# Ethereal spaces

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.

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## 99 thoughts on “Ethereal spaces”

1. CharlieM: “Euclid expresses by mathematical formulae only what can be described and constructed within the field of the “finite.” What I can state in terms of Euclid about a circle, a triangle or about the relations of numbers, is within the field of the finite, it is capable of construction in a sense-perceptible manner”.

Nope. Got that wrong too.

https://primes.utm.edu/notes/proofs/infinite/euclids.html

The Steiner quote was taken from a translation of a transcript of a lecture he gave in Amsterdam in 1904 so we cannot be sure word for word what he actually said. But his meaning is quite clear and you are missing the point (pun intended).

Here is how Euclid put it in Elements Book IX Proposition 20.

Prime numbers are more than any assigned multitude of prime numbers

And then he gave a proof. Basically he is saying that prime numbers continue without end. We also know that Zeno’s paradox gets us thinking about infinity, so the ancient Greeks were thinking about these things. But as stated here, prior to Geog Cantor’s time infinity was:

not a precisely understood mathematical concept

Euclid gives us a truth about prime numbers but he does not use this truth in any calculations concerning observable objects.

What Steiner is saying is that Newton and Leibnitz used the infinite point in their calculus formulae in order to calculate measurements for the slope of a tangent on a graph. In normal circumstances slopes can be calculated by taking the ratio of distances along the x and y axes, But because a tangent only has one point of contact, an infinitely small point, they had to factor in this infinity in their calculations. In other words they used something which is beyond the senses (the infinitesimal point) to obtain something which is observable (the slope of the graph).

Euclid never did anything like this when he laid down ways of dealing with observable geometrical entities. He remained within the observable while Newton and Leibnitz did not.

2. CharlieM: So long as you realise that they can also be taken seriously

That’s true. I’m sure if we were to study Zippy the Pinhead we could gain some information about the mind/s of its inventor/s. It all depends on whether we are reading something for amusement or reading something for the sake of learning and understanding.

3. CharlieM: Euclid never did anything like this when he laid down ways of dealing with observable geometrical entities. He remained within the observable while Newton and Leibnitz did not.

One of Euclid’s theorems is the “Triangle Angle Sum Theorem”, which proves that the internal angles of a triangle sum to 180 degrees.

As an abstraction, the theorem is true, despite the fact that it is not true for any observable triangle. That is to say, it is impossible to draw an actual triangle (an observable one) whose internal angles actually sum to 180 degrees.

4. timothya: One of Euclid’s theorems is the “Triangle Angle Sum Theorem”, which proves that the internal angles of a triangle sum to 180 degrees.

As an abstraction, the theorem is true, despite the fact that it is not true for any observable triangle. That is to say, it is impossible to draw an actual triangle (an observable one) whose internal angles actually sum to 180 degrees.

If it was Steiner’s contention that Euclid never broached the subject of infinity then he was wrong about that. Obviously the series of prime numbers is just one example of an infinite progression.

But in his constructions Euclid always stayed within the sense perceptible. Whereas Newton and Leibnitz in constructing a tangent to a curve had to factor into the equation the dimensionless point.

Euclid defined points, lines and so on in terms that can never be sense perceptible objects but then he gave instructions as to how to construct them using a straight edge and a compass. So his instructions were in how to construct representations of the things as defined.

I don’t know about you but in school I was taught to construct figures using a ruler, compass and protractor and I measured triangle to have internal angles that added up to 180 degrees. I know that this is an approximation that could never be exact but, for practical purposes it was good enough for Euclid and it was good enough for me.

5. CharlieM: I don’t know about you but in school I was taught to construct figures using a ruler, compass and protractor and I measured triangle to have internal angles that added up to 180 degrees. I know that this is an approximation that could never be exact but, for practical purposes it was good enough for Euclid and it was good enough for me.

No doubt if you took calculus, your teacher would have put drawings on the blackboard that would also have been helpful and “good enough for you.”

We’ve agreed that Steiner was wrong in what he said about Euclid. Next step is to see that the implications he and you want to draw from that false premise are also mistaken. It’s not much, but it’s a start.

6. walto: No doubt if you took calculus, your teacher would have put drawings on the blackboard that would also have been helpful and “good enough for you.”

By the time we were being taught calculus I had lost interest in school and I was absent more than I was present. So any teaching on the subject that I did receive was not really good enough for me to get a good grasp of it at the time.

We’ve agreed that Steiner was wrong in what he said about Euclid. Next step is to see that the implications he and you want to draw from that false premise are also mistaken. It’s not much, but it’s a start.

The next step would be to figure out if there is anything Steiner did say that you agree with. For instance what would you say about this passage

What I can state in terms of Euclid about a circle, a triangle or about the relations of numbers, is within the field of the finite, it is capable of construction in a sense-perceptible manner. This is no longer possible with the Differential Calculus with which Newton and Leibnitz taught us to reckon. The Differential still possesses all the properties that render it possible for us to calculate with it; but in itself as such, it eludes sense-perception. In the Differential, sense-perception is brought to a vanishing point and then we get a new basis — free from sense-perception — for our reckoning.

That quote begins from the part of the Steiner passage that you included here but your link did not touch upon it.

And there is a very important point to be considered in that link. Euclid did not use numbers in the abstract way that we do today. He understood them as count of units of length and he gives an excellent proof that prime numbers never terminate, however many prime number you can think of there will always be more to reckon with. (And don’t forget that for him prime numbers are quantities of units of length). But he does not make any practical use of this proof. The proposition is not used in the rest of the Elements.

And this is the main point that Steiner is making. Newton and Leibnitz use the imperceptible for practical purposes, Euclid does not do this.

7. In the book, “Catching the Light, The Entwined History of Light and Mind”, Arthur Zajonc discusses our progress from the static thinking of the past to a more dynamic,fluid thinking. He writes:

I would suggest that we see relativity theory as a reflection of an evolutionary movement of the human psyche towards true autonomy. The inflexible geometry of past science reflected the restricted structure of its imaginative base. By contrast the dynamic imagination of modern science liberates, but also endangers. The stability we once found without, we must now find within One of the most beautiful examples of this change occurred around the world’s geometry. Not since Brunellesschi invented linear perspective had the geometry of human experience undergone such a radical shift.
In the decades leading up to the period of relativity theory, the very architecture of space was revolutionized. Until then the mathematical imagination, and with it all scientific thinking, had been dominated by a single book. No text, other than the Bible, had done more to shape the thinking of the West than Euclid’s Elements. Since the invention of printing alone, more than one thousand editions have appeared. Yet the mathematical framework the Elements espoused grants an unfounded privilege to one view, excluding the very idea of non-Euclidean geometries.The roots of a more flexible attitude to geometry reach back to the Renaissance creators of linear perspective, but the development of their first insights into the modern discipline of “projective geometry” had to await the work of great mathematicians such as Poncelet, Cayley, and Klein. By the time of Einstein, non-Euclidean geometries and the even more comprehensive theory of projective geometry had broken the grip of Euclid on mathematical and spacial thinking, and a new imagination of space could be born.
To show something of the flavour offered by the new geometry, consider the circle and cube of Euclidean geometry. Once given, theses forms are fixed. The curvature of a circle is absolutely uniform and determined be its particular radius; similarly, the corner angles of a cube are always ninety degrees and all sides are of equal length. In projective geometries these invariants disappear. The circle and cube can undergo an infinity of metamorphic changes. The figure below exemplifies but one of the possibilities; you must imagine the multitude of others not drawn.

There is an image of a projective geometry structure in the book. He continues:

Olive Whicher’s lovely drawing allows us to see geometric forms as crystallizations born from light. Rays, like beacons, stream through space in an orderly fashion; their intersections defining and lengths embracing familiar geometrical figures. In addition, theses rays must be imagined in motion, the figures ever-changing, so that all stasis disappears. The space of forms is entirely mobile, in flux, a geometry of streaming metamorphic life, and mathematics is alive with the vital force of the modern imagination.

In the book he also quotes Einstein:

“It seems the human mind has first to construct forms independently before we can find them in things”, Albert Einstein.

In contrast to many skeptics he dared to believe, “that pure thought can grasp reallity, as the ancients dreamed.”

The expansion of Euclidean geometry into projective geometry, of rigid formulae for areas and volumes to the mobility of calculus, the image of DNA as sticks and beads twisted together to the realisation of the dynamic intra-cellular molecular activity; all this required a transformation of mind from holding dead thoughts to acquiring living thinking.

Researches in physics made this type of transformation in thinking a necessity. It is inevitable that biology will also follow suit, but in its own way.

8. CharlieM: The next step would be to figure out if there is anything Steiner did say that you agree with

I’m pretty sure I can find something to agree with in everybody who has ever written anything. That’s a very low bar. But to satisfy you, I’ll find some passage I like in Steiner and report back to you within the next couple of days.

The thing is, he wrote SO MUCH GARBAGE!

9. Many words, but little content.

Four people are given a copy of Euclid’s Elements in its Original Greek to examine. The first person is blind, the second is monolingual in English, the third can read the text but has very little mathematical understanding, and the fourth is an expert mathematician and can read ancient Greek.

The content of the book has vastly differing value depending on who is studying it.

10. CharlieM: The next step would be to figure out if there is anything Steiner did say that you agree with

I’m pretty sure I can find something to agree with in everybody who has ever written anything. That’s a very low bar. But to satisfy you, I’ll find some passage I like in Steiner and report back to you within the next couple of days.

The thing is, he wrote SO MUCH GARBAGE!

I was meaning anything he said in the passage I quoted, here, and that is why I suggested a specific section of it for you to give your opinion on.

But if you you do not want to stick with this passage and are prepared to read any of Steiner with as little preconceived ideas and prejudice as you can manage then carry on.

11. graham2:
Has Charlie explained what ‘ethereal’ is yet ?

In short there is a polarity between point-wise and plane-wise. Point-wise is to the physical as plane-wise is to the ethereal. The former is easier to quantify and measure because of its contractive, static, rigid nature. The latter is more difficult to quantify because of its expansive, mobile, fluidity. But we can determine its nature by the use of projective geometry. The former is more to do with matter building the latter form building.

12. CharlieM: I’m pretty sure I can find something to agree with in everybody who has ever written anything. That’s a very low bar. But to satisfy you, I’ll find some passage I like in Steiner and report back to you within the next couple of days.

The thing is, he wrote SO MUCH GARBAGE!

I was meaning anything he said in the passage I quoted, here, and that is why I suggested a specific section of it for you to give your opinion on.

But if you you do not want to stick with this passage and are prepared to read any of Steiner with as little preconceived ideas and prejudice as you can manage then carry on.

Preconceived, Shmeconseived.

I have no problem with this:

Economic life has recently, simply of its own accord, taken on quite new forms. Through one-sided activity it has asserted undue power and weight in human life.

But again, a good 87% of what the guy wrote was absolute garbage. Read better stuff.

13. I asked for a definition of ‘etherial and got this:

… Point-wise is to the physical as plane-wise is to the ethereal. …

World to Charlie: you don’t use the word being defined as a description of the word being defined. And word salad doesn’t describe anything anyway.

14. graham2:

I asked for a definition of ‘etherial and got this:

… Point-wise is to the physical as plane-wise is to the ethereal. …

World to Charlie: you don’t use the word being defined as a description of the word being defined. And word salad doesn’t describe anything anyway.

I was tempted to ask for a demonstration of this:

But we can determine its nature by the use of projective geometry.

It might be entertaining, but it might also be excruciating.

15. Like Scientology, knowledge of it could be fatal, so I would recommend caution.

16. Ooh, don’t go there. CharlieM gets upset when anyone compares Steiner to L. Ron Hubbard.

17. walto: No doubt if you took calculus, your teacher would have put drawings on the blackboard that would also have been helpful and “good enough for you.”

I have not followed all (actually any) of what Charlie M is saying, but as best I can tell from people who are quoting him, there is a confusion between mathematics and reality.

Theorems about the nature of triangles depend on the set of axioms one uses. Under Euclidean axioms one can prove that the interior angles of triangles sum to 180 degrees. Under other axiom systems, they do not. This all has to do with the (in)famous parallel postulate, of course.

Which axioms of mathematics apply to real spacetime at cosmological scales? That’s a problem for physics. I think the current view is that spacetime at those scales is flat (Euclidean). But I have not researched that and would be happy to see any informed corrections to my view.

There is a problem with that advice. In order to pick out what is ‘better stuff’ much more reading is required. If we have determined better stuff before we have read it then we are following our preconceived opinions, which is not a good thing.

Reading on the internet is like prospecting for gold. We have to dig through masses of worthless dirt in order to find the gold, but it can be very worthwhile and prove very rewarding if we are prepared to do the digging.

19. graham2: I asked for a definition of ‘etherial and got this:

… Point-wise is to the physical as plane-wise is to the ethereal.

World to Charlie: you don’t use the word being defined as a description of the word being defined. And word salad doesn’t describe anything anyway.

I have used other descriptive words that you do not mention. Words like, overarching, dynamic, formative, inward working planar forces.

Asking for a definition of the ethereal is like asking me to define another person.. No definition will come anywhere near getting to know that person through personal contact and regular interaction in their everyday lives.

If even dictionary definitions don’t do it justice, why do you think that I can define it to your satisfaction with a few words?

We can explain (explane) the forms of matter by pointing to the ethereal 🙂

20. keiths: I was tempted to ask for a demonstration of this:

But we can determine its nature by the use of projective geometry.

It might be entertaining, but it might also be excruciating

It needn’t be excruciating. For a start do you agree that Euclidean geometry is an example of an instance of projective geometry which is frozen in a single perspective? For example do you agree that a square in Euclidean geometry becomes an infinitely changing form in projective geometry? What appears as static becomes dynamic.

21. graham2:
Like Scientology, knowledge of it could be fatal, so I would recommend caution.

Life is fatal but we just get n with it anyway 🙂

22. keiths:
Ooh, don’t go there.CharlieM gets upset when anyone compares Steiner to L. Ron Hubbard.

I don’t mind comparisons, so long as they are educated comparisons 🙂

23. BruceS: I have not followed all (actually any) of what Charlie M is saying, but as best I can tell from people who are quoting him, there is a confusion between mathematics and reality.

Can you expand on that? If the confusion is mine then I want to know about it.

Theorems about the nature of triangles depend on the set of axioms one uses.Under Euclidean axioms one can prove that the interior angles of triangles sum to 180 degrees.Under other axiom systems, they do not.This all has to do with the (in)famous parallel postulate, of course.

Which axioms of mathematics apply to real spacetime at cosmological scales?That’s a problem for physics.I think the current view is that spacetime at those scales is flat (Euclidean). But I have not researched that and would be happy to see any informed corrections to my view.

I am happy if we are living in a flat universe. Davide Castelvecchi, the author in your link deems as strange something which I would say is not strange and is only to be expected. He writes:

Our intuition tells us that every square should close. The world is far stranger than our intuition would have us believe.

This is in relation to instructions he gives to walk and trace out the edges of a square so that we return to our starting point. This is a good example of how Euclidean geometry can be seen in relation to projective geometry. I would say projective geometry is closer to reality.

It is true that if I follow his instructions I will end up at my starting point. But only in relation to the surface I am walking on. Relative to the movement of the earth through space it is not true. My starting point has moved on somewhat.

For those of us who live in the UK and watch channel 4 there is a good example of this when they show their logo between programmes. The camera angle shows a group of disconnected objects, but when the camera pans round the objects are seen in an alignment which displays the figure 4. What is seen as a closed figure is in reality unattached objects.

With our eyes we see things from a certain perspective, but with our minds, especially when they are open, we can see things from other perspectives and get a better grasp of reality.

24. CharlieM,

For example do you agree that a square in Euclidean geometry becomes an infinitely changing form in projective geometry? What appears as static becomes dynamic.

Projective geometry is no more inherently dynamic than Euclidean geometry is.

To picture this, imagine a bright light shining on a wire-frame square and projecting its shadow onto a flat screen. The shadow is fixed. You can change it by moving the light, the square, or the screen, but then the change is coming from you. It isn’t inherent in the projective geometry.

Likewise, suppose you take the wireframe square and bend it so that it becomes a rhombus. The figure has changed, but it isn’t because Euclidean geometry is inherently dynamic. It’s because you altered the figure.

25. CharlieM,

For example do you agree that a square in Euclidean geometry becomes an infinitely changing form in projective geometry? What appears as static becomes dynamic.

Projective geometry is no more inherently dynamic than Euclidean geometry is.

To picture this, imagine a bright light shining on a wire-frame square and projecting its shadow onto a flat screen. The shadow is fixed. You can change it by moving the light, the square, or the screen, but then the change is coming from you. It isn’t inherent in the projective geometry.

Geometry is a tool, it is a human activity. I would hope we all understand that. The important point is our way of thinking when we practice these geometries.

When we practice Euclidean geometry we treat the point as fundamental and then lines and planes are defined in terms of points. When we practice projective geometry this is not the case. Here planes and lines are equally fundamental and each can be generated from the other, See here for an example of a plane being generated from straight lines.

A line can be thought of as the intersection of two planes and a point can be seen as the intersection of three planes or two lines.

Euclidean geometry with its measurements of lengths and angles is ideal for disciplines such as engineering where structure is important. Projective geometry is not concerned with measuring and counting but deals with the generation and relationships of forms. So Euclidean geometry is suited to lifeless matter whereas projective geometry lends itself more to the study of life. The mobility of thinking required to practice projective geometry is good training for the study of changing forms of living systems.

Likewise, suppose you take the wireframe square and bend it so that it becomes a rhombus. The figure has changed, but it isn’t because Euclidean geometry is inherently dynamic. It’s because you altered the figure.

And by the standards of Euclidean geometry it will remain a rhombus until it is altered from without. (unless you are using some kind of memory metal).

But by using projective geometry we can change the rhombus into a square and back to a rhombus again simply by changing our viewing point. For Euclidean geometry a square and a rhombus are two distinct figures, for projective geometry they are two aspects of the same figure. The former reduces the subject to its parts the latter views them holistically.

26. CharlieM:

The important point is our way of thinking when we practice these geometries.

That isn’t what you said earlier. There you claimed that Euclidean geometry itself is static, whereas projective geometry is dynamic…

For a start do you agree that Euclidean geometry is an example of an instance of projective geometry which is frozen in a single perspective? For example do you agree that a square in Euclidean geometry becomes an infinitely changing form in projective geometry? What appears as static becomes dynamic.

…which is incorrect.

keiths:

Likewise, suppose you take the wireframe square and bend it so that it becomes a rhombus. The figure has changed, but it isn’t because Euclidean geometry is inherently dynamic. It’s because you altered the figure.

CharlieM:

And by the standards of Euclidean geometry it will remain a rhombus until it is altered from without…

But by using projective geometry we can change the rhombus into a square and back to a rhombus again simply by changing our viewing point.

In both cases, the figure remains static unless you change something. Neither Euclidean nor projective geometry is inherently dynamic, as I’ve already explained.

For Euclidean geometry a square and a rhombus are two distinct figures, for projective geometry they are two aspects of the same figure.

No, the projections themselves are figures in their own right, and they’re distinct projections of the same figure onto distinct planes.

27. CharlieM:

The important point is our way of thinking when we practice these geometries.

That isn’t what you said earlier. There you claimed that Euclidean geometry itself is static, whereas projective geometry is dynamic…

That’s why I’m glad you’re back 🙂 You get me to think more carefully about what I am trying to say. So let me put things another way. Projective geometry is about transformations and polarities. We have seen how the movement of a line generates an ellipse. Various forms can be generated by projections of a moving point of perspective as in the video I linked to. We can take a quadrilateral and project it so that we can see the edges and angles changing and we can stop the movement when the projection assumes the shape of a square as defined by Euclid, (“a square is that which is both equilateral and right-angled”). If the quadrilateral that we began with was a square then there is only a single orientation of the projected plane in which the figure obtained will be a square and that is at right angles to line from the point of viewing (the centre of perspectivity) to the primary vanishing point. Lines in projective geometry extend to infinity, lines in Euclidean geometry are more limited and bounded by points.

For a start do you agree that Euclidean geometry is an example of an instance of projective geometry which is frozen in a single perspective? For example do you agree that a square in Euclidean geometry becomes an infinitely changing form in projective geometry? What appears as static becomes dynamic.

…which is incorrect.

Okay, so do you agree that practicing projective geometry encourages us to think in terms of movement and how figures change whereas practicing Euclidean geometry is more about defining the properties of figures and thinking about how they are constructed?

keiths:

Likewise, suppose you take the wireframe square and bend it so that it becomes a rhombus. The figure has changed, but it isn’t because Euclidean geometry is inherently dynamic. It’s because you altered the figure.

CharlieM:
And by the standards of Euclidean geometry it will remain a rhombus until it is altered from without…

But by using projective geometry we can change the rhombus into a square and back to a rhombus again simply by changing our viewing point.

In both cases, the figure remains static unless you change something. Neither Euclidean nor projective geometry is inherently dynamic, as I’ve already explained.

Well yes, It’s all about the point of view of the observer. Changing perspective is a major part of practising projective geometry in the same way that measurement is a major part of Euclidean geometry.

CharlieM: For Euclidean geometry a square and a rhombus are two distinct figures, for projective geometry they are two aspects of the same figure.

No, the projections themselves are figures in their own right, and they’re distinct projections of the same figure onto distinct planes.

Yes but the importance lies in the relationships. For example, projective geometry differs from Euclidean geometry in that parallel lines meet at infinity and the only way to understand how parallel lines meet at infinity is to imagine the lines moving in relation to each other. As soon as this happens the point of intersection can be measured as being a finite distance along the lines. As the angle of incidence increases the point of intersection moves closer to the centre of rotation. If we imagine two horizontal, parallel lines, we can envision the top one rotating clockwise around a point, the point of intersection will move through the field of vision from right to left. It will disappear on the left and reappear on the right as it passes through infinity.

Your statement that, “the projections themselves are figures in their own right”, is worth thinking about in more detail and I might comment on it further if I don’t get distracted.

28. CharlieM,

We have seen how the movement of a line generates an ellipse. Various forms can be generated by projections of a moving point of perspective as in the video I linked to.

You can dynamically generate an ellipse in Euclidean geometry, too. All you need to do is a) specify the two foci; b) specify a third non-collinear point; c) form a triangle T with the three points as its vertices; and d) generate all possible triangles having the same perimeter as triangle T while sharing the two foci as vertices. Connect all the non-focal vertices and you will have an ellipse.

The physical equivalent of this is perhaps easier to visualize. Pound a couple of nails into a flat piece of wood, one at each focus. Place a loop of string over the nails. (The loop must be longer than twice the distance between the foci.) Stick a pencil inside the loop and pull it taut. Move the pencil all the way around the two foci while keeping the string taut. The curve traced will be an ellipse.

Lines in projective geometry extend to infinity, lines in Euclidean geometry are more limited and bounded by points.

You’re confusing lines with line segments. Lines are infinitely long in Euclidean geometry, which is why non-parallel lines are guaranteed to intersect. Line segments can be non-parallel without intersecting.

Okay, so do you agree that practicing projective geometry encourages us to think in terms of movement and how figures change whereas practicing Euclidean geometry is more about defining the properties of figures and thinking about how they are constructed?

No, because movement is no more inherent to projective geometry than it is to Euclidean geometry. I already showed how you could generate an ellipse dynamically using Euclidean geometry. Here’s another example: take the upper vertex of the first triangle below and “slide” it infinitely far to the right while maintaining the same height. Along the way, you will form a series of triangles including the triangle on the right. All of the triangles you form — and there will be infinitely many — will have the same area, since the base and height remain the same. This is true even when the perimeter becomes thousands or millions of times that of the original triangle.

29. Charlie,

Suppose Steiner had argued for the dynamism of Euclidean geometry over projective geometry, rather than vice-versa. Would you now be arguing that Steiner got it wrong?

I highly doubt it. I think it’s the source that matters to you, not the content.

The Dear Leader Must Not Be Questioned.

30. keiths: No, the projections themselves are figures in their own right, and they’re distinct projections of the same figure onto distinct planes.

Euclid describes a circle as a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Those who study projective geometry describe a circle, an ellipse a parabola or a hyperbola. in terms of a conic sections.

For Euclid the separate figures are distinct but for those who practise projective geometry the conic section is the primary concept from which the figures are generated.

Euclid sees only one way of producing a circle, point-wise, whereas projective geometry can be used to form a circle from either of two directions, as a series of points composing a line as in Euclid or as a series of lines oriented about points, point-wise and plane-wise. This adds a dimension not found in Euclid.

The figures that make up conic sections can be seen as distinct but when the connection between them is understood then our knowledge increases accordingly. The aim of education is to understand subjects by making connections. Frog spawn, tadpoles and frogs can be seen as distinct entities but in reality they are all one creature.

Transformation is to projective geometry what metamorphosis is to frogs and what evolution is to life. As above, so below.

31. CharlieM,

We have seen how the movement of a line generates an ellipse. Various forms can be generated by projections of a moving point of perspective as in the video I linked to.

You can dynamically generate an ellipse in Euclidean geometry, too. All you need to do is a) specify the two foci; b) specify a third non-collinear point; c) form a triangle T with the three points as its vertices; and d) generate all possible triangles having the same perimeter as triangle T while sharing the two foci as vertices. Connect all the non-focal vertices and you will have an ellipse.

The physical equivalent of this is perhaps easier to visualize. Pound a couple of nails into a flat piece of wood, one at each focus. Place a loop of string over the nails. (The loop must be longer than twice the distance between the foci.) Stick a pencil inside the loop and pull it taut. Move the pencil all the way around the two foci while keeping the string taut. The curve traced will be an ellipse.

And that demonstrates that Euclidean geometry is equivalent to projective geometry applied to flat planes and solids. It is a limited form of projective geometry.

An actual example from the Elements that you could have given is definitions 19 and 20:

19. The axis of the cone is the straight line which remains fixed and about which the triangle is turned.
20. And the base is the circle described by the straight in which is carried round.

From the accompanying guide:

Euclid knew a parabola as a “section of a right-angled cone.” It was Apollonius who named them ellipse, parabola, and hyperbola.

Euclid didn’t investigate conic sections any further.

Lines in projective geometry extend to infinity, lines in Euclidean geometry are more limited and bounded by points.

You’re confusing lines with line segments. Lines are infinitely long in Euclidean geometry, which is why non-parallel lines are guaranteed to intersect. Line segments can be non-parallel without intersecting.

Here is Euclid on lines:

Definition 2. line is breadthless length.
Definition 3. The ends of a line are points.

He does not say that lines need necessarily be infinitely long.

Okay, so do you agree that practicing projective geometry encourages us to think in terms of movement and how figures change whereas practicing Euclidean geometry is more about defining the properties of figures and thinking about how they are constructed?

No, because movement is no more inherent to projective geometry than it is to Euclidean geometry. I already showed how you could generate an ellipse dynamically using Euclidean geometry. Here’s another example: take the upper vertex of the first triangle below and “slide” it infinitely far to the right while maintaining the same height. Along the way, you will form a series of triangles including the triangle on the right. All of the triangles you form — and there will be infinitely many — will have the same area, since the base and height remain the same. This is true even when the perimeter becomes thousands or millions of times that of the original triangle.

This is an application of projective geometry to a Euclidean triangle. If Euclid demonstrated moving the apex of a triangle through infinity then can you give me a link?

The fact that any triangle in this series will have the same area is obvious because the formula half base times height has nothing to do with the length of any other side.

32. keiths:
Charlie,

Suppose Steiner had argued for the dynamism of Euclidean geometry over projective geometry, rather than vice-versa.Would you now be arguing that Steiner got it wrong?

I highly doubt it.I think it’s the source that matters to you, not the content.

The Dear Leader Must Not Be Questioned.

Yes I would be arguing that he got it wrong. But I don’t know what he thought of the relationship apart from the fact that he regarded Euclid as a great teacher of geometric truths. In fact I don’t know if Steiner ever said or wrote anything comparing Euclidean geometry to projective geometry. I do know that he was very enthusiastic about synthetic geometry, and both of the above are forms of synthetic geometry.

He regarded the study of synthetic geometry as great way of training the mind and Euclidean geometry as the best place to start.

33. CharlieM:

Yes I would be arguing that he got it wrong.

I highly doubt that. You weren’t even willing to admit that he got tomatoes and black people wrong, remember?

Steiner:

Tomatoes have no desire to step outside of themselves, no desire to step outside the realm of strong vitality. That’s where they want to stay. They are the least social beings in the entire plant kingdom. They do not want anything from strangers, and above all, they do not want any fertilizer that has gone through a composting process; they reject all that. This is the reason that they can influence what works independently within the human or animal organism.

And:

In the Negro the posterior brain is specially developed. That goes through the spinal cord and can work over all the light and warmth that is in him.

Hence all that is connected with the body and metabolism is strongly developed in the Negro. He has, as one says, a strong desire-life, instinctive life. And since he actually has the sun-like, light and warmth, on the surface of his skin, his whole metabolism proceeds as if there were a cooking by the sun itself in his interior. Hence comes his desire-life. There is really a continuous cooking going on within him, and what stokes the fire is the posterior brain.

Sometimes man’s organization throws off further byproducts. That is to be seen just in the Negro. The Negro not only has this cooking in his organism, it not only boils there, but he also has a frightfully crafty and observant eye. He peers craftily and very observantly. You can easily take this as a contradiction. But it is like this: If there in front is the nerve of the eye [see drawing], the nerves go just into the posterior brain; they cross there [see drawing]. The nerve goes into the posterior brain, and since that is specially developed in the Negro therefore he peeps out so craftily, is such a sly observer of the world. If one begins to understand the matter, it all becomes clear. But modern science does not make such studies as we do and so it knows nothing about these things.

34. Exactly.

I say read other (better) thinkers, and Charlie balks. I’m just prejudiced against Steiner, he says. There’s not the slightest willingness even to contemplate the possibility that the guy is widely seen as a quack by academia simply because he WAS a quack. The lamb’s bladders and tomatoes all must be ignored because he wrote a few passages that make Charlie feel good.

We see this desperate need for gospel not only in Bible-thumpers.if they give it up….what will they be left with?

I mean, I have my own heroes too. Everyone does, probably. What’s hard is to accept that none of them carried the golden tablets–even the ones whose writings dispel our anxieties. Charlie likes Steiner. Fine. But the jury has long been in on that guy.–and it was not a kangaroo court.

35. And this is important….why?

Just as it is important for the body to have a healthy diet and to exercise so I believe it is important for the health of the mind to take in information and stretch the mind. I know that my thinking will be challenged here and I’ll need to expend some effort to figure out if I am justified in my thinking. I know there is a good deal of selfishness involved in this, but that’s just the way it is.

And of course I believe that learning geometry is important if we want to understand the world around us.

36. CharlieM:
Yes I would be arguing that he got it wrong.

I highly doubt that. You weren’t even willing to admit that he got tomatoes and black people wrong, remember?

Steiner:

Tomatoes have no desire to step outside of themselves, no desire to step outside the realm of strong vitality. That’s where they want to stay. They are the least social beings in the entire plant kingdom. They do not want anything from strangers, and above all, they do not want any fertilizer that has gone through a composting process; they reject all that. This is the reason that they can influence what works independently within the human or animal organism.

And:

In the Negro the posterior brain is specially developed. That goes through the spinal cord and can work over all the light and warmth that is in him.

Hence all that is connected with the body and metabolism is strongly developed in the Negro. He has, as one says, a strong desire-life, instinctive life. And since he actually has the sun-like, light and warmth, on the surface of his skin, his whole metabolism proceeds as if there were a cooking by the sun itself in his interior. Hence comes his desire-life. There is really a continuous cooking going on within him, and what stokes the fire is the posterior brain.

Sometimes man’s organization throws off further byproducts. That is to be seen just in the Negro. The Negro not only has this cooking in his organism, it not only boils there, but he also has a frightfully crafty and observant eye. He peers craftily and very observantly. You can easily take this as a contradiction. But it is like this: If there in front is the nerve of the eye [see drawing], the nerves go just into the posterior brain; they cross there [see drawing]. The nerve goes into the posterior brain, and since that is specially developed in the Negro therefore he peeps out so craftily, is such a sly observer of the world. If one begins to understand the matter, it all becomes clear. But modern science does not make such studies as we do and so it knows nothing about these things.

To argue for or against the above statements would take a lot of knowledge of the background context. I don’t see the relevance to the topic and the point of doing this here although I would say that you are right to have concerns about some of these statements.

I would prefer to discuss the ethereal, space, counter-space, polarity, infinity and other such topics relevant to the thread.

37. Exactly.

I say read other (better) thinkers, and Charlie balks. I’m just prejudiced against Steiner, he says. There’s not the slightest willingness even to contemplate the possibility that the guy is widely seen as a quack by academia simply because he WAS a quack. The lamb’s bladders and tomatoes all must be ignored because he wrote a few passages that make Charlie feel good.

We see this desperate need for gospel not only in Bible-thumpers.if they give it up….what will they be left with?

I mean, I have my own heroes too. Everyone does, probably. What’s hard is to accept that none of them carried the golden tablets–even the ones whose writings dispel our anxieties. Charlie likes Steiner. Fine. But the jury has long been in on that guy.–and it was not a kangaroo court.

38. CharlieM:

To argue for or against the above statements would take a lot of knowledge of the background context.

When I originally quoted that tomato passage to you, we had the following amusing exchange:

keiths:

I have subdivided that Steiner passage into numbered “verses”. For each of them, could you tell us

a) whether you believe the “verse” is true; and
b) why or why not?

Here’s the passage:

1) Tomatoes have no desire to step outside of themselves,

2) no desire to step outside the realm of strong vitality.

3) That’s where they want to stay.

4) They are the least social beings in the entire plant kingdom.

5) They do not want anything from strangers,

6) and above all, they do not want any fertilizer that has gone through a composting process; they reject all that.

7) This is the reason that they can influence what works independently within the human or animal organism.

CharlieM:

Sorry but I don’t know tomatoes well enough to answer any of these questions.

keiths:

So as far as you’re concerned, the jury’s still out on whether tomatoes “desire to step outside of themselves” or “want anything from strangers”?

LOL.

Now, you write:

I don’t see the relevance to the topic and the point of doing this here although I would say that you are right to have concerns about some of these statements.

The relevance is that you’re looking to have your thinking challenged here, as you stated above. One of the biggest problems with your thinking, in my opinion, is your reluctance to challenge the Dear Leader, as demonstrated by the exchange I just quoted. I’m not going to belabor the point here, but if you want to improve your thinking, this is an issue you’ll need to address.

39. keiths:

You’re confusing lines with line segments. Lines are infinitely long in Euclidean geometry, which is why non-parallel lines are guaranteed to intersect. Line segments can be non-parallel without intersecting.

CharlieM:

Here is Euclid on lines:

Definition 2. line is breadthless length.
Definition 3. The ends of a line are points.

He does not say that lines need necessarily be infinitely long.

Just as Darwin isn’t the final word on evolutionary theory, Euclid isn’t the final word on Euclidean geometry, despite its name.

Euclid’s terminology could be confusing and his axioms were insufficient. These flaws have been corrected in modern Euclidean geometry.

Here’s a pictorial representation of the modern terminology:

40. Charlie,

This all started because you said:

But we can determine its nature [the nature of the ethereal] by the use of projective geometry.

So far you’ve made some questionable assertions about the inherent dynamism of projective geometry, but you haven’t begun to justify your claim above or to embark on a determination of the nature of the ethereal.

Please do so, but keep in mind that justifying and asserting are two different things, as are determining and asserting.

41. keiths: The relevance is that you’re looking to have your thinking challenged here, as you stated above. One of the biggest problems with your thinking, in my opinion, is your reluctance to challenge the Dear Leader, as demonstrated by the exchange I just quoted. I’m not going to belabor the point here, but if you want to improve your thinking, this is an issue you’ll need to address.

I’ll answer this briefly but I don’t want to dwell on it as it will take us too far off topic and it would be more suitable in a thread of its own.

Okay, here’s a brief summary of Steiner’s position which is logical and consistent even if not easy to believe. Although he stressed that he never wanted anyone to accept anything that they could not confirm for themselves.

Steiner claimed to have access to higher realms of existence and he gave instructions as to how this might be achieved by anyone willing to follow the path. The instructions can be found in the book, Knowledge of the Higher Worlds And Its Attainment
The earth consists of four kingdoms, mineral, plant, animal and human. Humans can be said to have four lower principles, 1. ego, 2. astral ‘body’, 3. etheric or life ‘body’ and 4. physical body, demonstrated by the fact that we have self-consciousness, feeling-consciousness, life principle and obviously a physical body.

We share the last three principles with animals, the last two with plants and the last one with the mineral kingdom.

Now all of these kingdoms have all four principles but it is only in humans that the ego has descended and condensed enough to align with the individual organism. In all the other kingdoms the higher principles are in the higher realms and are more diffuse. Plant egos exist in realms which we do not have access to with our everyday consciousness.

Tomato plants can be seen to have physical bodies and they are obviously living but they do not appear to be conscious let alone self-conscious. These principles remain in the higher realms and they have not condensed into earthly individuality. The physical body which belongs to the tomato ‘ego’ consists of all tomato plants in their entirety. It could be said that tomato plants are the cells of the tomato being.

Only those who have conscious access to these higher realms can come to know the tomato ego.

Now because this is so hard for our modern minds to believe, it is easy to dismiss as a fantasy invented by Steiner. But if we really looking for a sure method to confirm or deny what is said above, without having to prejudge it, then we can follow his instructions in the book, Knowledge of the Higher Worlds And Its Attainment and see where it takes us.

Unless we are willing to do this then all we have is opinions about what he said. It is obvious that I would like Steiner to be reliable, but I cannot speak with any authority on this, and that is why I haven’t answered your list of questions about tomatoes. I plead ignorance about the tomato. I have made its acquaintance but I don’t know it well enough to speak to 🙂

“There are more things in heaven and earth, Horatio…”

42. keiths:
keiths:

CharlieM:

Just as Darwin isn’t the final word on evolutionary theory, Euclid isn’t the final word on Euclidean geometry, despite its name.

Euclid’s terminology could be confusing and his axioms were insufficient.These flaws have been corrected in modern Euclidean geometry.

Here’s a pictorial representation of the modern terminology:

Do you agree that Euclidean geometry is one-sided in that it define a circle in terms of points and lines, when it is equally correct to define it in terms of lines and plains? This polarity must be taken into account in order to get a more complete understanding of what a circle is.

Of course it is. Nothing else, really. The actual point of all this Euclid business is Steiner, Steiner, Steiner. It’s why you brought it up.

44. CharlieM:

It is obvious that I would like Steiner to be reliable…

Painfully obvious.

Problem is, your desire is so strong that it barely allows you to question Steiner at all, much less the way you would question, say, L. Ron Hubbard.

Are you seeking Steiner, or are you seeking truth? If the former, then we’ll leave you to your blind hero worship. If the latter, then you’ll need to question Steiner’s output rigorously and vigorously. “I like it, therefore I believe it” just won’t cut it.

45. Charlie,

Do you agree that Euclidean geometry is one-sided in that it define a circle in terms of points and lines, when it is equally correct to define it in terms of lines and plains?

No. We don’t need more than one definition.

As for generating a circle, Euclidean geometry offers more than one way, including as a conic section.

46. Charlie,

Still waiting for you to get around to this:

But we can determine its nature [the nature of the ethereal] by the use of projective geometry.

Again:

So far you’ve made some questionable assertions about the inherent dynamism of projective geometry, but you haven’t begun to justify your claim above or to embark on a determination of the nature of the ethereal.

Please do so, but keep in mind that justifying and asserting are two different things, as are determining and asserting.

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