Look up at the dark clear moonless night-time sky. What do we see? Points of light arranged against a deep dark background. I propose that in the points of light we see physical substance, matter, and in the darkness we are looking into the encompassing ethereal realm.
There are certain fundamental processes in the universe, one of which is expansion and contraction. Goethe observed this in plants and in crystallisation out of solution we see a contraction of a substance into its solid state.
Likewise the points of light we see in the night sky are processes of matter condensing out of the surrounding ethereal space. The ethereal creates a void in which matter forms and cosmic space which is void of matter is actually filled with etheric activity.
The processes of expansion and contraction are taking place at all levels, as above, so below. Our physical eyes allow us to see the stars and other heavenly bodies but it takes more than physical senses to see the etheric.
And don’t you see that generating a circle in this way is no different in essence from any other way in Euclidean geometry, by using a central point and a radius.
keiths:
CharlieM:
Of course it’s different, and no, it doesn’t require the use of a central point and a radius. You simply take a plane that’s perpendicular to the axis of the cone. The intersection of that plane with the cone is a circle.
Charlie, earlier:
Charlie, now:
You haven’t given any reasons for thinking that “the ethereal” even exists, much less that projective geometry gives us insights into its nature.
The “reason” is that Steiner said so. That’s enough.
The ether is substantiated in polar bears. obviously.
The axis of the cone is the centre point of the circle. Anyway its not that important. The fact is that Euclidean geometry was ideally suited to measuring the earth in three dimensions. Descartes took this further which enabled the use of Cartesian coordinates to make precise measurements using relative positions within these right angled planes. He developed analytical geometry from the synthetic geometry of Euclid. Desargues took Euclidean geometry further in the direction of synthetic geometry but it took a back seat to the analytical geometry of Descartes.
Now projective geometry is coming back into prominence. It has expanded on the rigidity of Euclidean geometry with the latter’s starting ‘point’ and reliance on the right angle.
I’ll ask again. The peripheral plane is equally as fundamental as the central point. Do you agree?
Maybe you could answer the question I asked keiths. The peripheral plane is equally as fundamental as the central point. Do you agree?
Very good. A comic brings some light hearted relief to the discussion on conics.
And obviously the distinctive appearance of a polar bear is influenced by the surrounding environment.
Darwinian theory is concerned with the evolution of life and you have highlighted the way in which geometry mirrors the evolution of consciousness.
Euclid gave us the details of how humans developed ways of measuring the world of experience using right angles, points, lines and planes. We see this way of measuring demonstrated in the Old Testament where we read statements such as:
But with the progression to projective geometry our thinking advances from the notion that space is just the container in which matter exists to the idea that spacial measurements are not absolute. We now think of space is much more fluid than used to be taught in the older geometry. The older geometry is the geometry suited to lifeless matter, projective geometry is the geometry suited to living processes. If we want to understand life then we should try to understand projective geometry. Our consciousness is evolving.
To continue from my last post. Euclidean geometry is the geometry of space, projective geometry is the geometry of time.
And the activity in studying living forms such as that of plants can reveal to us the equivalent dual nature of observation. Look at any plant growing in nature and you will see it as it is situated in space. Do what Goethe did and see it with the mind’s eye changing through time and you will see it situated in its metamorphosis. The plant seen through the eyes, and then the plant understood in its becoming are comparable to the study of Euclidean geometry taken further through the study of projective geometry.
Here is an article on tissue regeneration:
This is the abstract:
Some animals have the ability to regenerate certain tissues and plants are masters of regeneration. From the above paper it can be seen that the tissues are regenerated to form structures that were originally formed in the normal development of the animal. The regenerated tissue is formed from a different source and by a different pathway. So it is the finished form that is common in both cases and may not be the originating cells. This is the case in newt lens cells which develop from pigmented cells. Also limbs have been found to regenerate from different cell types than those which form the limbs during normal development.
This is a good indication that these animals retain the formative forces which in more advanced animals weaken as the animal becomes mature. These formative forces are holistic in nature using different parts to achieve the same overall result. The overarching overall form is key to the process. I equate formative forces to ethereal forces working in the opposite direction to expansive growth. These two principles working in polarity shape the living organism.
Please define “fundamental” and “equally as fundamental.”
Shit. I totally forgot about that. Thanks.
They are basic and from which all other geometrical figures can be derived.
Euclid defines a straight line in terms of points, but it could equally be defined in terms of planes.
In terms of planes you say? Bah, what a narrow minded take on lines when you can describe lines in terms of four dimensional equations. Do you realize how much reality you’re missing by restricting yourself to lowly 3D planes?
I am talking about geometry of space that we can experience, not algebra.
Of course you are quite correct, at least mathematicians recognise that there are higher dimensions.
dazz, to CharlieM:
It’s a shame that Steiner was such a pedestrian thinker.
keiths:
CharlieM:
The axis of the cone is a line, not a point. And although the center point of the circle lies on the axis, it is not used in generating the circle. The circle is generated by intersecting the plane with the cone.
And you can’t experience the super-ether reality unless you think in 4 dimensions. Just give it a try and get back to me, you’re gonna shit bricks when you get past 10 dimensions!
LOL, that’s not algebra, it’s still geometry. Shouldn’t you at the very least learn the basics before you start pontificating about “thinking” in terms of this or that mathematical framework?
No, they don’t.
keiths:
CharlieM:
I’m asking you for an argument, not a Socratic dialogue.
You’ve said:
Please present a persuasive argument that a) “the ethereal” even exists, and b) that projective geometry provides insights into its nature.
CharlieM:
Time appears nowhere in the axioms of projective geometry, and it cannot be derived from them.
I’m still reeling from the polar bear thing…..
They bear the truth of the ether.
I’ll get me coat
Get mine too, willya?
To determine if the section is a going to result in a circle you still need to know that the base of the cone produced by the section is perpendicular to the axis of the cone. Anyway IMO it is too petty to spend much time arguing over whether conic sections are classed as Euclidean or projective geometry especially as Euclidean geometry is just a special case of projective geometry where the parts are given more attention than the whole. For projective geometry it is equally true that the line (axis) belongs to the point (centre) as it is that the point lies on the line. That is the nature of its duality.
IMO if there are equations with unknowns then it is algebra.
And there are others who would agree with me:
It would help if instead of constantly arguing about which hole to put the pigeon in, we made an attempt to understand the creature itself.
Dazz in the previous reply
Where did you get the idea of these extra dimensions you are joking about?
I have to admit that I have utterly lost the thread here. Can you please explain–as simply as you can–how you get from any of this stuff about Euclidean geometry to ether?
Premises, valid conclusions, that sort of stuff. Thanks.
CharlieM:
The section doesn’t produce a cone. The section is produced by intersecting the cone and a plane.
Which is why I keep asking you to make your case:
I see that walto is asking for the same thing:
keiths and walto, thankyou for at least reading what I have to say. I am trying to make the case that the ethereal and the physical are two poles which between them give form to living matter. A living organism, like a growing crystal is a condensation of substances from its surroundings. The difference being that the crystal grows by adding extra substance to its surface, while the living organism takes substances into itself, breaks them down and rebuilds from the inside out so to speak. The breaking down is physical and the building back up into specific tissues and organs is a formative process. I can see that the word “ethereal” is off-putting so I’d like to replace it with the phrase, “formative principle”.
I gave a link to an article about lens regeneration in newts which shows that the form generated is not dependent on the way it is built up from bodily processes as these processes do not have to follow the same path for the production of the lens. There is a formative principle at work which is not reliant on the path taken. No one has commented on this.
It’s very difficult to make a case that projective geometry has anything to do with formative principles unless a person actually experiences the thinking involved in practising exercises in this geometry and understanding how the various forms morph into each other and how it is the processes of transformation is more primal than any fixed form. It is not the process itself but the type of thinking required that is the link. Goethe was carrying out that type of thinking when he said that he had discovered the archetypal plant.
In my next post I will link to a video that attempts to explain this in a way that is easy to understand.
Imagine you have a physical cone. You cut through the cone to make a circular section then you will end up with a truncated base and a smaller cone with a circular base. That is what I mean. I like to keep things true to what would happen in the real world as much as possible.
Here is the video I promised. (A Journey into the 4th Dimension – Perspective [Part 1])