They wimped out by not rotating the tesserect around the fourth dimension. It is simple, just set up a 4x4 matrix, rotate the four dimensional node positions of the hypercube and then project back to 3d and then project to 2d. Come on guys, I expect more from you all.
mimis dog-us Correct me if I'm wrong but... I think, it isn't. Our mind is 3-dimensional. You have to kind of simulate the fourth dimension through animation. But the object itself is still 3-dimensional like when you draw a cube on a flat piece of paper, it's 2-dimensional. It looks like a cube. But it's still flat.
Extend a line (1d-cube) perpendicular to its length for a distance equal to its length, and you get a rectangular surface (2d-cube). Extend the surface perpendicular to the surface for a distance equal to its length and you get a 3d-cube. Now at this step, you have to extend the cube perpendicular to its volume for a distance equal to its length and you get a 4d-cube. Of course we cann't point to a direction perpendicular to a volume since we are just humble 3d creatures with no feeling of the 4th dimension. Similar with some poor 2d creatures that are not able to point to the height direction.
But because gods are generous, there is a way for us to take a small glimpse to this 4th dimension. How? By seeing the projections of 4 dimensional objects in our 3d world. This is the hypercube that we see at the end of this gif.
In order to understand this concept we need to subtract 1 dimension. Imagine that we are 2 dimensional creatures leaving in a flat world. How we could see a 3d cube? Well, imagine that a 3 dimensional hyper being, keeps the cube above our flat world and points to it with a light. Its shadow falling to our flat land is the projection of the 3d cube. It would have the shape of the cube that we all had to draw in a paper when we were at school.
Now if we add 1 dimension to the previous story, we can maybe understand a little more what this mysterious hypercube is.
Let us think that a 4d space is consisted of infinite (not countable) 3d spaces one "above" the other. In each of these 3d spaces we would be the same, as we are now. So we could say that our 4d shape would be an infinite sequence of our bodies, one "above" the other, without any communication between them, and the 4d body to have a finite 4d-volume. And for this 4d-creature, we 3d-people would be of zero measure, so we would not exist. When we see the base of a cube, we realize this because in our mind, this flat surface has a tiny thickness, so, even the bottom is a 3d object. I think that we can't conceive a 2d-object, namely with zero height. Ofcourse, all the above, is a feelling of mine, an imagination, no prove at all. But I liked to write down this "dream".
First, in about 1978 or so, Tom Banchoff had a special machine made to rotate the hypercube around. It cost about $180,000. My friend Tony Robbin who is an artist when to see this, and it influenced his art from that point onward. Now there are a myriad of hypercube programs that are really cheap and 1000s of times more powerful.
Second, I always feel competed to show this video (Yes there is a line missing, see the sequel): Drawing the Hypercube # 1
Here is another description of a four dimensional cube. First, hold up your index finger. Keep it straight, but bend it at the last knucle (closest to the hand) and watch it make up to a ninety degree angle. Wiggle it to and fro. Imagine that the tip traces out a quarter circle which, for this argument, we identify with an interval of numbers. Now do the same with your middle finger. Smile sheepishly at the person in the coffee shop who thinks you are flipping her off. Now move both fingers independently. The set of configurations consist of two angles each between 0 and 90 degrees. Each traces an interval, combined they describe a square, rectangle or other surface (say 1/4 of the surface of a doughnut). Now consider your annular finger along with the other two. Find a point in the room. Move your index finger proportionally to the distance from the east wall. Move your middle finger proportionally from the south wall. Move your annular finger proportionally from the floor. These three fingers effective describe a point in the room. Now think of a time in the future within the next 6 hours. Move your baby finger proportionally from upright to a position to represent that time.
Your four fingers have a set of configurations that represents a point in space.
More graphically, pretend you are a drummer. Left foot highhat, right foot bass drum, left hand snare, and right hand cymbal. Consider a measure of time. Now tap each limb independently within that time. The configurations represent a 4 dimensional space.
Many things are multidimensional. Thinking about them in this fashion can be helpful.
I'd rather say, no matter how many things you do simultaneously, they are all just 4-dimensional. You move your limbs in three dimensions and the movement (along the timeline) is number 4. Even if you had 10 arms and 20 legs, it's still 4-dimensional, just representing different values of the 3 space dimensions at the same value of number 4.
You could however, like you suggested, Scott Carter, consider them to be completely independent values. Then you'd have four dimensions per limb. One should, probably, due to relativity which not only applies to whole individuals but to every single particle in this universe. So, yes, your hand and your feet exist in different timeframes as long as you move them relatively to one another.
Corina Marinescu is impressive in her compassionate take on the comments and that's something definitely likeable!! Keep spreading cheer in your unique and charming way... :)
Like someone commented, this animation still shows the 4-D as just a 3-D instance since perceiving fourth dimension is an exercise of thought. Representing it tangibly visually is difficult. However, hypercubes are practically 3-D when static.
Then again, like Scott Carter and Hendrik Wiese have suggested, observing regular movements over a period of time by mentally drawing out a trace of a few points (dots/nodes) on the subject (person/object that moves about) would create a space-time version of that subject which is truly in 4-D but this entity as a tangible whole exists only in the mind of the observer.
Considering the observer's influence on the existence of the higher dimension adds more dimensions recursively as far as we wish to continue observing, counting and mentally seeing the effects of each addition.
In short, 3-D is in space and the rest are in time and the observing mind(s). In plurality, none of any two higher dimension entities would be identical when considered at the same dimensional degree of thought.
Wowww!!
ReplyDeleteIt's always 3 dimensional
ReplyDeletenic animation
ReplyDeleteThey wimped out by not rotating the tesserect around the fourth dimension. It is simple, just set up a 4x4 matrix, rotate the four dimensional node positions of the hypercube and then project back to 3d and then project to 2d.
ReplyDeleteCome on guys, I expect more from you all.
tom gross I have no idea what you expect from the "guys" ...I happen to be no "guy" and I like this ;)
ReplyDeleteJeremy Fields I used to think it would be cool if they invented chalk that didn't squeak
ReplyDeleteCorina Marinescu We can do this. check the wikipedia page for Plane_of_rotation the matrix math is good fun. And it produces cool images: http://en.wikipedia.org/wiki/File:Tesseract.gif
ReplyDeleteJeremy Fields actually, I am or was pretty proficient with abacus and slide rule. Computers are wonderful though.
ReplyDeletewow corina... totally awsome.
ReplyDeleteSo much swag
ReplyDeleteSalvatore Perciante It is not. It's always two-dimensional.
ReplyDeleteI am new in this community. To be frank, I don't understand how it is possible to conceive or to imagine a 4 space-dimension object.
ReplyDeletemimis dog-us Correct me if I'm wrong but... I think, it isn't. Our mind is 3-dimensional. You have to kind of simulate the fourth dimension through animation. But the object itself is still 3-dimensional like when you draw a cube on a flat piece of paper, it's 2-dimensional. It looks like a cube. But it's still flat.
ReplyDeletemimis dog-us you should follow John Baez- he explains in great detail all these math - "brain pinch" questions ;)
ReplyDeleteHere is one of his posts about a truncated tesseract
https://plus.google.com/117663015413546257905/posts/eaUv6rhqv56
Corina Marinescu
ReplyDeletethank you very very much. Because, every time I said that to my friends, they looked at me suspiciously.
Extend a line (1d-cube) perpendicular to its length for a distance equal to its length, and you get a rectangular surface (2d-cube). Extend the surface perpendicular to the surface for a distance equal to its length and you get a 3d-cube. Now at this step, you have to extend the cube perpendicular to its volume for a distance equal to its length and you get a 4d-cube. Of course we cann't point to a direction perpendicular to a volume since we are just humble 3d creatures with no feeling of the 4th dimension. Similar with some poor 2d creatures that are not able to point to the height direction.
ReplyDeleteBut because gods are generous, there is a way for us to take a small glimpse to this 4th dimension. How? By seeing the projections of 4 dimensional objects in our 3d world. This is the hypercube that we see at the end of this gif.
In order to understand this concept we need to subtract 1 dimension. Imagine that we are 2 dimensional creatures leaving in a flat world. How we could see a 3d cube? Well, imagine that a 3 dimensional hyper being, keeps the cube above our flat world and points to it with a light. Its shadow falling to our flat land is the projection of the 3d cube. It would have the shape of the cube that we all had to draw in a paper when we were at school.
Now if we add 1 dimension to the previous story, we can maybe understand a little more what this mysterious hypercube is.
Let us think that a 4d space is consisted of infinite (not countable) 3d spaces one "above" the other. In each of these 3d spaces we would be the same, as we are now. So we could say that our 4d shape would be an infinite sequence of our bodies, one "above" the other, without any communication between them, and the 4d body to have a finite 4d-volume. And for this 4d-creature, we 3d-people would be of zero measure, so we would not exist. When we see the base of a cube, we realize this because in our mind, this flat surface has a tiny thickness, so, even the bottom is a 3d object. I think that we can't conceive a 2d-object, namely with zero height.
ReplyDeleteOfcourse, all the above, is a feelling of mine, an imagination, no prove at all. But I liked to write down this "dream".
So much quality. Oh those pixels
ReplyDeleteTwo comments:
ReplyDeleteFirst, in about 1978 or so, Tom Banchoff had a special machine made to rotate the hypercube around. It cost about $180,000. My friend Tony Robbin who is an artist when to see this, and it influenced his art from that point onward. Now there are a myriad of hypercube programs that are really cheap and 1000s of times more powerful.
Second, I always feel competed to show this video (Yes there is a line missing, see the sequel): Drawing the Hypercube # 1
Thanks Scott Carter ;)
ReplyDeleteJeremy Fields, thanks for the link! It's really interesting.
ReplyDeleten
ReplyDeleteHere is another description of a four dimensional cube. First, hold up your index finger. Keep it straight, but bend it at the last knucle (closest to the hand) and watch it make up to a ninety degree angle. Wiggle it to and fro. Imagine that the tip traces out a quarter circle which, for this argument, we identify with an interval of numbers. Now do the same with your middle finger. Smile sheepishly at the person in the coffee shop who thinks you are flipping her off. Now move both fingers independently. The set of configurations consist of two angles each between 0 and 90 degrees. Each traces an interval, combined they describe a square, rectangle or other surface (say 1/4 of the surface of a doughnut). Now consider your annular finger along with the other two. Find a point in the room. Move your index finger proportionally to the distance from the east wall. Move your middle finger proportionally from the south wall. Move your annular finger proportionally from the floor. These three fingers effective describe a point in the room. Now think of a time in the future within the next 6 hours. Move your baby finger proportionally from upright to a position to represent that time.
ReplyDeleteYour four fingers have a set of configurations that represents a point in space.
More graphically, pretend you are a drummer. Left foot highhat, right foot bass drum, left hand snare, and right hand cymbal. Consider a measure of time. Now tap each limb independently within that time. The configurations represent a 4 dimensional space.
Many things are multidimensional. Thinking about them in this fashion can be helpful.
Very interesting Scott Carter =)
ReplyDeleteThat was a cool reading, danke!
I'd rather say, no matter how many things you do simultaneously, they are all just 4-dimensional. You move your limbs in three dimensions and the movement (along the timeline) is number 4. Even if you had 10 arms and 20 legs, it's still 4-dimensional, just representing different values of the 3 space dimensions at the same value of number 4.
ReplyDeleteYou could however, like you suggested, Scott Carter, consider them to be completely independent values. Then you'd have four dimensions per limb. One should, probably, due to relativity which not only applies to whole individuals but to every single particle in this universe. So, yes, your hand and your feet exist in different timeframes as long as you move them relatively to one another.
Mind blowing...
Corina Marinescu is impressive in her compassionate take on the comments and that's something definitely likeable!! Keep spreading cheer in your unique and charming way... :)
ReplyDeleteLike someone commented, this animation still shows the 4-D as just a 3-D instance since perceiving fourth dimension is an exercise of thought. Representing it tangibly visually is difficult. However, hypercubes are practically 3-D when static.
Then again, like Scott Carter and Hendrik Wiese have suggested, observing regular movements over a period of time by mentally drawing out a trace of a few points (dots/nodes) on the subject (person/object that moves about) would create a space-time version of that subject which is truly in 4-D but this entity as a tangible whole exists only in the mind of the observer.
Considering the observer's influence on the existence of the higher dimension adds more dimensions recursively as far as we wish to continue observing, counting and mentally seeing the effects of each addition.
In short, 3-D is in space and the rest are in time and the observing mind(s). In plurality, none of any two higher dimension entities would be identical when considered at the same dimensional degree of thought.