The Circle and its Hyperbola
Real Plane View
Complex Plane View
Graph of a Circle
Graph of a Hyperbola
These are two views of the same function: a circle on the left, and a more esoteric shape called a hyperbola on the right. _ They come as a set.

You'll notice flat lines to the left and right of the circle, that's the hyperbola; and the flat line at the center of the hyperbola, is the circle.

They lie in perpendicular planes, (planes at right angles), so that when you face one directly, you see the other edge on. . Below is a unified view.

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Mathematically, the hyperbola can be thought of as an anti-circle, or as a circle turned inside-out in the conceptual sense;

for where the circle is closed, temporary though uninterrupted, and constant in form,

the hyperbola is open, endless though interrupted, and variable.

These sister-shapes are so basic to math that one could, i imagine, fill whole books with their various properties and effects...

But here I'm interested in a philosophy.

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You see, much has been made of the idea of life-after-death; not least of all the promises of many religions.

Annihilation, more than just depressing in the abstract, can be downright terrifying when one believes it's at hand.

And one of the reasons i believe as firmly in math as i do, and that it's not at odds with spirituality ... is that it offers a path not around, but through death.

One which preserves life, while respecting how frightening the point of death can be in its approach.

The mathematical basis is this: nothing is temporary. . Functions which appear to end on one plane, have jumped to another.
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Speaking poetically for a moment, but not pointlessly:

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The circle, growing at the center of the hyperbola,

opens and closes in the "Real Plane" -- it is born, lives and dies.

Hidden in the flat lines to its sides, are its before and afterlives...

influences and consequences which bend to meet the circle;

as emotions and their actions seek and stem from us.

Where the way we felt yesterday, travels today.

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Never more curved than where it and the circle meet;

the farther the hyperbola travels, the straighter its limbs become.

Change ... is focused where the "imaginary" touches the "real".

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3D graph of a Circle and Hyperbola
And the camera pans to where Rod Serling would stand, and say, "Point for consideration".

That the real plane in math may be analogous to the material plane in life; and the imaginary plane in math, to the spiritual. . The third axis would be time.

I can't prove this, but hold that it's not unprovable, (or at least untestable). . It's untestable with material things: a telescope, an accelerator...

The result, (i imagine), would come by monitoring thinking minds through real-world, (preferably), situations.

(Though itself a scary thought, for the power this implies)

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But let's put power aside ... to take the analogy further.

In life, if we are like circles, self contained and looking out at the universe;

in the afterlife we could be like hyperbolas, open, forever looking back at the circles that were our lives.

And if in life our bodies are material, and our thoughts and feelings subliminal;

in the beyond, it would follow that our thoughts and feelings will be material and the objects surrounding them, subliminal.

Dreamspace, so to speak.

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The emotions we radiated into our surroundings, become us in the beyond. . (Again, "for consideration").

Religions often traffic in the idea of heaven and hell. . It may be unnecessary.

If you've ever caught yourself in reflection, by surprise in a crowd, and not recognized this person until you reacted...

you will understand that sometimes we are truly transparent; and it is essential to be at peace with ourselves and with what we create.

No-one would want to look back forever upon less.

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Imagine it's already happening.

That the spirit would not be so much a lump sum entity that one must "pass through the eye of a needle";

but rather an innate part of nature, a part which says "I exist", being focused by our minds and radiating outward upon the world.

That force, that part saying "I exist", is in my vision God.

And we honor God by giving existence to things, by loving and by preserving existence.

In complement to this we are also given the power to destroy, and the hunger to do so; sometimes for its only sake.

That hunger is my vision of Satan, (i don't say this lightly).

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( It's a complex world ... in which one must destroy to create, and create to destroy.

I don't argue that one can live, or should attempt, a life of pure good.

And those religions which do are intentionally placing an unclearable bar before their believers, to control them. )

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I imagine that as our actions ripple out through spirit space ... it would be through our souls. . It would be our souls.

And life, which is short but feels long, (and vise-versa), could be thought of as time in the material womb.

When our point of consciousness reaches the point of death, into which no thought or memory can be carried;

it will or would be born onto the spirit-plane to inhabit the soul we created.

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It may be the dead are able to put some things to right, to rectify and to visit, for the hyperbola is curved as it meets the circle;

and curvature implies change and the ability to change ... but not necessarily the will to do so, and time would be of the essence.

The hyperbola is immortal, but buys this immortality at the price of slowly freezing against two lines. . Lines don't change.
And it turns out those lines are important, so let's put them up.

To the left you see the same hyperbola as before, (from a little bit farther out),

with the addition of its framing lines in light blue.

Mathematically we might say the limbs of the hyperbola tend to their lines.

Philosophically, the figure implies that, all things being equal,

no-one can act outside their experience.

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For the influences of a life are gathered from the left half of the hyperbola,

and the expressions of a life radiate into the right; together they are a 'soul'.

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To some extent this is a good, reassuring thing, in accord with our common sense.

One can buy a musical instrument with favorite songs firmly in mind, but not immediately play them.

Without practice, the ability to apply the notes we hear is outside of our experience.

Yet even those who become skilled cover-players would I think find it difficult to improvise like their subjects

so that if they were playing behind one curtain, and the subject behind another, educated fans could be fooled --- (reality-show prospect).

Our collected experiences and the ways they interact within us, make us unique.

(And can make different periods of our lives unique from each-other.)

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But are we simply a place through which experiences pass? .No greater than the sum of our parts?

The somewhat demoralizing answer implied in the metaphor, (which may or may not apply), would be:

"Yes, no greater in sum... but this does not speak for complexity or community."

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-In rootcurve mathematics one looks at the way in which curves which resolve to "simple" shapes

in one direction ... can together construct shapes which, though noticable and memorable

in another direction, do not resolve (or resolve easily) to the mathematically simple.

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Also, various character and cultural traits do seem to pass from host to host.

(Not always the traits and hosts intended) .

As capacitative conductors and remixers of information, we support an ecology of memes,

info-viruses (which often include memes), collective or group-thought-organisms and perhaps ...

manners of "holographic life form" among and between us.

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All things are not equal, nor all lives equally lived.

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The circle is a balanced example of a larger class called the ellipse.

A workable formula for an ellipse at the origin of the real plane is:

y equals (plus or minus) (the real part of) the square root of

(the radius squared, minus compression squared times x squared).

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'Compression', to paraphrase Forrest Gump, 'is as compression does.'

If it's greater than 1, it makes the circle shorter.

If it's less than one, it expands it --- fractional compression.

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The effect can be seen at right.

As a reference, the blue circle-hyperbola pair is uncompressed.

The narrowest of the ellipses is compressed by the square root of 6, (about two and one half),

and the widest of them by the square root of one quarter, (exactly one half).

Between these are many intermediate states and it can be seen that their sister, 'elliptic-hyperbolas', change with them.

The curious thing, is that the widest of the ellipses, which lasts the longest and has the greatest area,

is joined with the smallest and narrowest of the elliptic-hyperbolas, (when considered as a fraction of the complex plane).

And that the narrowest of the ellipses, which is briefest and has the least area, is coupled with the greatest of the elliptic-hyperbolas.

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To me, the following is implied: . Those periods of our lives, be they minutes, years, or the whole of our time;

where we are open broadly to chance and influence, powerful emotions, are by their structure --- compressed, abbreviated.

Trying to revolve with the One. . Walking into the sun. . My eyes will hurt my skin will burn but I will know what I have learned.

We're born this way. . Are called to return from time to time. . What we're doing in the spirit, is larger than the material realities reveal.

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It also serves as a warning, for in those times when we walk with an open heart, when we leave the blinders off, to the extent which we can...

we inevitably encounter those, those people or systems who'll say, "Now you have found me, disregard all else."

Fundamentalist religious orders do this. . The evangelist will challenge a person to open their heart, to a message, to The God;

then instruct them to close their heart around it --- to see only that vision as true, all else as misleading.

In this way fundamentalism diminishes the soul it claims to save. . For as we narrow our compass, we narrow our spirit as well.

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To a baby, all things are true. . They will not naturally look away, exclude information. . Until we have shown them to.
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(Image generated in Mathcad 13. Quality is uneven.)
Having looked at the effects of modifying shape, let's look at that of modifying size alone.

To the left is a set of circles with their hyperbolas,

varying almost smoothly from a radius of 1/4, to a radius of 2.

For reference, the heavy curve is radius 1. . (The dot at the center is an artifact.)

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The thing i note here, is that whereas a larger circle on the real plane,

necessarily means a smaller hyperbola, (as a percentage of the imaginary);

the difference is comparitively minor. . By implication:

The spiritual penalty for being rich and powerful, is by itself not so heavy;

when the great and small feel the same way, which can happen.

However, it would be dangerous (in my opinion) to overemphasize this.

Wealth and power give those who hold them options, (to the degree that these are held).

A person with the ability to exercise options - to influence or possibly gain from a situation,

even a bad situation - is that much less likely to feel the same way about it as one who can not.

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Also to be considered is the natural, and understandable, tendency to create a personal environment.

Wherein to apply a selective vision of the world also implies selective blindness;

To pursue selective goals, implies selective exclusions and protections intended to guard those goals.

Though people of all classes will do this, and for many reasons, it might fairly be said:

That the wealthy or powerful can pursue material visions and goals farther for having this advantage;

But such a pursuit risks the construction of longer, narrower life-curves as seen above.
Thank you for reading.

This is one set of interpretations, based on amateur math and personal experience.

Yours may differ.