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Ellipses (to come)
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Parabolas (to come)
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Hyperbolas (to come)
This and the related sections are about math and how you can use it in noetic spaces. If you're not comfortable with math, not to worry. This is not a rigorous exercise. Rather, we will take a softer, Moon-Mercury approach to develop a feel for mathematical spaces.
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Mathematical Intuition
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You don't need to study math to practice spirituality. But, in terms of developing vision, it is very helpful to have some degree of facility with mathematical spaces. Some superastral spaces are highly mathematical but difficult to tune into, much less comprehend. Without some background in math, you will perceive a general structure to the space but not see much more.
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For this purpose, you don't need a high level of formal math understanding, although that helps. Remember that math exists on multiple levels. On one level, it is a highly technical symbolic language that takes years of study to learn its rules and vocabulary. This is the level that most people find distasteful because they never learned it properly.
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The "grammar", or rules of math, being a closer reflection of archetypal spaces, are much more powerful than the rules of ordinary spoken languages, however; they can unlock additional rules and spaces and insights into reality. This is the great seduction of math for so many intellectuals and scientists.
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We are after a more fluid, more intuitive level that approaches direct, packed insight. Some people have an innate mathematical intuition even though they can’t explain at a formal level how they achieved their insights. You can sometimes feel this ability in them as children even before they learn to speak as a certain, beautiful superastral structure in their column. But even in terms of developing basic noetic vision, math is very helpful. Some examples:
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With an exploration of shapes and geometry, you can develop a “geometric vision” where you noetically intuit the shapes in everyday objects like, buildings, pieces of fruit, trees and landscapes. The world comes alive in an unexpected way as a deep, archetypal layer reveals itself. Eventually you will find yourself perceiving the geometrical spaces in abstract things don’t have an apparent shape, like music, writing and ideas.​
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By studying algebra, you learn to abstract noetic perception completely from visual form. This allows you to gain familiarity with formless noetic spaces, and eventually deep insights into archetypes.
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By studying projective geometry, you can gain insight to how packed noetic forms translate across different dimensions. You also gain insights into how much of your own, normal 3D vision is itself an abstraction.
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By studying calculus you intuit another dimension of dynamic force of noetic spaces. For example, you feel the circle not as something static, but as a dynamic force that makes your consciousness move or bend in a certain way. Your understanding of space and time may converge, so that you feel the temporal aspect of space (as well as the more common spatial experience of time).
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Math is not commonly studied in spiritual traditions, and the ones that do tend to focus on geometry and numbers. I wouldn't read too much into this. The emphasis on geometry and number by mystics of ancient Greece like Pythagoras and Plato, for example, was partly reflection of where they were in the evolution of mathematical development. It would be a mistake to think that the other areas of math aren’t equally or more fruitful in terms of obtaining spiritual insights. It’s quite likely that if Pythagoras lived in medieval times he would have been a sacred algebraist as well as a geometer, if he lived in the Enlightenment era he would have seen the sacred virtues of calculus, and if lived now he would be a spiritual computer scientist, etc.
Shapes and Geometry in Non-Physical Spaces
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Shapes or geometric patterns can appear in non-physical spaces. These can range widely, from a general sense that there is a geometry to the space to clear images of shapes. You might also feel the geometry topologically, as a defining contour to the very structure of the shape. You might also feel things like curves or angularity at a more abstract level.
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Often when you see a clear shape or geometric pattern, there is a feeling of intelligence or significance. Determining the implications of these spaces is often difficult, however, and requires greater discernment.
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Traditionally, the appearance of sharp geometric shapes or patterns in a non-physical space is associated with something highly superastral. I think that a lot of people find this fascinating, and would love to explore high spaces where you see these things. But, as with other aspects of esoteric vision, it helps to see these types of things in the ordinary world and activities first, and then apply them to non-physical spaces.
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To really understand a shape or pattern, you want to go beyond its image. You want to reach its essence, the noetic feeling, for example, of what makes a circle a circle. A shape’s image is just a limited manifestation of something deeper.
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How would a circle manifest in a one-dimensional space? If you are in a space with no extension at all, could you feel the difference between a circle and a square?
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Remember also that as we move into higher dimensions, there are more complexities and possibilities because the archetype has more "room" to manifest. For example, you might think a sphere as a 3D version of a circle. But couldn't you make the argument that a cylinder, cone or a torus is just as much a 3D version of a circle as a sphere? And as you will see below, even in the 2D Euclidean plane there is a way of regarding an ellipse and a parabola as a version of a circle as well. My point is that all of these things are reflections of something much more abstract.
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Generally, if you perceive a shape in a space in a non-physical space, try and can get to an even higher, more packed level of emanation of the space that is closer to the archetype. For it is that which has the effect on consciousness. Visualizing a circle isn’t going to do much, but getting in touch with the archetype behind circular shapes can be profound.
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In the next sections, what we'll do is explore some basic math ideas about common shapes and geometry concepts as a doorway to developing your intuition. Don't be distracted by the math theory itself; you want to learn and get a feel for it but it's not an end in itself for this purpose.
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Curves and Conic Sections
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Rather than exploring specific shapes at first, let's start with something much more generic: curves and (straight) lines. It's not easy to see clear images of polygons or circles in a non-physical space. But perceiving a general curvature or angularity to a space is not so difficult. You can then source that quality to a more specific geometric perception. In terms of mapping, you can start with asking whether a space (or an object in the space) is "smooth" or "sharp" without fixating on what that means exactly.
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In terms of exploring curved spaces and shapes, a good place to start is conic sections. Conic sections are just various 2D slices of a double cone. Ancient Greek mathematicians who were originally interested in these shapes for theoretical reasons but they ended up having many practical applications. The classic conic sections are the circle, ellipse, parabola and hyperbola.
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Try the following if you are having trouble seeing how the conic sections are alike. Take an object with a circular shape like a coffee mug, and slowly tilt it away from you. In your vision you will see it become an ellipse, and then a parabola once it is close to a 180 degree tilt. You will realize that you actually see the conic sections all the time in your visual field, changing shape as you move around.​
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The classic conic shapes share a focus and a periphery. The periphery is the actual line of the shape, and you can think of the ‘focus’ as the center point of the shape. Circles and parabolas have one focus, and ellipses and hyperbolas have two foci. ​
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There is something fundamental about the relationship between the focus and the periphery of the conic. You can think of the focus (or foci) as creating the bend in the line that creates the curve of the conic shape. It’s this “pull” towards the center, the centripetal force, that you want to perceive with these shapes from an archetypal perspective.
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Sine Waves and Sinusoidal Motion
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Let's now explore some features of sine waves. Sine waves have many special properties, and studying them is also helpful for getting an intuition into curves and curved spaces, particularly once you add the dimension of time.
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Like conic sections, sine waves have a focus and a periphery, although it works a bit differently. Sine waves oscillate between the periphery and a center point (the focus). The periphery marks the peak and trough of the sine wave, and the center point is the middle.
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This is easy to see in the following gif. Notice the oscillation across one dimension (the black dot) as well as two dimensions (the sine wave itself):
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Sine wave motion, or sinusoidal motion, is everywhere. The ripples that spread when you throw a rock in a ponds when, the oscillations on a fixed spring, the sound waves when you pluck a guitar- these are all sinusoidal. But even more abstract phenomenal like the cycle of day and night relative to the year (the amount of daylight relative to night time increases disproportionately as you get closer to a solstice) and the flow of the breath (your breath slows down at the end of the exhalation and inhalation phases, and is at the maximum speed in the middle of the breath) are sinusoidal.
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Moreover, you can recast or transform (many) non-sinusoidal waves as a series of sinusoidal waves. This is the idea behind Fourier analysis, which has countless applications in modern technology. ​
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Here is a slowed down version:
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Notice the rhythm of the pulsing movement. If you take the perspective of a point moving along a sine wave (this is easier with the 1D wave at first), your acceleration and speed are also in constant oscillation. As you move towards the focus (the center of the vertical line), you accelerate and your speed is at its maximum. As you move towards the top, you decelerate and your speed slows to a halt as you change direction. This change in acceleration and speed is what gives it the curved, smooth, whip-like feeling (as opposed to something like a triangular wave).
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There is a water element to sinusoidal movement. Think of how an ocean wave builds and crashing. Again, you want to catch that characteristic pulsing, whip-like motion around a center point which is the essence of sinusoidal motion. ​
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Next time you channel release or work on etheric exercises, try moving the energy with a sinusoidal motion rather than a straight line motion. See if you a notice a difference.
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In calculus, sine waves also have the following amazing property. If you take the derivative of a sine wave, you obtain another sine wave. (Actually it's a cosine wave, which is just a sine wave moved a quarter cycle back from the first sine wave). If you take the derivative of a derivative of a sine wave, you get a sine wave that is a half cycle back, or basically a sine wave that moves inversely to the first sine wave. That is, when the first sine wave is at its peak, its second derivative is at its trough.​
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What that means is when you have something changes over time in a sinusoidal pattern, its rate of change is also in sinusoidal motion and its rate of change is also in sinusoidal motion, etc. No kind of other wave has this property. ​
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For example, consider a simple pendulum going back and forth in a semicircle:
Its path of motion (its position over time) is sinusoidal. Its speed (the change of motion, or first derivative) is also sinusoidal: it reaches max speed at the bottom of the pendulum swing, and zero speed at the two horizontal top points of the pendulum swing. Its acceleration, or change of speed (the second derivative), is also sinusoidal and happens to be directly inverse the path of motion: acceleration is zero at the bottom of the swing (when the position is at its maximum distance from the starting point) and maximum at the top of the swing (when the pendulum returns to the starting point on the x- axis). And the rate of change of the rate of acceleration is also sinusoidal…kind of dizzying when you think about it.
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This is one of the reasons the Fourier transform is so powerful; by decomposing waves with complex patterns into series of sine waves, you can take advantage of this special property of sine waves for analysis.
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Ponder this strange quality of sine waves- how the different rates of change have this fractal-like, recursive mirroring of each other that extends infinitely. Again, there's a noetic quality here you want to pick up that is fundamental to sine waves.​
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Circles
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Some works on sacred geometry present the circle as the fundamental, static shape. The one, the uncreated, the unchanging, etc. But hopefully you can see how there is actually a lot of dynamism with the circle, and that the circle is just one manifestation of something much deeper.​​
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Like conic sections and sine waves share the following properties: (1) balance or tension between a focus and periphery, (2) a constant change of speed and acceleration which results in the characteristic smoothness of motion of the curve and (3) notes or echoes of a special flavor of reflective or recursive infinity.
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Just as the circle is one of the conic sections, you can also regard it as a special type of sine wave. Actually, it's a combination of two sine waves, one tracking horizontal path of the circle and the other vertical path. The following graphic should make this clear:​​
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So the special properties of sine waves apply in many ways to circles. The difference is that circles are a closed loop, while sine waves are open. But with the graphic, try again to get the noetic feel of the sinusoidal motion. And how the center of the circle, or its focus, pulls the path of the line in that smooth, sinusoidal way.
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Although the gifs are animated, noetically you can still feel the smoothness of sine wave or circle even if there's no motion. It's not dependent on time.
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Ellipses (to come)
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Parabolas (to come)
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Hyperbolas (to come)
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