I'm in the process of reading The Wave, and in Chapter 5 I have been surprised to encounter an idea I've played with in my mind before: The idea that a line, in its ultimate extension, actually meets itself and forms a circle.
I formed the first parts of this idea in math class in high school a couple years ago. Having drawn a standard 2-dimensional grid, with both an X and Y axis extending in the positive directions (making an "L"), I started thinking about what it meant to say that each axis "extends to infinity". To give myself something tangible to work with, I imagined an "infinity point" at the "end" of each axis, beyond all of the finite numbers.
So, hypothetically, each axis extends to a point, representing "number infinity". Recall that a point on a graph is represented by two numbers, in the format [x,y]. X is horizontal distance from zero and Y is vertical distance from zero. So you can have point [1,1] when you move one to the right and one up and draw a point there, [3,5] when you move right three and up five, and so on. Well, since we are considering infinity as a point on the graph, we can do the same with it. We can imagine a point, [Infinity, Infinity] in the upper right, beyond all of the numbers in both directions.
Well, this gives the graph space a strange and fascinating property! Recalling that lines are infinite in length, any line that is diagonal on this graph must extend to infinity in both X and Y, and therefore touch the point, [Infinity, Infinity]! This implies that a line is indeed curved in nature, but this can only be seen by stepping outside the limitation of finite values. A finite measurement, no matter how large, will always show the lines as straight, but if you instead ignore (set aside) the finite values and extrapolate to where you know the line is heading you find that, to reach toward infinity, the lines must curve. Not only that, but they must curve toward one another. This takes us towards understanding what the Cassiopaeans were trying to explain, I think.
I used only lines passing through [0,0], the origin, here. However, even parallel diagonal lines must also curve toward meeting at infinity. The box represents the limitation of considering only finite values - within the box, the lines must appear straight because they must follow the rules of slope (the ratio of y to x).
However, I have only used one part of the graph, and lines extending in one direction. It becomes more difficult for me to comprehend once I include the negative values along with the positive ones, although it is very fun to try. So I warn that from this point on my reasoning my seem strange(-er) to some, but it my mind it seems to make sense and goes a ways toward understanding what the Cs were saying. First, the symbol the Cs were using to describe "perpendicular realities". Two concentric circles, connected by seven spokes, like a wagon wheel, which the Cs said "Designates union of perpendicular realities.". Like this, as I understand:
If we insist on a finite or linear interpretation, the outer circle must disappear from view when its size exceeds perceivable or imaginable values, or if we stretch our view to keep the large circle within our view, the smaller circle must shrink until it disappears.
So, if we think linearly, the circle disappears from our view; we can't imagine where it goes or what happens. So, let's turn back to the idea of treating infinity like a point. Since the circle expands outward from something we are already using a centerpoint as a reference. So as the circle an spokes expand out forever, they approach infinity. We can consider that since each spoke represents its own perpendicular reality, maybe each of them is kind of like its own axis, like the dimensions of X and Y (and Z, for 3D), except now there are seven dimensions. So as those lines expand outward with the circle, they are all approaching infinity, and therefore they must curve toward each other as they approach this hypothetical intersection. The simplest way to represent this is a sphere. You can imagine one of those beach balls, with a circle at each end, with lines going around the ball connecting the circles. The expanding circle keeps expanding toward infinity but never reaches it, so the "sphere" keeps expanding as the spokes get longer.
My feebly heroic or heroically feeble attempt to draw the concept. Imagine the outer circle expanding as far as you can see in your mind, and try to imagine how space must curve for it to continue toward infinity, following the arrows. Remember that our representation of infinity as a point is symbolic. After all, "There is no point."!
Curiously, we've gone from a circle expanding in 2D space to a sphere expanding in 3D space. We could maybe keep expanding it if only we could imagine a next, 4D shape, then 5D, etc., but I don't think that would useful for producing anything but headaches right now. It may relate, however, to how this expansion, with no beginning or end, can contain everything.
So, anyway, these are my musings which I hope will be useful, somehow, and interesting. Do I have something here? Have I explained/reasoned logically?
[EDIT: Added pictures I forgot to attach when I first posted.]
[EDIT2: Added more pictures which I had meant to add originally.]
I formed the first parts of this idea in math class in high school a couple years ago. Having drawn a standard 2-dimensional grid, with both an X and Y axis extending in the positive directions (making an "L"), I started thinking about what it meant to say that each axis "extends to infinity". To give myself something tangible to work with, I imagined an "infinity point" at the "end" of each axis, beyond all of the finite numbers.
So, hypothetically, each axis extends to a point, representing "number infinity". Recall that a point on a graph is represented by two numbers, in the format [x,y]. X is horizontal distance from zero and Y is vertical distance from zero. So you can have point [1,1] when you move one to the right and one up and draw a point there, [3,5] when you move right three and up five, and so on. Well, since we are considering infinity as a point on the graph, we can do the same with it. We can imagine a point, [Infinity, Infinity] in the upper right, beyond all of the numbers in both directions.
Well, this gives the graph space a strange and fascinating property! Recalling that lines are infinite in length, any line that is diagonal on this graph must extend to infinity in both X and Y, and therefore touch the point, [Infinity, Infinity]! This implies that a line is indeed curved in nature, but this can only be seen by stepping outside the limitation of finite values. A finite measurement, no matter how large, will always show the lines as straight, but if you instead ignore (set aside) the finite values and extrapolate to where you know the line is heading you find that, to reach toward infinity, the lines must curve. Not only that, but they must curve toward one another. This takes us towards understanding what the Cassiopaeans were trying to explain, I think.
I used only lines passing through [0,0], the origin, here. However, even parallel diagonal lines must also curve toward meeting at infinity. The box represents the limitation of considering only finite values - within the box, the lines must appear straight because they must follow the rules of slope (the ratio of y to x).
However, I have only used one part of the graph, and lines extending in one direction. It becomes more difficult for me to comprehend once I include the negative values along with the positive ones, although it is very fun to try. So I warn that from this point on my reasoning my seem strange(-er) to some, but it my mind it seems to make sense and goes a ways toward understanding what the Cs were saying. First, the symbol the Cs were using to describe "perpendicular realities". Two concentric circles, connected by seven spokes, like a wagon wheel, which the Cs said "Designates union of perpendicular realities.". Like this, as I understand:
June 17 said:
If we insist on a finite or linear interpretation, the outer circle must disappear from view when its size exceeds perceivable or imaginable values, or if we stretch our view to keep the large circle within our view, the smaller circle must shrink until it disappears.
same session said:
So, if we think linearly, the circle disappears from our view; we can't imagine where it goes or what happens. So, let's turn back to the idea of treating infinity like a point. Since the circle expands outward from something we are already using a centerpoint as a reference. So as the circle an spokes expand out forever, they approach infinity. We can consider that since each spoke represents its own perpendicular reality, maybe each of them is kind of like its own axis, like the dimensions of X and Y (and Z, for 3D), except now there are seven dimensions. So as those lines expand outward with the circle, they are all approaching infinity, and therefore they must curve toward each other as they approach this hypothetical intersection. The simplest way to represent this is a sphere. You can imagine one of those beach balls, with a circle at each end, with lines going around the ball connecting the circles. The expanding circle keeps expanding toward infinity but never reaches it, so the "sphere" keeps expanding as the spokes get longer.
My feebly heroic or heroically feeble attempt to draw the concept. Imagine the outer circle expanding as far as you can see in your mind, and try to imagine how space must curve for it to continue toward infinity, following the arrows. Remember that our representation of infinity as a point is symbolic. After all, "There is no point."!
same session said:
Curiously, we've gone from a circle expanding in 2D space to a sphere expanding in 3D space. We could maybe keep expanding it if only we could imagine a next, 4D shape, then 5D, etc., but I don't think that would useful for producing anything but headaches right now. It may relate, however, to how this expansion, with no beginning or end, can contain everything.
So, anyway, these are my musings which I hope will be useful, somehow, and interesting. Do I have something here? Have I explained/reasoned logically?
[EDIT: Added pictures I forgot to attach when I first posted.]
[EDIT2: Added more pictures which I had meant to add originally.]
