANNOT visualize it. plain and simple.
some people just dont have the visual or mathematical mind power to envision more than what they see in front of them. o well, their loss right?!
honestly i dont understand people and their inability to see 4d objects
I guess you mean 'visualize', rather than see.
It's a cognitive limitation... simple 3D visualization is instinctive, requiring no abstract thought. Increasing complexity requires more and more abstract thought.
All people have a limit, even in 3D (or 2D). Have you ever tried to solve a complex 2D puzzle like this one (http://www.ex.ac.uk/cimt/puzzles/slideblk/century.htm) in your head?
With nontrivial 4D objects, it's much harder.
Try this, for example:
How many different 4D shapes can be constructed with exactly 4 hypercubes?
How many of those shapes are reflections of another?
planaria 031604, 07:37 PM i have a qeustion
lets assume : a 4d cube is a 3d cube that can move in 1 time dimension,, but does that mean the cube can move in any direction or can it only move in a straight line .. ? ?
The assumption is false, unless you can equate time units with length units, and the 3D cube exists motionless for a length of time equal to the width of the cube. Before an after the time segment, the 3D cube must not exist.
Throwing in relativity, in which "motionless" is only meaningful for a specified frame, complicates things. I think you'd need to use Minkowski space.
I'm not sure how a 4D cube in Minkowski space would appear... I don't know if the concept is even meaningful.
planaria 031704, 02:48 AM what it doesnt exist ? 3d cubes move in time all the time..
i guess my next question would be , where do you seperate time and space from reality in order to do this .. because in reality a 3d cube with 1 time dimension can move in any direction it wants .. (except back and forth in time at the same time i suppose) .. so i guess to answer that riddle about how many different 4d shapes can be constructed with exactly 4 hypercubes is a hugely large number .. since in essence looking for away at a 4d hypercube it could look like a vibrating string or a string in a loop .. or whatever shape .. personally i equate motion with time.. so that can create a lot of shapes .. for more than i have the memory or experience for.
that 2d puzzle is easy if you cheat just rotate the whole thing 90 degrees :) ( i would need to play a lot of rubix cube to get the skill to figure that 2d problem out)
what it doesnt exist ? 3d cubes move in time all the time..
Making it a long 4D prism, not a 4D cube.
i guess my next question would be , where do you seperate time and space from reality in order to do this .. because in reality a 3d cube with 1 time dimension can move in any direction it wants .. (except back and forth in time at the same time i suppose) ..
Making complex 4D shapes. Thinking about time as a fourth dimension is a bit restrictive in this discussion. Just forget time, and think about a 4th spatial dimension instead.
so i guess to answer that riddle about how many different 4d shapes can be constructed with exactly 4 hypercubes is a hugely large number .. since in essence looking for away at a 4d hypercube it could look like a vibrating string or a string in a loop .. or whatever shape .. personally i equate motion with time.. so that can create a lot of shapes .. for more than i have the memory or experience for.
No, a hypercube is a very specific shape. It's not flexible at all.
A cube has six square faces, a hypercube is a has eight cubic blocks (a word I chose to describe the next step up from a face. ie vertex, edge, face, block).
A cube has 6 faces, 12 edges, 8 vertices.
A hypercube has 8 blocks, 24 faces, 32 edges, and 16 vertices
On a cube, each square face is surrounded by four other square faces on planes at right angles to the first face.
On a hypercube, each cubic block is surrounded by six other cubic blocks in spaces at right angles to the first block.
Now do you see why visualizing 4D is hard?
planaria 031804, 02:38 AM uhm actually, there really very easy to visualize.. with practice.
but i understand more now why people have such a hard time with them
Now do you see why visualizing 4D is hard?[/QUOTE]
Blindman 031804, 05:37 AM uhm actually, there really very easy to visualize.. with practice.
Try visualizing a spinning sphere in the 4th D.. Way to easy with your talent, it looks like a sphere.
My camera set to long exposer sees your 4th D with ease.
Do you include perspective in your view of the 4th D. Should future time not be smaller even out of focus.
Then again maybe you have a very unique ability to actually see the 3d projection of a 4D object. One could learn this skill with computer aided visualizations but you are still restricted by the projection matrix.
How do you define a 4D lens?
Mathematics and psychology don't mix well.
uhm actually, there really very easy to visualize.. with practice.
So, can you answer the riddle about how many different 4d shapes can be constructed with exactly 4 hypercubes? It's not a large number...
planaria 031904, 04:35 AM So, can you answer the riddle about how many different 4d shapes can be constructed with exactly 4 hypercubes? It's not a large number...
well this riddle is a bit open ended .. for instance can the shapes be redundant shapes but different in terms of direction? do the faces have to actually touch ? lets say we make sure faces have to touch that still leaves us a cube that is in essense 4 cubes .. each cube has 6 faces thats 24 possible faces each x 4 is almost 100 possible combinations (although this is also reduced if you take out redundant shapes due to direction) .. thats alot of shapes to visualize ... and overly tedious
i gave up drawing them at about 20 something .. and im not a math wiz just a visualization person .. so yes i give up on the tedium .
planaria 031904, 04:52 AM Do you include perspective in your view of the 4th D. Should future time not be smaller even out of focus.
well it all depends on where you are looking ,
Then again maybe you have a very unique ability to actually see the 3d projection of a 4D object. One could learn this skill with computer aided visualizations but you are still restricted by the projection matrix. how am i restricted by the projection matrix ? im very curious about this.
How do you define a 4D lens?
well there are a couple ways to define a 4d lens .. for instance people are 4d lenses .. because they see objects 3dimensionally and store this information in memory .. so you can for instance see the progression of a 3dimensional object as you live.. a building being built and being demolished.. you can as a 4d lens go back and forth through your memory seeing individual slices of this 4d object.
a 4d camera . would be maybe a stereoscopic camera , with a very long exposure time.. something that would allow someone to see 3 dimensions and 1 time.
you could also have a 4d lens which does a a spheroidal laser scan of its surroundings and stores each as a 3dimensional frame in a hardrive or something.. and this way you could make a holographic movie. .
i think a 5dimensional (2 time 3 dimensions) lens would be neat.. one that would allow you to see a multiverses floating around and each universe that you could see with the lens would have multiple histories connected to it .. because the objects are 3 dimensional and have 2 dimensions you would be able to see a sortof .. wave or topography of the multiple histories of a possible universe.. so universes would be split up into seperate things and the alternate histories of each universe would only interconnect between itself.
and interestingly also you would be able to tell which universes were 5 dimensional and which were 4 simply by looking at how fat they are.
Mathematics and psychology don't mix well.
well they do a little bit.. for instance creating computer graphics requires a little bit of math and a little bit of psychology
well this riddle is a bit open ended .. for instance can the shapes be redundant shapes but different in terms of direction? do the faces have to actually touch ? lets say we make sure faces have to touch that still leaves us a cube that is in essense 4 cubes .. each cube has 6 faces thats 24 possible faces each x 4 is almost 100 possible combinations (although this is also reduced if you take out redundant shapes due to direction) .. thats alot of shapes to visualize ... and overly tedious
i gave up drawing them at about 20 something .. and im not a math wiz just a visualization person .. so yes i give up on the tedium .
Sorry, I should have been more specific.
Each shape is made of 4 hypercubes, attached squarely together  every hypercube must have at least one block (six faces) coincident with a block of another hypercube in the shape.
I hadn't thought of only connecting faces  that would be unstable, like connecting cubes at the edges.
If a shape can be rotated to look like another shape, then those two shapes are considered to be the same. If a shape can be reflected but not rotated to look like another shape, then those two shapes are considered to be different.
There are less than 20.
You still seem hung up on time as the 4th dimension... are you aware that when people talk about the difficulty of visualizing 4D objects, they're talking about 4 spatial dimensions?
planaria 032004, 10:54 PM well assuming what you meant by six faces be coincident with eachother meaning that one of the possible cubes in a hypercube are in the same position as another hypercubes cube.. then i would say 7 because your sortof limiting the possibilities that the extra dimensions allow into essentially a 3 dimensional space..
i dont see why your so hung up on thinking of time and space as seperate things ..
in my mind atleast in this context they are virtually the same thing ..
also to answer an old question how to measure time in a 4d object with 1 time dimension .. it would be like any current method of measurment .. that is to say it will completely depend on how fine you want to cut time up into artificial pieces.. in essence there is no true way to measure time because it cannot be quantitized.. . its like pi you will never find a pattern to quantitize it .. time is infinitely smooth .. just like a perfect circle is infinitely circular.
well assuming what you meant by six faces be coincident with eachother meaning that one of the possible cubes in a hypercube are in the same position as another hypercubes cube..
then i would say 7 because your sortof limiting the possibilities that the extra dimensions allow into essentially a 3 dimensional space..
Very good.
Interestingly, the number of 4D shapes that can be made from exactly four hypercubes is one less than the number of 3D shapes that can be made from exactly four cubes.
What about five hypercubes?
The number of 4D shapes that can be made from exactly five hypercubes (same restrictions as above) is much more than the number of 3D shapes that can be made from exactly five cubes (27).
Why? What's different in this situation? Is this situation still readily visualizable?
in my mind atleast in this context they are virtually the same thing ..
Which leads to confusion, unless you're comfortable with Minkowski Space.
For example:
what it doesnt exist ? 3d cubes move in time all the time..
i guess my next question would be , where do you seperate time and space from reality in order to do this .. because in reality a 3d cube with 1 time dimension can move in any direction it wants .. (except back and forth in time at the same time i suppose) .. so i guess to answer that riddle about how many different 4d shapes can be constructed with exactly 4 hypercubes is a hugely large number .. since in essence looking for away at a 4d hypercube it could look like a vibrating string or a string in a loop .. or whatever shape ..
This is an example of confusion caused by trying to treat time as a spatial dimension. Unless you're comfortable with things blinking in and out of existence, changing shape and position as they do so, time isn't a good thing to use for 4D thinking. Particularly as 4D becomes more interesting when thinking about a 4D object in motion (particularly rotation), which can't be done if time is your 4th dimension.
synergy 032504, 03:44 PM okay, i know this may have come up in numerous conversations over the years, but WHAT IF the universe is a 4D sphere, where the big bang is the 4D center and the present is at a certain 4D radius and the future is at a larger 4D radius. Then travelling toward the 4D center is moving back in time, and freezing the 4D expansion of space actually halts time. Just a thought. :cool:
That would make it a 4D cone (or maybe a shuttlecock sort of shape??) rather than a 4D sphere.
invert_nexus 051304, 05:39 AM ...assuming what you meant by six faces be coincident with eachother meaning that one of the possible cubes in a hypercube are in the same position as another hypercubes cube...
Is this how 4d objects stack? I'm reading this to mean that one of the 3d cubes of one hypercube overlap a 3d cube from another hypercube. That's an odd concept.
I suppose you could say when you stack regular cubes that a 2d square from one cube overlaps a 2d square from another cube. But there is actually a space between them, even if extremely small. How is this represented in 4d? Do you have a link to a rough visualization of this?
Peter2003 051404, 02:15 PM According to Savov's theory of interaction 3D is the smallest number of dimension in which the allbuilding interaction remains always finite and so exists.
cephas1012 051404, 11:00 PM Here is a fun idea.
Imagine you can move through time at any rate you want fowards or backwards. But restrict your motion through one of the spacial dimensions so that you can only move foward in it. Speeds are then measured by the amount of distance in this spacial dimension you have passed through. For example suppose I went backwards in time at a rate of 20 seconds per meter. I would be seeing different cross sections of the universe at different times.
What's the point? Well I was thinking about the arrow of time and thought maybe I could understand it better if I switched which dimension we were passing through. I thought it kind of fit here since this is about visualization and all.
I was going to make equations to transfrom moving objects like this as well. I'll get to work on that, then maybep people will understand what I am talking about.
Fallen Angel 051404, 11:39 PM also to answer an old question how to measure time in a 4d object with 1 time dimension .. it would be like any current method of measurment .. that is to say it will completely depend on how fine you want to cut time up into artificial pieces.. in essence there is no true way to measure time because it cannot be quantitized.. . its like pi you will never find a pattern to quantitize it .. time is infinitely smooth .. just like a perfect circle is infinitely circular.
From what I remember in my physics readings, there's a theory out there that mentions quanta of time. So time is not necessarily infinitely smooth. And I really wouldn't doubt that there exist quanta of time, if you think of the universe as a step function, stepping from one state to the next, it is easy to visualize quantum, or the shortest time possible.
chaoticorder 060604, 04:02 AM well to make see that dimension is actually more than what you are perceiving .it was Lebesgue who gave dimension it's modern definition. in essence he defined dimension as a continuous function. remember the general concept of dimension does not have specific names for each dimension. so when you talk about 4D forget about words like TIME and space and distance. as humans our eyes are very primitive, they only see 2 dimensions, the 3rd dimesion is a learned behavior, and to learn 4th dimension requires that you forget about the concept of right and left which is a very difficult for humans to do and after hard work it usually can be done in very short intervals. why is that the case, well you need to imagine your body floating in space and the moment you achieve that then you tend to try to hold on to something because you think you are going to fall so are back to 3D i mmediately. now when you try to a 4D object crossing your 3D world is like adding of cross sections to get the full image unfortunately if your brain does not have that image already stored in your brain you see nothing. have you ever experience turning your head because you saw or you felt a shadow that is the crossing of a 4D object that your brain was able to recognize. although we live in 3D we really have movement in 2D. so goto your dreams to experience how we really move in 3D
so do you feel the amount of fredom and movement you have now imagine if were able to move on to the 4D. in 2D to see your entire home you will have to walk through every room to see it all.
in 3D you can fly through every room to see it all , in 4D you do not have to move at all to see it all. to really see 4D objects it requires that your brain add up an infinite number of cross sections very fast and very quickly without any hesitation, a task almost impossible for humans to do. i hope this helps you.
chaoticorder 060604, 04:41 AM and extra note.
the 4D can be interpredted in many particular types of problems
1) the Euclidean way (a hypercube, hypersphere) in other words your regular geometic extensions of the circle , the square, and so on.
2) as the coordinate system for imaginary equations such as (X+ i)2=0, so you let x=real, y=i, z=real and w=i
3) x=lenght, y=widht, z=depth and d=time
4)as defined by hamilton, he define the properties that would make a 4D system be a field . in other words it has an identity, it has inverses except for 0., the associative property, the distributibe property and ..... property hold. and so on . but be careful here the definition of inverses is not like in the real numbers (it's more like a hyperbolic coordinate system)
so you see that these are specific usage of a 4D system, but trying to see 4D objects in general has nothing to do with these specific definitions of the 4tth dimension in general.
a good way see the fourth dimension is to draw a cube in 3D but instead of using dotted lines to connect the sides that you do not see, use solid lines. and stare at it . What do you see?
you should be able to see a box sometimes you see the bottom sometimes you see the inside, guess what that flipping of the box occurs in 4D. not possible in 3D unless you move the box up and down with your own hands, but if that movement happens just by staring at it . that can only happen if the flips in 4D. easy example of seeing the power of 4D.. ENJOY
pgriesbach 072204, 11:50 AM For a truly unique visual experience of 4D Hypershift, check out my gallery/hyperspace website at www.okapistudio.netfirms.com (Specifically, the Hyperspace Gallery and the righthand page links to hyperspace diagrams). There is also a free download of a neat little Spin Ship.
patty (artist)
pgriesbach 092404, 07:28 PM New hyperspace web site URL:
http://www.hyperspace.netfirms.com
RawThinkTank 092704, 09:16 AM I have absolutely no problems imaging a 4D object in my mind, Its like a over exposed photograph. The only thing I dont get is why should time flow in forward or backward directions , can time travel in any other directions too ?
synergy 092704, 05:10 PM I once toyed with four dimensional tictactoe. It was a normal tictactoe diagram, but each cell was split into three subcells. It was played with crayons. Three x's in a row would be like this:
red x green x blue x all in the same cell (different subcells) or,
red x green x blue x in three cells in a row  the color order matters (reverse okay) or,
red x red x red x in three cells in a row (or three blue, or three green)
Since the color order matters, it represents a wellordering of the fourth dimension. Think "red is past, green is present, blue is future".
In this sense, any quality like color or smell can be made to represent a dimension, assuming an order is placed on the different values.
Just thought you might like another mathematician's perspective.
Aaron
synergy 092704, 07:00 PM I meant three dimensional tictactoe
Aaron
synergy 092704, 07:03 PM I later used a 4X4 matrix, with 4 subcells, and you had to get 4 in a row (with 4 colors) to win. This version was more challenging, on both offense and defense. Well worth playing!
Aaron
Dinosaur 092904, 10:15 PM Consider a 4D cube and a 4D sphere.
At each corner of a 4D cube there are 4 straight lines, each of which is perpendicular to the other three. Forget about visualizing the entire cube, I have trouble visualizing one corner of it. The first three perpendicular lines are easy, but I have a lot of trouble trying to visualize a direction for the fourth. I find it difficult (impossible?) to believe any person who claims to be able to visualize that corner.
Now consider the 4D sphere, but first think about 2D creatures on the surface of a 3D sphere. If they could draw a big enough triangle and measure the interior angles, they would discover that the sum of the angles was more than 180 degrees. This would give them a clue that they were not on a plane.
If the sphere was really huge, they might not be able to measure the angles of a big enough triangle. Suppose they decide to start at some point and build larger and larger circles. From our 3D point of view, they would be drawing or painting circles centered on what might be considered the North Pole of the sphere.
They would start by drawing the largest circle possible for them, using a long string anchored at the North Pole. Then they would go to many points on the circle they had just drawn and measure out fixed distances perpendicular to the circle. By connecting a bunch of points thus measured, they would build (draw) a larger circle. They could go to many points on this next circle and similarly build (paint) a larger circle.
If we watched this process with our 3D vision, we would see them making larger and larger circles until they drew the equator. From then on, they would be making smaller circles. At some point (near the South Pole), they would realize that they were inside the last circle they had drawn. From their 2D point of view, this would be mind boggling, but it would not seem strange to us.
Now imagine being 3D creatures on the 3D surface bounding a 4D sphere. Imagine building larger and larger concentric spheres. Such creatures would end up inside the last 3D sphere they had built. If I were one of them, I would find it quite mind boggling. I cannot visualize that 4D sphere. I find it impossible to believe any one who claims to be able to visualize the 4D sphere.
BTW: In N dimensions, the longest diagonal of a unit hypercube is SquarreRoot(N). In 441 dimensions, a hypercube with sides one unit long has diagonals 21 units long. The surface of a 441D hypersphere inscribed in this Hypercube, touches each face of the hypercube, but is ten units from each corner. I find this object a bit difficult to grok.
Interestingly, the volume of a unit hypersphere increases with dimension up to 7, then decreases...
A 3D sphere has more 3D volume than a 2D sphere (a circle) has 2D volume, but a 10D sphere has less 10D volume than a 9D sphere has 9D volume.
Dinosaur 093004, 11:19 AM [b}Pete:[/b] I am surprised that the volume does not decrease sooner. Are you sure of your formulae? A unit hypersphere has (½)<sup>n</sup> as a factor. That exponential gets small in a hurry.
One must be careful with hyper dimensional volumes. A unit hypercube has one unit of hypervolume in all dimensions. If the unit is feet, the hypervolume is always one hyperfoot (1<sup>n</sup>). If you measure in inches, the voume is 12<sup>n</sup>, which grows without bound as n increase. Using yards as units, the hypervoume is (1/3)<sup>n</sup>, which approaches zero as n increases.
Are you sure of your formulae?
No. For me, this is 'Net Knowledge only!
Google search (http://www.google.com.au/search?hl=en&ie=UTF8&q=hypersphere+volume&spell=1)
To me, the realization that extra dimensions were actually actions or flavors of the third dimension is when everything became clear.
Anyone able to explain the concept of a 4D sphere? A URL would be helpful, especially if there are representative visualizations.
Also, anyone care to comment on the two following barely related paragraphs:
Infinity is the context of eternity and existence. Infinity, eternity, and existence may be represented by an equilateral triangle with infinity at the base. The space outside of the triangle is imaginary.
If quantum physics reveals the existance of multiple dimensional planes in fact, then that may necessitate the invention of a new number system to describe the fourth and further dimensions, and not simply the extension of imaginary numbers.
Thanks,
D.
http://dennisys.com/
Anyone able to explain the concept of a 4D sphere?
A circle is the set of all points in a 2D space that are the same distance from a specified point.
A sphere is the set of all points in a 3D space that are the same distance from a specified point.
A nsphere is the set of all points in an nspace that are the same distance from a specified point.
A URL would be helpful, especially if there are representative visualizations.
You can imagine 4space, but you can't visualize it... the best you'll find is a projection onto 3space, which isn't terribly useful for a 4sphere, since the 3D projection is simply a 3sphere.
Infinity is the context of eternity and existence. Infinity, eternity, and existence may be represented by an equilateral triangle with infinity at the base. The space outside of the triangle is imaginary.
Bollocks. It's too misconceived to even be classed as wrong.
If quantum physics reveals the existance of multiple dimensional planes in fact, then that may necessitate the invention of a new number system to describe the fourth and further dimensions, and not simply the extension of imaginary numbers.
More bollocks
Joey_21 100504, 05:54 PM Even though I can explain what dimensions higher than the 3rd are, I cannot visualize these objects. I'm speaking of 4 SPATIAL dimensions. First let me explain the difference between a SPATIAL dimension and a VARIABLE dimension. A variable dimension is an independent factor (such as time) that effects the outcome of certain events or ideas. Time is NOT the 4th dimension!! Time is A 4th dimension. It is a VARIABLE 4th dimension.
As for spatial dimensions, back in algebra class we learned that points can be plotted on the xy plane. The xy plane is 2dimensional (hence, x and y dimensions). In multivariable calculus (and basically in your environmental surroundings) you'll learn that there are 3 dimensions of space usually denoted x, y, and z.
There exists a pattern in the properties of all ndimensional spatial directions: each axis forms a 90° angle with all the other axes. That is, each and every axis of ndimensional space is perpendicular to each and every other axis of the same ndimensional space.
For example: The xaxis in the xy plane is perpendicular (forms a 90° angle) with the yaxis, and vice versa. The x and y axes form 90° angles with the zaxis in 3 dimensions (this works associatively with all other axes). Another way of looking at this is, if you ignore the zaxis, you have an xy plane, if you ignore the y axis, you have the xz plane, and if you ignore the x axis, you have the yz plane.
Now, in 4thdimensional space, you'd have four axes of space (perpendicular to each other) to visualize. Let's call them w, x, y, and z. It could also be said that the wx plane is perpendicular to the xy plane is perpendicular to the yz plane is perpendicular to the wz plane is perpendicular to the xz plane is perpendicular to the wy plane.
For ndimensional space, nCr(n, 2) = n!/(2!*(n2)!) (where ! represents the factorial function) represents the number of planes that ndimensional space can be seperated into, and they all have the same property: They're all perpendicular to each other!
Being perpendicular is all fine and well for 2d or 3d space, but when it comes to FOUR seperate axes (for 4 dimensions) being perpendicular to each other, the visualization process becomes a mess. Which is exactly the reason I can't think of an easy way to visualize objects in higher dimensions, but I do hear there are books.
Some people have written books to explain the ideas of looking from a lower dimension into a higher one. I hear Flatland: A romance of many dimensions
http://www.math.brown.edu/~banchoff/gc/Flatland/
is a good source for some of these transitions.
planaria 100504, 08:47 PM look i dont know about any of you but i have no problem visualizing 4 dimensional objects .. spatial variable whatever its all fine ..
IT is simply a more complicated object .. so the more and more dimensions you add be it spatial or variable .. only adds to the complexity so it gets incrementally harder to visualize but thats it ..
and to people who say a projectioni is not a real thing then i say you will never be able to see anything because what you see is nothing but photons..
what you feel is nothing but sensation ..
our brains simulate everything .. so projecting a 4d object onto a 3d space is perfectly fine ..
when people say they cant visualize 4d objects .. i say you simply havent spent enough time trying ..
your brain is capable of it if you believe it is.. dont be limited by what your math teachers tell you .,.,
its like saying you cant understand a 4d object .. ofcourse you can .. and if you can understand it you can also visualize it .. it all comes from the same place..
my imagination conquers your intellect :P
So a sphere in the 4th dimension would still be spheriod. That is, it shouldn't have any granularity. OR a sphere is not possible in the 4th dimension. Seems that if there is a functional 4th dimension, that a working 4 dimensional lens is not out of the question. That is, while we are working with multidimensional attributes in physics, it might be easier to see them, than it is to produce or manipulate them. It helps to know what we are working with. High energy physics has not found the elusive first particle, and my guess is that they never will. Mathematics gives us a clue that any particle can be halved, infinitely, but observing a 4th dimension should be much easier to attain. My guess is that a virus, for example, is like an iceberg. It gets its intelligence from a source in another dimension. What we see as a virus is simply the skin of an interface, much like the magnetic loops of the sun are the visible artifacts of something that we cannot see, which is the behemoth magnetic structure that creates them. It might be worth the effort to attempt to produce the lens.
Infinity is the context of eternity and existence.
Infinity, eternity, and existance define the identity of the universe.
The universe is defined as the set whose members are infinity, eternity, and existance.
Bollocks? Ill defined? Got anything better?
Dinosaur 100604, 10:05 AM 4D and higher dimensional spaces can be dealt with mathematically. Some Analytical Geometry courses include working with more than 3D spaces. Most Differential Geometry courses deal with higher dimensional spaces.
The equations are not difficult to work with if you have some patience and are willing to spend time studying. However, all the mathematical knowledge you can develop does not help you visualize hyperdimensional objects.
You can easily set up equations for 4D spheres, cylinders, cones, toruses (tori?), and other more bizarre shapes. Studying the equations does not help much.
BTW: My spell checker does not have a spelling for the plural of torus. Does anybody know what it is?
While not a genius, I am reasonably intelligent. I spent a lot of time over a period of about ten years trying to develop the ability to visualize 4D objects. I knew that 5D and beyond had to be impossible to visualize, but I had hopes for 4D objects.
It was fun trying, and I developed some insights, so I did not consider the time wasted. I became convinced that just maybe Einstein, Bohr, Hawking, and others of their mental abilities might be able to visualize 4D objects, but I doubt it. We lesser mortals just cannot do it.
It is like a totally color blind person (do such exist?) trying to visualize colors. Such a person might get a clue by understanding the optical equations resulting in rainbows, and considering color analogous to the tones produced by a piano. However, she/he would not be able to visualize color the way that a person with normal vision can.
BTW: Suppose that you and I have the same color vision. Whenever I point to an object and specify its color, you agree with me. Can we ever be sure that our internal perceptions agree? Perhaps when I see something we agree is red, my internal image would look green to you or vice versa.
planaria 100604, 03:01 PM dinosaur your metaphor about color blind people trying to figure out color intrigued me so i thought about it. what i am trying to explain to you is more along the lines of someone who has a highly developed skill who strives for ever more complex goals.
besides if you look back in this topic someone actually gave me a 4dimen sional problem to solve it was something about how many shapes can be made from 4 hypercubes.. and through visualizing and drawing i figured this problem out not through mathematics .. i even figured out a shorthand "visual language" for the problem. i will scan this and post it for you if you have your doubts..
what i am trying to say is that it is possible to visually understand a 4d object .. my brain is not hampered by the 3d (1 variable) space around me because physics and science and imagination can show you many more spectacular things than what you see everyday.
for example, you can "visualize" a couch .. but in fact what you are seeing is only the tip of the iceburg of that couch, it would be much more difficult to say visualize this couch at the molecular scale. if you could visualize that you would be able to predict where indivual molecules (not types of molecule) would be.. or try visualizing how a couch is built this is a little easier .. you can try to visualize but only if you truly understand how it is made may you visualize the couch from the inside.
i understand and think about higher dimensional objects all the time.. my brain is a visual brain i think visually therefore i can see 4d objects.. thats really the jist of it ..
basically the only thing holding me back from visualizing higher dimensional objects past 7.. is they simply take too long for my patience to construct them in my head .. and really there are for more interesting problems (like visualizing how molecular structures interact) then this one.
and just because you cant do it doesnt mean i cant :)
Dinosaur 100604, 03:28 PM Planaria: As posted previously, I find it difficult to believe anyone who claims to be able to visualize 4D objects. I flat out disbelieve anyone who claims to be able to visualize 5D, 6D objects.
planaria 100604, 03:34 PM so how would you like me to prove it :)
planaria 100604, 06:31 PM and extra note.
the 4D can be interpredted in many particular types of problems
1) the Euclidean way (a hypercube, hypersphere) in other words your regular geometic extensions of the circle , the square, and so on.
2) as the coordinate system for imaginary equations such as (X+ i)2=0, so you let x=real, y=i, z=real and w=i
3) x=lenght, y=widht, z=depth and d=time
4)as defined by hamilton, he define the properties that would make a 4D system be a field . in other words it has an identity, it has inverses except for 0., the associative property, the distributibe property and ..... property hold. and so on . but be careful here the definition of inverses is not like in the real numbers (it's more like a hyperbolic coordinate system)
so you see that these are specific usage of a 4D system, but trying to see 4D objects in general has nothing to do with these specific definitions of the 4tth dimension in general.
a good way see the fourth dimension is to draw a cube in 3D but instead of using dotted lines to connect the sides that you do not see, use solid lines. and stare at it . What do you see?
you should be able to see a box sometimes you see the bottom sometimes you see the inside, guess what that flipping of the box occurs in 4D. not possible in 3D unless you move the box up and down with your own hands, but if that movement happens just by staring at it . that can only happen if the flips in 4D. easy example of seeing the power of 4D.. ENJOY
i find this coordinate system  method of looking at a hypercube very interesting .. it reminds me of how i draw pictures, when i look at a piece of paper i "see" a coordinate system in the fibers of the paper .. if you look at this coordinate system long enough you can see patterns emerge ,, like faces .. shadows , different shapes etc. the interesting thing is that since the coordinate system is so complex many different patterned shapes can emerge from the same location .. so assuming im not going out on a limb with that coordinate explanation im sortof drawing in 4d .. except i cant draw all the shapes arising from that one area i can pick and choose things i see and draw them ..
this idea of shapes arising from a coordinate system is also kindof like emergence in complexity theory no ? or am i just mixing stuff up ..
*me needs to start imaging how a 5d coordinate system cube would work.. *
examples of my art can be found on http://www.palnkton.deviantart.com
look for the drawings the 3d stuff is just playing around ..
As a side note: I found a 1996 book by David Berlinski titled "a tour of the calculus" at HalfPriced Books. It is a narrative that serves well as a quick refresher, besides being both entertaining and enlightening.
Dinosaur 100804, 02:27 PM Planaria: There is no way to prove or disprove assertions relating to our perceptual images. If somebody claims that he dreams in color, I am inclined to believe him whether I dream in color or not. Could such a simple assertion be proven? Of course not. I cannot prove that another person has consciousness, although I believe that others do.
Some claims about perceptions I am willing to believe. Others seem dubious and still others seem too outrageous to consider. Claiming the ability to visualize 4D objects is dubious at best. When I was attempting to develop the capability, I had some vague glimmers of understanding. Perhaps somebody else would consider those vague images an ability to visualize a 4D object. I did not consider them close to such an ability. A claim to visualize 5D or 6D objects is worst than dubious, I claim that it is flat out impossible.
There is an interesting type of game or challenge requiring 3D visualization, an analogue of which might be usable for testing the ability to visualize 4D, 5D, et cetera objects. The test involves several perspective views of a cube and a plane figure consisting of 6 squares, which can be folded to make a cube. The question to be answered is which (if any) of the perspective views match the unfolded pattern. A variant is to show one perspective view and several unfolded patterns, asking which (if any) of the plane figures matches the perspective view of the cube.
At least some (or all) of the squares shown have various designs like a star, a cross, a tartan plaid, different solid colors, a landscape photo, whatever. Note that the perspective view shows only three of the six squares. Note also that the standard plane figure is a cross, which is not the only possible plane figure for the unfolded cube. A typical series of games or challenges uses more than one of the possible plane figures. The easiest variations use the cross.
Some people (typically artists and draftsmen) find this test very easy. Others find it very difficult. A few find it almost impossible. As for other intellectual or physical challenges, almost all people improve with practice.
Perhaps, a similar test could be created using a tesseract (4D hypercube) unfolded into a 3D object consisting of eight 3D cubes, or a 5D hypercube unfolded into some number of tesseracts. I am not sure how to display the folded up 4D hypecube, and the unfolded 5D hypercube boggles my mind as much or more than the folded up 5D hypercube.
BTW: I think a 4D creature in a universe with 4D optical laws would be able to see more than 3 faces of a 3D cube at the same time without using mirrors or some other gadget. A person with true 4D perceptual awareness should be able to visualize a view of a 3D cube showing more than 3 faces.
I have a program which includes the above game as well as several others. Each individual game has about 20 variations, starting with the easiest and ending with a very difficutlt variation. My girl friend's grandson delighted in the cube visualization game and became so good that it became boring. He now plays with another of the games which involves getting a car out of a parking lot. The lot is packed solid with cars, limousines, & trucks except for one empty space the size of a car.
planaria 100804, 09:09 PM http://www.deviantart.com/deviation/11263772/ click on image for the fullsize image.. *note for simplicity most cubes were drawn in an orthogonal prespective*
ok well i figured i should put my money where my mouth is and actually draw out some visualizations of hypershapes ..
now i could also draw what you were saying of a 4d cube where its a 3d cube that you can see both sides of .. this is very easy you simply draw two cubes, shade them oppositly , then cross your eyes and your steroscopic vision will allow you to see both sides of a shaded cube..
this is mostly to show you my ideas on hypercube visualization i dont think its anything that new .. i made a "shorthand" method of drawing general hypershapes which i describe over at the page..
and if you will look to the far left underneath the blue hypercube which im not quite sure as to how many dimensions it is .. is it 6 ? i get confused with numbers not images :) anyway there is a 3d cross of 6 cubes.. this is what i believe to be the answer to unpacking a 4d hyper cube i further unfolded that 3d cross *its to the right of it* which was very easy .. i imaging a 5d hyper cube would be much the same way except you might have to have certain parts not connected ..
basically what your talking about is in 3d world called uv mapping.. where you take each face of a 3d object and flatten it in a sensible way so that you can put textures on the 3d shape.. i was thinking prolly a more proper way to do this would be to use the vectors instead of faces for this problem since when you do it by face you replicate vectors.. but both ways can work i actually did the face version and not the vector version to show you its possible.
why does a 4d cube convert to a 3d cross ? well because a 3d cube converts to a 2d cross thats why .. each time you flatten all your doing is figuring out a way to represent a higher dimensional shape in a lower dimension and this problem only needs you to lower by 1 so yea..
i hope this agrees with your ideas of visualization as well.. i thought this was a pretty fun problem actually ..
Dinosaur 100804, 11:19 PM Planaria: Your version of an unfolded 4D hypercube is a 3D cross consisting of only six 3D cubes. A 4D hypercube unfolds into a 3D figure consisting of eight 3D cubes. This discrepancy suggests that your ability to visualize a 4D hypercube is a bit suspect.
The eight 3D cubes representing an unfolded 4D hypercube can form various 3D objects. You can start constructing some of these 3D objects by stacking four of the cubes to form what can be viewed as a four story building (actually a 1X1X4 parallelopiped). The four additional cubes can be attached to the four exposed faces of the cube forming any one of the four stories. All of these objects are symmetric crosslike structures.
Starting with the four story building object, various other representations can be constructed by attaching the four additional 3D cubes in various unsymmetric positions. Not all such objects represent an unfolded 4D hypercube. I think that each of the additional four 3D cubes must be attached to a different side of the four story building.
The situation is analogous to unfolding a 3D cube into six squares. The crosslike figure is not the only possible configuration for the six squares. You can start with four squares forming a 1X4 rectangle and attach the remaining two squares, one on each long side of the rectangle at any one of the four possible positions. If you attach the 5th & 6th squares to the same long side of the rectangle, the resulting figure cannot be folded to form a cube.
planaria 100904, 01:58 AM actually you can make a 4d cube with this many cubes..
if you look at this instead as 2 2d crosses and you use each 2d cross to construct a cube the edges that connected and made the 3d cross create the necessary hyperness that connects the two cubes .. meaning those edges are what create the cubes that connect to the cubes.
if you really really need me to i will draw it out step by step.
Dinosaur 100904, 05:37 PM Planaria: You are a piece of work. You claim to be able to visualize 4D, 5D, & 6D objects. You also claim you could visualize 7D objects if you had enough patience.
honestly i dont understand people and their inability to see 4d objects
i understand and think about higher dimensional objects all the time.. my brain is a visual brain i think visually therefore i can see 4d objects.. thats really the jist of it ..
basically the only thing holding me back from visualizing higher dimensional objects past 7.. is they simply take too long for my patience to construct them in my head After making such claims you direct us to your web site where there is an erroneous perspective view of an unfolded 4D hypercube.
As mentioned in my previous post, a 4D hypercube unfolds into a 3D object consisting of eight 3D cubes, while the object shown at your site consists of only six 3D cubes. Your last post seems like some attempt at obfuscation.
actually you can make a 4d cube with this many cubes..
if you look at this instead as 2 2d crosses and you use each 2d cross to construct a cube the edges that connected and made the 3d cross create the necessary hyperness that connects the two cubes .. meaning those edges are what create the cubes that connect to the cubes.The above does not make sense. Perhaps English is not your native language.
If it is so easy to "visualize" then why don’t you draw it? That’s what they mean by hard to visualize.
planaria 101004, 04:38 PM cato .. http://www.deviantart.com/deviation/11263772/
this was posted earlier btw ..
and dinosaur i will post a detailed drawing explaining how my method works. what i realize is that your method deals with the problem in terms of cubes and i deal with the problem in terms of vertice/edges/faces a subtle difference but important nonetheless. actually both methods work your method constructs my method folds and joins. dont worry if this explanation is confusing the drawing will clarify.
Its not really 4d, its just 3d that you give some qualifications to. I understand what you mean, and how to think of those shapes as 4d. However, what I meant was that you can’t do something like say... draw x, y, z, and t axes all orthogonal to each other.
planaria 101004, 08:41 PM thats true but you also cant draw a 3d cube with all orthogonal lines..
you can only do that in 3d space. you can draw a 2d square with all orthogonal lines however.. because that is the realm 2d lives in ..
this bring up a thought however, you can represent everything orthogonally if you note the vertex coords next to each vertex, for instance if you draw a 2d square but then on the 4 visible vertice you simply write both xyz coords for the vertice that is theoretically behind that vertex you can represent.. show a 3d cube on a 2d surface with all orthogonal lines correctly ..
now this should work with a hypercube as well correct ? the number would just jump up to 4 coords for each visible vertex.. ?
this should also work if your thinking about a hyper cube in terms of faces or cubes.. although this starts to shift from visualization to a combination of visualization and notation..
Dinosaur 101404, 09:39 PM Planaria: The following looks like obfuscation or an attempt to baffle with bull**it rather than dazzle with brilliance.
what i realize is that your method deals with the problem in terms of cubes and i deal with the problem in terms of vertice/edges/faces a subtle difference but important nonetheless. actually both methods work your method constructs my method folds and joins. dont worry if this explanation is confusing the drawing will clarify.While awaiting the drawing, I am wondering about the difference between dealing with cubes instead of vertices, edges, & faces.
I still claim that your erroneous 3D perspective view of an unfolded 4D cube indicates a lack of the ability to visualize a simple 4D object such as a tesseract (4D cube). If somebody showed me a 2D object consisting of 4 squares and claimed that it represented an unfolded cube, I would doubt that he understood what a cube looked like.
Infinity, eternity, existence
You can plot a hologram in space easy enough, but what if you plotted it in time as well. Wouldn't you get a ghost of the positional features in the time references, if somehow you could colorcode the different plots.
The 4th dimension may not exist as we might expect from our current mathematics because we have modeled the world in three dimensions, except for the spacetime thing, which is simply an extension of the current logic as much as the xyz plane is an extension of the positional features of space. Spacetime simply goes it one better by including time as a 4th dimension, when it is really an extension of the two dimensional coordinate system into the third plane, not into a 4th. Rather, it is a combination of the two extensions that may bring us a 4th dimensional plane, the wxyz plane, and in that is existence as well as space and time.
We are at a place in time, but we are also an apple.
That's my view to date. I'm still working on it.
What do you think?
Even though I can explain what dimensions higher than the 3rd are, I cannot visualize these objects. I'm speaking of 4 SPATIAL dimensions. First let me explain the difference between a SPATIAL dimension and a VARIABLE dimension. A variable dimension is an independent factor (such as time) that effects the outcome of certain events or ideas. Time is NOT the 4th dimension!! Time is A 4th dimension. It is a VARIABLE 4th dimension.
As for spatial dimensions, back in algebra class we learned that points can be plotted on the xy plane. The xy plane is 2dimensional (hence, x and y dimensions). In multivariable calculus (and basically in your environmental surroundings) you'll learn that there are 3 dimensions of space usually denoted x, y, and z.
There exists a pattern in the properties of all ndimensional spatial directions: each axis forms a 90° angle with all the other axes. That is, each and every axis of ndimensional space is perpendicular to each and every other axis of the same ndimensional space.
For example: The xaxis in the xy plane is perpendicular (forms a 90° angle) with the yaxis, and vice versa. The x and y axes form 90° angles with the zaxis in 3 dimensions (this works associatively with all other axes). Another way of looking at this is, if you ignore the zaxis, you have an xy plane, if you ignore the y axis, you have the xz plane, and if you ignore the x axis, you have the yz plane.
Now, in 4thdimensional space, you'd have four axes of space (perpendicular to each other) to visualize. Let's call them w, x, y, and z. It could also be said that the wx plane is perpendicular to the xy plane is perpendicular to the yz plane is perpendicular to the wz plane is perpendicular to the xz plane is perpendicular to the wy plane.
For ndimensional space, nCr(n, 2) = n!/(2!*(n2)!) (where ! represents the factorial function) represents the number of planes that ndimensional space can be seperated into, and they all have the same property: They're all perpendicular to each other!
Being perpendicular is all fine and well for 2d or 3d space, but when it comes to FOUR seperate axes (for 4 dimensions) being perpendicular to each other, the visualization process becomes a mess. Which is exactly the reason I can't think of an easy way to visualize objects in higher dimensions, but I do hear there are books.
Some people have written books to explain the ideas of looking from a lower dimension into a higher one. I hear Flatland: A romance of many dimensions
http://www.math.brown.edu/~banchoff/gc/Flatland/
is a good source for some of these transitions.
planaria 101604, 01:29 AM hi there ,,
i have been unable to get to a scanner ..
here is the picture..
http://www.deviantart.com/view/11465146/
I like the cubes marked 4D especially because of the shadows that help illustrate the different sides, and the ability to see through them. It would be helpful to colorcode the planes (of existence I might add, as well, those being not only the planes of the inner cube, but also of the outer cube). I begin to see how well various particles and forces of atomic structure could be illustrated, animated, and maybe even manufactured.
hi there ,,
i have been unable to get to a scanner ..
here is the picture..
http://www.deviantart.com/view/11465146/
Infinity, eternity, existence
You can plot a hologram in space easy enough, but what if you plotted it in time as well. Wouldn't you get a ghost of the positional features in the time references, if somehow you could colorcode the different plots.
The 4th dimension may not exist as we might expect from our current mathematics because we have modeled the world in three dimensions, except for the spacetime thing, which is simply an extension of the current logic as much as the xyz plane is an extension of the positional features of space. Spacetime simply goes it one better by including time as a 4th dimension, when it is really an extension of the two dimensional coordinate system into the third plane, not into a 4th. Rather, it is a combination of the two extensions that may bring us a 4th dimensional plane, the wxyz plane, and in that is existence as well as space and time.
We are at a place in time, but we are also an apple.
It's the next day.
I don't think this goes far enough. The combination of the two extensions would still lack the breadth of a borderless plane of existence, and I'm not sure it could be described simply by adding a single reference (w). Could be that it would require more than one.
When I think of xy I think of x as time and y as space. z would be existence, and the combination of the three would enable a 'map' of the the three dimensional universe in part or in whole. In other words, the universe is time, space, and existence, and each of those is a domain of its own. Hence, the nature of three persons in God, as declared in some Christian religions.
DenniSys.com
Dinosaur 101604, 10:48 PM Planaria: Once again you are showing a perspective view of a 3D object consisting of six 3D cubes forming a cross, and claiming (at least implying) that it represents an unfolded 4D cube. This is an erroneous representation, clearly indicating that you cannot really visualize a 4D cube.
Again, I point out that a 4D cube unfolds into a 3D object consisting of eight 3D cubes.
Note one of the last drawings in your montage: It shows a 3D cube inside of a larger 3D cube with the corners of the two cubes connected. This is a well known representation of a tesseract (4D cube). A proper interpretation of that representation clearly indicates the eight 3D cubes comprising a 4D cube, namely the inner 3D cube, the outer 3D cube, and the six cubes connecting the faces of those two cubes. I do not claim to be able to visualize a 4D cube, but my concept of it is better than yours.
BTW: It boggles my mind when I try to visualize how four straight lines can be mutually perpendicular, which is what occurs at each vertex of a 4D cube. It boggles my mind even more when I realize that five planes meet at each vertex, and that each plane is perpendicular to the other four.
planaria 101704, 03:18 PM i just told you that i drew My way of unfolding a hypercube.. my way deals with hypercubes not in a method of connecting Cubes together but a method that connect Vertice and Edge together. i drew that common method of showing a hypercube to show you what i thought you were talking about, i could see that from this angle of a hypercube that you could easily create the solution of 8 cubes necessary for creating the hypercube.. however from the view i was looking at i seem to have created a different solution.
my method creates cubes as they are being constructed into a hyper cube so the vertices and edges that are assembled Create the extra cubes needed. they are a consequence of the folding action.. i could even make a more slimmed down version of this folding excersize if you said that you could also not reproduce any vertice or line.
if youll note the drawing after the last folding is a hypercube with 8 cubes.. and then there are rotated drawings of that same cube to show it is so . can you see that from this drawing?
look even you yourself said there are many solutions to a problem such as unfolding geometrical objects (such as 3d cubes as was the example) can you not fathom that i have found a method that requires 6 cubes instead of 8 ?
it is you who cant visualize my ideas not the other way around.
****Again, I point out that a 4D cube unfolds into a 3D object consisting of eight 3D cubes.*****
again i point out that in a previous post i AGREED with you that your method works , but that also my method works i do not understand how you cannot see this after i went through the trouble of drawing it out for you .. can you tell me what confuses you atleast? you only seek to dissprove..
****BTW: It boggles my mind when I try to visualize how four straight lines can be mutually perpendicular, which is what occurs at each vertex of a 4D cube. It boggles my mind even more when I realize that five planes meet at each vertex, and that each plane is perpendicular to the other four.****
try to visualize a 3d cube with all perpendicular lines, you cant and yet you can visualize a 3d cube just fine..
Dinosaur 101704, 04:04 PM Planaria: I give up. I will not waste more of my time on this nonsense. If you still think a 4D cube unfolds into a 3D object with six 3D cubes, there is no convinincing you that you are dead wrong. Mathematics is not a matter of opinion: You can be wrong. The following is nonsense.
look even you yourself said there are many solutions to a problem such as unfolding geometrical objects (such as 3d cubes as was the example) can you not fathom that i have found a method that requires 6 cubes instead of 8 ?Yes, I understand that you found a way to unfold a 4D cube in six 3D cubes. You have claimed this several times and have now shown me the same perspective view of the erroneous object at least twice. What I also understand is that you are mistaken when you claim that the unfolded object represents a 4D cube.
You can Google 'tesseract' and find Java applets that fairly illustrate various concepts surrounding the subject. Maybe it is more important to describe the use of such information. We usually approach a problem and then find a solution. The establishment of mathematics affords an organized method of problem solving.
Mathematics itself, as with most science, is useless without application. Once the problem is described the utilization of the methods define the meaning of the organizational elements. Basically, it helps to have a problem in mind before trying to visualize a solution, and with that the solution may be reduced to its necessary level of complexity, wherein elements are given meaning.
You can define the xy coordinate system as anything you want and visualize it any way you choose within the limits of its usefulness, so that the assignments that you make become important only with reference to the methods used and the outcomes they produce.
What use is it to be able to define a four, five, or more dimensional system if there is no application to apply it. The answer is that there are applications that are suited to the methods, and each application may take a discrete or unique viewpoint on the meaning of the organizational elements. By understanding the application the visualization then becomes a done deal. So the importance exists within the concept or concrete object that you are attempting to model.
If I am trying to plot a belief rather than an apple, then I see a belief system made of ideas which have behavioral significance with an anatomic source. This belief object determines the use I make of the xy coordinate system, and having assigned meaning to the coordinates I can visualize some sense of the object.
The study of mathematics is concerned with assigning the coordinate system the values of (x) as space and (y) as time, so that those planes of existence can be measured. I am concerned that the term 'planes of existence'' contains the phrase 'of existence' as giving existence a meaning and identity of its own where time and space are subordinate components. I want to give existence an equal share and build it into the coordinate system, because the three elements, and no others, comprise the universe as we know it. That is, you would be hard pressed to build an index of importance for the three objects.
planaria 101804, 04:42 PM Planaria: I give up. I will not waste more of my time on this nonsense. If you still think a 4D cube unfolds into a 3D object with six 3D cubes, there is no convinincing you that you are dead wrong. Mathematics is not a matter of opinion: You can be wrong. The following is nonsense.Yes, I understand that you found a way to unfold a 4D cube in six 3D cubes. You have claimed this several times and have now shown me the same perspective view of the erroneous object at least twice. What I also understand is that you are mistaken when you claim that the unfolded object represents a 4D cube.
what you dont realize is my erroneous object is your tesseract just in a different view .. your tesseract and my drawing are the same thing just in different states of visualization , i know this because for one thing there are several java applets that will cycle between them in a smooth tranformation ..
just admit that were both right :)
