- #1

nightcleaner

This thread is for discussion and presentation of results in the attempt to visualize higher dimensions.

My current work involves identifying stucture in the 4d world of spacetime equivalence. Marcus showed me that structures in spacetime are related to the Pascal triangle, and selfAdjoint helped by pointing out that Pascal's triangle can be calculated from the N-choose-k math. All this is interspersed in Marcus' two-worlds thread on this board.

The Pascal triangle is formed by lining up numbers and then adding the two above on the next line below. The second and subsequent lines also start with the first two numbers, like so:

You see the one and one in the first line add to the two in the second line. In the third line, two plus one is three. In the fourth line, one plus three is four, three plus three is six, and so on.

Marcus and selfAdjoint told us in the Two Worlds thread that you can calculate these numbers by a method known as N-choose-k, in which you have a group of N objects, and you pick them up in as many ways as possible in groups of k. So if you have three N objects, A,B,C, and you pick them up two k ways, obveously you can pick A,B, A,C, or B,C. In this method, A,B is counted as the same group as B,A.

There is a formula for this method, and it involves factorial numbers. These numbers are written as n! where n is a positive integer. 3! is the product of 3 and all the positive integers below it, 1x2x3=6=3!. Another example,

4!=3!x4 is 6x4=24. It is conventional that 0!=1=1!.

The formula for finding the numbers in the Pascal triangle is

[tex]N!/k!(N-k)![/tex]

where N is the row number and k is the column number, counting the first column as zero. For example, in the fourth row of the Pascal Triangle, in the first column, we have 4!/0!(4-0)! is 24/24which equals one. The second number in the fourth row is 4!/1!(4-1)! = 24/3!=24/6=4. And so on.

There is also a tetrahedral form of the Pascal triangle, where there are three ones at the top, and ones around a two in the next layer, and ones around threes around a six in the next layer.

Now to get back to business. There is an idea for thinking of dimensionality in terms of the least structure that can fully represent the dimension in question. For example, as in plane geometry, a point has zero dimension. It has no size that can be measured, and until higher dimensions are invoked, there is no way to talk about where the point is located. The point is called in this system a zero dimensional simplex.

The one dimensional simplex is a line, which has two end points. The only thing you can measure in a line is how long it is, so it is one dimensional. In geometry, a line is thought of as having only one point in cross section, so it is not the same thing as a wire strung between two poles, which has, as well as length, a diameter or thickness. Note also that a line in this sense has to be "straight," because there is no other dimension defined for it to curve into.

The two dimensional simplex is a triangle on a flat plane. It has three apex points, three lines, and one flat surface. We count it as one surface, because it has no thickness, so if you define one side, you have defined the whole thing. From here on, we will be talking about apex points so that the word point can still be general enough to use in dividing higher dimensional line. Surfaces or faces will be exterior faces to the structure.

The three dimensional simplex is a tetrahedron. It has four apex points, six lines, and four surfaces.

Note that each dimension is represented by adding one point to the simplex for the dimension beneath it. Zero dimension is one point, one dimension is two points, and two dimension is three points. We have to specify in this system that the added point is not in the same dimension as the points before it. The third point is non-co-linear with the first two points. That is, no single straight line can be drawn to connect all three points of the 3simplex.

Now we can make a table to show the relationships between the simplexes of zero, one, two, and three dimensions.

Now we can see the Pascal triangle developing in this analysis of dimensionality. It seems reasonable to assume that we can continue the process started here to find the component parts of higher dimensional simplices. Below I have carried the table a few steps further.

We now have three methods for finding the number of lower dimensional simplices in a higher dimensional simplex. We can use the Pascal triangle, or we can count using the N-choose-k method, or we can calculate using the formula for the N-choose-k method.

Can we extract any meaning from the numbers in the higher dimensionalities? And can we use the meaning to form some kind of visualization? We have seen how a one dimensional 1simplex (line) can be built up from two zero dimensional 0simplices. And we have seen how a two dimensional 2simplex can be built up from three 1simplices. We may easily be able to imagine the three dimensional tetrahedron built up from four 2simplices, joined by three 1simplice lines at four 0simplex apices.

Now how do we go on to visualize the four dimensional structure analogous to the tetrahedron? I have called it a hypertetrahedron here, for the simple reason that I do not know what else to call it. Probably we will have to find new words for these higher dimensional simplices. For now, I think I will use the convention of merely notating them as 5simplex and so on, even though that seems to me to lack a certain desireable elegance.

Now it is possible to represent a 3simplex (tetrahedron) in two dimensions by a number of methods. One can draw a fair perspective drawing, which looks like a three sided pyramid on a triangular base. Of course the base is only distinguished in the perspective, and really it is no different from the other sides. Also in the perspective, to create the illusion of depth, we do not draw all the lines the same length, but the ones that go down into the perspective or up out of it are drawn as shorter lines. That way we can arrange all the apices of the simplex in one two dimensional plane, and it seems quite natural for us to do so.

We should remember however that this method of perspective drawing is not really natural. Instead, it was invented in the renaisance by a scientist, DaVinchi, if i remember my history. He strung a frame with wires in squares, and fixing an observation point, he drew what he saw in each square, thus making a fair perspective. It seems simple to us now, but in the fifteenth century things were different. If you look at portraits of people from before the renaisance, you will see that they were drawn flat. Landscapes from that period look especially odd to us now, but at that time, it was thought natural to show distant objects as small, close objects as large, and no thought was given to the way lines between them looked. Small distant objects might be represented under the feet of a large close figure, so that today it looks to us as if the figure is floating in air above the tiny houses, churches, and people below.

We learned to draw and to view drawings in three dimensional perspective, and later we learned to make a series of such drawings, like a cartoon, to represent the passage of a fourth dimension, time. Now when we watch movies or television, we can see three dimensions on a two dimensional screen, and watch the flow of time as the two dimensional images flicker past us too fast to notice that they do not really, in themselves, one at a time, move at all. In fact, on television, if you understand how the image is formed, you can see that the data stream is really linear, a single line of information that is divided up across your television screen to give the illusion of a two dimensional image. So you see we can actually represent at least up to four dimensions even with a single line of data, a one dimensional object. We just have to know how to look at it.

Then the question, again, is how can we learn to see four dimensional objects? Is there some way to establish a frame and a viewpoint, like DaVinchi did, that will give us vision in 4d?

The viewpoint seems easy enough. A point is a point in any dimension up to four, as we have already seen, so we know what we mean when we choose one well enough. But what is a frame?

We might think of a picture frame, which is a sort of border to enclose and protect a picture. What is inside the frame is not the same thing, usually at least, as what is outside the frame. Inside may be a landscape painting, of mountains, trees, lakes and streams. Outside may be the wall of your living room. That would be a frame in two dimensions. Real frames have thickness, of course, but that is not part of the functionality we are discussing here. A picture of a picture frame is really as much a frame as the frame the picture of the picture frame is hung in. It may have the third dimension, but it doesn't really need it to be itself. Two dimensions are sufficient to a frame of this kind.

But a three dimensional frame needs its thickness as a part of its definition. The third dimension must also be limited. A frame in three dimensions is like a room, perhaps, or like a world, or like a universe. A frame in three dimensions is everything that fits inside some three dimesnional shape. It could be any shape that is convenient to our purpose. We might think of a train as a frame, as Einstein did in Special Relitivity, or as an elevator, as in Einstein's General Relitivity. But the simplest frame in three dimensions is probably the sphere, just as the circle is the simplest frame in two dimensions.

And what then is the four dimensional analog of the sphere?

Often when trying to visualize in four dimensions we think of something like M.C. Escher's strange castles, or Salvatore Dali's cross or his melting timepieces. But perhaps the simplest view of the fourth dimension is as a process, a blurring of the frame of reference, as in a cartoon the artist may draw a ball moving through the air as a sort of blurred series of images of itself, a sort of line that we recognise as the visual effect of an object pursueing a course through a limited amount of time and space: a four dimensional frame. We only see a part of the ball's movement in the picture, not its entire movement history. We only see the ball park, the player and the player's mit, not the entire universe. These regions of timespace are the frame of the drawing.

In any four dimensional visualization we will have to consider a point of view, and a frame enclosing three dimensions of space and one of time.

The tetrahedron fits inside a sphere. The sphere is the simplest representation of a frame in three space. If we want to represent a 3space frame in four space, we have to consider our point of view and for how long we are going to be watching. If we establish a point of view that is moving along with the 3frame sphere, for a limited amount of time, then the sphere does not seem to change at all. It looks just like a sphere in 3d, even though we are looking at it in 4d. But if the sphere is moving relitive to our point of view, then it takes on a blurr. If the movement is directly toward us or away from us, we will see the sphere getting larger or smaller. But if it is moving across our field of view, tangential to us at some distance, then it will appear like some kind of line. Imagine the line of a tracer bullet fired into the air. Your eye keeps the light for a while, and you do not just see a point of light where the bullet is, but you see a line. That is a four dimensional object you are looking at, from your point of view, and in the frame provided by the target range and the length of flight and burn of the tracer round.

A sphere moving in a straight line is probably the simplest image of a 4d object. But it does not reveal much structure, beyond that found in the first dimension. Really it is more. The sphere contains all the structures found in the lower dimensionalities, six lines, four apices, four surfaces. As it moves through time, all of those subspace structures must have continuity. So a line in a certain orientation will trace out a sheet within the line of motion of the sphere. We call this a world sheet. Of course, if the line is moving along its own length, it is still a line, a world line, I suppose.

Be well,

nc

42 views as of last edit

My current work involves identifying stucture in the 4d world of spacetime equivalence. Marcus showed me that structures in spacetime are related to the Pascal triangle, and selfAdjoint helped by pointing out that Pascal's triangle can be calculated from the N-choose-k math. All this is interspersed in Marcus' two-worlds thread on this board.

The Pascal triangle is formed by lining up numbers and then adding the two above on the next line below. The second and subsequent lines also start with the first two numbers, like so:

Code:

```
1,1
1,2,1
1,3,3,1
1,4,6,4,1
```

You see the one and one in the first line add to the two in the second line. In the third line, two plus one is three. In the fourth line, one plus three is four, three plus three is six, and so on.

Marcus and selfAdjoint told us in the Two Worlds thread that you can calculate these numbers by a method known as N-choose-k, in which you have a group of N objects, and you pick them up in as many ways as possible in groups of k. So if you have three N objects, A,B,C, and you pick them up two k ways, obveously you can pick A,B, A,C, or B,C. In this method, A,B is counted as the same group as B,A.

There is a formula for this method, and it involves factorial numbers. These numbers are written as n! where n is a positive integer. 3! is the product of 3 and all the positive integers below it, 1x2x3=6=3!. Another example,

4!=3!x4 is 6x4=24. It is conventional that 0!=1=1!.

The formula for finding the numbers in the Pascal triangle is

[tex]N!/k!(N-k)![/tex]

where N is the row number and k is the column number, counting the first column as zero. For example, in the fourth row of the Pascal Triangle, in the first column, we have 4!/0!(4-0)! is 24/24which equals one. The second number in the fourth row is 4!/1!(4-1)! = 24/3!=24/6=4. And so on.

There is also a tetrahedral form of the Pascal triangle, where there are three ones at the top, and ones around a two in the next layer, and ones around threes around a six in the next layer.

Now to get back to business. There is an idea for thinking of dimensionality in terms of the least structure that can fully represent the dimension in question. For example, as in plane geometry, a point has zero dimension. It has no size that can be measured, and until higher dimensions are invoked, there is no way to talk about where the point is located. The point is called in this system a zero dimensional simplex.

The one dimensional simplex is a line, which has two end points. The only thing you can measure in a line is how long it is, so it is one dimensional. In geometry, a line is thought of as having only one point in cross section, so it is not the same thing as a wire strung between two poles, which has, as well as length, a diameter or thickness. Note also that a line in this sense has to be "straight," because there is no other dimension defined for it to curve into.

The two dimensional simplex is a triangle on a flat plane. It has three apex points, three lines, and one flat surface. We count it as one surface, because it has no thickness, so if you define one side, you have defined the whole thing. From here on, we will be talking about apex points so that the word point can still be general enough to use in dividing higher dimensional line. Surfaces or faces will be exterior faces to the structure.

The three dimensional simplex is a tetrahedron. It has four apex points, six lines, and four surfaces.

Note that each dimension is represented by adding one point to the simplex for the dimension beneath it. Zero dimension is one point, one dimension is two points, and two dimension is three points. We have to specify in this system that the added point is not in the same dimension as the points before it. The third point is non-co-linear with the first two points. That is, no single straight line can be drawn to connect all three points of the 3simplex.

Now we can make a table to show the relationships between the simplexes of zero, one, two, and three dimensions.

Code:

```
Dim. Simplex s0 s1 s2 s3
zero point 1 0 0 0
one line 2 1 0 0
two triangle 3 3 1 0
three tetrahedron 4 6 4 1
```

Now we can see the Pascal triangle developing in this analysis of dimensionality. It seems reasonable to assume that we can continue the process started here to find the component parts of higher dimensional simplices. Below I have carried the table a few steps further.

Code:

```
Dim. Simplex s0 s1 s2 s3 s4 s5
zero point 1 0 0 0 0 0
one line 2 1 0 0 0 0
two triangle 3 3 1 0 0 0
three tetrahedron 4 6 4 1 0 0
four hyper tet. 5 10 10 5 1 0
five ? 6 15 20 15 6 1
six ? 7 21 35 35 21 7
```

We now have three methods for finding the number of lower dimensional simplices in a higher dimensional simplex. We can use the Pascal triangle, or we can count using the N-choose-k method, or we can calculate using the formula for the N-choose-k method.

Can we extract any meaning from the numbers in the higher dimensionalities? And can we use the meaning to form some kind of visualization? We have seen how a one dimensional 1simplex (line) can be built up from two zero dimensional 0simplices. And we have seen how a two dimensional 2simplex can be built up from three 1simplices. We may easily be able to imagine the three dimensional tetrahedron built up from four 2simplices, joined by three 1simplice lines at four 0simplex apices.

Now how do we go on to visualize the four dimensional structure analogous to the tetrahedron? I have called it a hypertetrahedron here, for the simple reason that I do not know what else to call it. Probably we will have to find new words for these higher dimensional simplices. For now, I think I will use the convention of merely notating them as 5simplex and so on, even though that seems to me to lack a certain desireable elegance.

Now it is possible to represent a 3simplex (tetrahedron) in two dimensions by a number of methods. One can draw a fair perspective drawing, which looks like a three sided pyramid on a triangular base. Of course the base is only distinguished in the perspective, and really it is no different from the other sides. Also in the perspective, to create the illusion of depth, we do not draw all the lines the same length, but the ones that go down into the perspective or up out of it are drawn as shorter lines. That way we can arrange all the apices of the simplex in one two dimensional plane, and it seems quite natural for us to do so.

We should remember however that this method of perspective drawing is not really natural. Instead, it was invented in the renaisance by a scientist, DaVinchi, if i remember my history. He strung a frame with wires in squares, and fixing an observation point, he drew what he saw in each square, thus making a fair perspective. It seems simple to us now, but in the fifteenth century things were different. If you look at portraits of people from before the renaisance, you will see that they were drawn flat. Landscapes from that period look especially odd to us now, but at that time, it was thought natural to show distant objects as small, close objects as large, and no thought was given to the way lines between them looked. Small distant objects might be represented under the feet of a large close figure, so that today it looks to us as if the figure is floating in air above the tiny houses, churches, and people below.

We learned to draw and to view drawings in three dimensional perspective, and later we learned to make a series of such drawings, like a cartoon, to represent the passage of a fourth dimension, time. Now when we watch movies or television, we can see three dimensions on a two dimensional screen, and watch the flow of time as the two dimensional images flicker past us too fast to notice that they do not really, in themselves, one at a time, move at all. In fact, on television, if you understand how the image is formed, you can see that the data stream is really linear, a single line of information that is divided up across your television screen to give the illusion of a two dimensional image. So you see we can actually represent at least up to four dimensions even with a single line of data, a one dimensional object. We just have to know how to look at it.

Then the question, again, is how can we learn to see four dimensional objects? Is there some way to establish a frame and a viewpoint, like DaVinchi did, that will give us vision in 4d?

The viewpoint seems easy enough. A point is a point in any dimension up to four, as we have already seen, so we know what we mean when we choose one well enough. But what is a frame?

We might think of a picture frame, which is a sort of border to enclose and protect a picture. What is inside the frame is not the same thing, usually at least, as what is outside the frame. Inside may be a landscape painting, of mountains, trees, lakes and streams. Outside may be the wall of your living room. That would be a frame in two dimensions. Real frames have thickness, of course, but that is not part of the functionality we are discussing here. A picture of a picture frame is really as much a frame as the frame the picture of the picture frame is hung in. It may have the third dimension, but it doesn't really need it to be itself. Two dimensions are sufficient to a frame of this kind.

But a three dimensional frame needs its thickness as a part of its definition. The third dimension must also be limited. A frame in three dimensions is like a room, perhaps, or like a world, or like a universe. A frame in three dimensions is everything that fits inside some three dimesnional shape. It could be any shape that is convenient to our purpose. We might think of a train as a frame, as Einstein did in Special Relitivity, or as an elevator, as in Einstein's General Relitivity. But the simplest frame in three dimensions is probably the sphere, just as the circle is the simplest frame in two dimensions.

And what then is the four dimensional analog of the sphere?

Often when trying to visualize in four dimensions we think of something like M.C. Escher's strange castles, or Salvatore Dali's cross or his melting timepieces. But perhaps the simplest view of the fourth dimension is as a process, a blurring of the frame of reference, as in a cartoon the artist may draw a ball moving through the air as a sort of blurred series of images of itself, a sort of line that we recognise as the visual effect of an object pursueing a course through a limited amount of time and space: a four dimensional frame. We only see a part of the ball's movement in the picture, not its entire movement history. We only see the ball park, the player and the player's mit, not the entire universe. These regions of timespace are the frame of the drawing.

In any four dimensional visualization we will have to consider a point of view, and a frame enclosing three dimensions of space and one of time.

The tetrahedron fits inside a sphere. The sphere is the simplest representation of a frame in three space. If we want to represent a 3space frame in four space, we have to consider our point of view and for how long we are going to be watching. If we establish a point of view that is moving along with the 3frame sphere, for a limited amount of time, then the sphere does not seem to change at all. It looks just like a sphere in 3d, even though we are looking at it in 4d. But if the sphere is moving relitive to our point of view, then it takes on a blurr. If the movement is directly toward us or away from us, we will see the sphere getting larger or smaller. But if it is moving across our field of view, tangential to us at some distance, then it will appear like some kind of line. Imagine the line of a tracer bullet fired into the air. Your eye keeps the light for a while, and you do not just see a point of light where the bullet is, but you see a line. That is a four dimensional object you are looking at, from your point of view, and in the frame provided by the target range and the length of flight and burn of the tracer round.

A sphere moving in a straight line is probably the simplest image of a 4d object. But it does not reveal much structure, beyond that found in the first dimension. Really it is more. The sphere contains all the structures found in the lower dimensionalities, six lines, four apices, four surfaces. As it moves through time, all of those subspace structures must have continuity. So a line in a certain orientation will trace out a sheet within the line of motion of the sphere. We call this a world sheet. Of course, if the line is moving along its own length, it is still a line, a world line, I suppose.

Be well,

nc

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