4d object visualization ..
- Thread starter planaria
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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.
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.
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.
[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.
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.
Xmo1
Registered Senior Member
Anyone able to explain the concept of a 4-D 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/
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/
A circle is the set of all points in a 2D space that are the same distance from a specified point.Xmo1 said:
A sphere is the set of all points in a 3D space that are the same distance from a specified point.
A n-sphere is the set of all points in an n-space that are the same distance from a specified point.
You can imagine 4-space, but you can't visualize it... the best you'll find is a projection onto 3-space, which isn't terribly useful for a 4-sphere, since the 3D projection is simply a 3-sphere.
Bollocks. It's too misconceived to even be classed as wrong.
More bollocks
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 x-y plane. The x-y plane is 2-dimensional (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 n-dimensional spatial directions: each axis forms a 90° angle with all the other axes. That is, each and every axis of n-dimensional space is perpendicular to each and every other axis of the same n-dimensional space.
For example: The x-axis in the x-y plane is perpendicular (forms a 90° angle) with the y-axis, and vice versa. The x and y axes form 90° angles with the z-axis in 3 dimensions (this works associatively with all other axes). Another way of looking at this is, if you ignore the z-axis, 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 4th-dimensional 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 n-dimensional space, nCr(n, 2) = n!/(2!*(n-2)!) (where ! represents the factorial function) represents the number of planes that n-dimensional 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.
As for spatial dimensions, back in algebra class we learned that points can be plotted on the x-y plane. The x-y plane is 2-dimensional (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 n-dimensional spatial directions: each axis forms a 90° angle with all the other axes. That is, each and every axis of n-dimensional space is perpendicular to each and every other axis of the same n-dimensional space.
For example: The x-axis in the x-y plane is perpendicular (forms a 90° angle) with the y-axis, and vice versa. The x and y axes form 90° angles with the z-axis in 3 dimensions (this works associatively with all other axes). Another way of looking at this is, if you ignore the z-axis, 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 4th-dimensional 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 n-dimensional space, nCr(n, 2) = n!/(2!*(n-2)!) (where ! represents the factorial function) represents the number of planes that n-dimensional 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.
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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
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
Xmo1
Registered Senior Member
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.
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 hyper-dimensional 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.
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 hyper-dimensional 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.
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
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
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chaoticorder said:
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 ..
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.
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.