--

Yet, as an abstraction, meaning can be hard to pin down.

Because a fact by itself doesn't usually mean very much.

It's a point on the map, a location... a tree in the woods.

With enough of them, ways start to appear by which they can be connected.

These patterns become more visible than the points themselves. .But whether they're more viable---is a dilemma.

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Why look here? .At a group of mathematical curves that, together, resemble a flower ...and an erotic flower at that.

Because the parallels are adept. .If "Aphrodite's Curves", to which these are related, represented the ability of sexual reproduction...

"Elegant Pursuits", whether this flower be botanical or animal, represents the intent; usually much more immediate than plans.

This is a thing of meaning. .If intent is recognized, and acted upon, it will influence its surroundings in obvious ways.

Even unacted on, it influences its surroundings in subtle ways. .But if unrecognized, intent becomes virtual.

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Not necessarily virtuous,"Elegant Pursuits", sketches an obvious curve: the passage with it's tongue-like fold;

out of a multitude of obscure ones: root functions running from back to front. .(See below).

It is the root functions that are demonstratable facts.

The curves which run side-to-side, which seem so obvious at first glance;

may not even be mathematically modelable---without restating the roots involved.

--

What this may have to do with meaning...

is perhaps about as much as our brains have to do with consciousness.

The one is a collection of facts, individualizable down to its constituent points.

The other is the shape of those occurrances within. .Difficult, if not impossible, to subdivide.

I don't know if our spirits have a reality without the network of cells which support them.

There are mathematical reasons, (in the complementary nature of circles and hyperbolas),

to believe that anything that currently exists always has, and in some form, always will.

But circles and hyperbolas are definable mathematical objects.

Is meaning also a "definable mathematical object---" .Or has mathematics constructed, "Meaning", as an unsolvable?

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"I think, Sebastian, therefore I am."_ (Pris, Blade Runner).

--

This is more than a defensive argument. .

What is within us has real world effects. .And what can change its surroundings, must to some extent be real.

If a sense of belief can raise an altar, and summon to it a crowd; that sense of belief, at least, is real.

But if the practical test for the reality of meaning, is that it should influence its surroundings...

one is presented with the falling-tree paradox: without a witness, would there be sound?

--

This dilemma was first put to me, and the class---I think rather gladly---by a sixth-grade teacher.

It's intended to be unanswerable. .Without anyone there, and this can be expanded to include listening devices;

the best one can do is say, "I think so... .that's a lot of wood to break and then come down quietly."

(The intellectual slight-of-hand was placing the burden of proof on the listener.)

As one grows older, that paradox fades and reappears in other forms---other debates.

Discussion of "Intelligent Design", in its classic sense, comes down to similar howdoweknowthisisso?isms.

"How do we know this isn't?"

I don't believe in the miracles, yet I look at a cell and believe it wants to live; at a virus and belive it wants to kill.

Intentions. .These things have no mind---can they have intention? .Can it make a difference to assign one?

--

Assigning intentions brings us to a second paradox, a mirror to the first, which I'll call "La Madonna".

People have seen the Blessed Virgin, in laundry hung out to dry.

Weeping parishoners have visited the image of Jesus in a hurricane-damaged wall.

These people are convinced that what they're seeing is divine. .Ponderable: does this make it so?

If there is a crash in the woods, does it make a fallen tree?

--

Things which swirl around the ideas of meaning, intention, and how we carry them:

There are faces we wear in public, often at variance with the way we feel;

if the public face is what is recorded, what gets results; is the private one less real?

There are places we visit in our sleep, which we see flashes of when we're tired;

we recognize them, and yet they are nowhere but within us. .Are they real?

If we see the dead in a dream, is it then rain of recollection?

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"Not everything that counts can be counted. .And not everything that can be counted, counts." .Albert Einstein.

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To put this in a mathematical context, please view the figure below.

Running from front to back, and emphasized by heavy lines, are a family of root curves.

They arc and twist through space like the stations of a dimensional rail-line.

These are real, independent facts. .They can stand alone. .They are eternal.

--

But what of the routes the trains, so to speak, don't run?

Rotational curves which loop around and over the fold---so obvious to the eye.

I can find no definition, save to restate their component roots; and it may be they have no definition...

Can non-mathematical functions persist in the absence of what built them?

--

Consider an oval, (unfortunately I have no illustration).

With three tacks and a loop of string, an oval can be drawn.

It has no known mathematical formula... .But we see it.

And now for a look at the mathematics.

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When I was first looking for a way to describe these, I decided to use the term "root envelope",

and many of the cards I made call them that. .This reflects a misunderstanding on my part,

first of the term "envelope", and perhaps also of the curve itself.

Mathematically, by the American Heritage Dictionary, an "envelope" is:

A curve or surface that is tangent to every one of a family of curves or surfaces.

As I have not built it from tangents, nor demonstrated that it can be;

"Elegant Pursuits" should not be considered an envelope in the standard sense.

--

In looking about for a new term, one I could use without breaking conventions,

I briefly called them a "root curtain", (which I still use to describe "Aphrodite's Curves"),

before settling on the expression "rootcurve flower", for elegant pursuits for I think, obvious reasons.

As far as I have looked, these terms are free for me to define mathematically. .And so I define them, as:

--

"Theoretically contiguous three dimensional shapes constructed of a set of rootcurves from a family of functions."

Qualifying Statement: "Not all rootcurves from said family need be included; as some will strike different shapes."

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--

"Elegant Pursuits", in their various views, are a "rootcurve flower" of the function set:

seed ^degree + (0 through 2 times) seed + 1 = 0

"Degree" runs from 1, which produces a line up the center of the fold, to 6,

(for the curves which are shown as a floral arrangement); or 6 1/2, for the close-up.

The difference is in how close the flower comes to closing at the top.

Were degree to be increased to infinity, the flower would close.

--

If I had the skills, and if Mathcad were agreeable, I would go for that effect. .There's a couple technical reasons why I didn't:

One involves the program's default lighting algorithm, which produced a nicely mottled effect as one climbs away from the fold.

It's not supposed to do that, but in this context it works, up to about 6 3/4, when it suddenly returns to hard lighting.

The second involves its root-seeking algorithm, which starts picking up new shapes shortly after 6 1/2.

--

New shapes? .Aye. .When dealing with roots in complex space, where these functions are,

it's important to remember that a square will have 2 roots, a cube 3, and so on---so that a hex will have 6.

Thus by the time the program gets that far, there are a number of shapes to choose from.

--

How does it choose? .That has to do with the value given to it as seed, I've been using (3/2 + 3/2i).

Mathcad takes this, and in a black-box kind of way, starts moving it around, (warmer-warmer, colder-colder),

looking for a root, (a solution to the equation), while keeping the other variables constant.

-

"Complex space", and the magical-mystical "i":

"Real space", X/Y space, doesn't allow for the even roots of negative numbers.

In day to day mathematics, this is not usually a limitation, but for seeking roots it surely is.

This led to the discovery of "i", the square root of -1; which is so fundamentally different that it has its own axis.

On the X/I plane, any root for any number can be placed, and the three: X, Y, and I, allow the plotting of root curves, as above.

F.Y.I. : The X axis runs vertically in the figures, the I axis horizontally, and the Y from back to front.

The rotation of the flowers is just a presentational thing, they're all the same.

--

-

-

"Elegant Pursuits" serves as an expanded look at the first degree of "Aphrodite's Curves",

--

whose formula is: . seed ^{(to the degrees 1 through 7.18)} + (4/3) seed + elevation = 0

--

Their similarity is no co-incidence. .They do, essentially, share one cross section:

(where the first degree coefficient is 4/3, and the elevation is 1). .Apart from that they' re their own sets.

For Pursuits, degree has been trimmed and elevation fixed at 1; while the first-degree coefficient has been made variable.

When its value is 0, the roots form a circle, which is dimly visible at the base of the upward-facing, and the reversed, "Gothic" views.

(It was edited out of the other three.)

As the value increases the bottom folds inward, while the curve itself widens;

until when the value is 2, a bulb has formed which faces into the channel.

--

As I've said, these curves are remarkably resistant to analysis, (mine at any rate),

but for those who wish to study curves, though not this one, not yet...

you will find an excellent resource in the MacTutor Archive.

http://www-history.mcs.st-andrews.ac.uk/history/index.html

(School of Mathematics and Statistics, University of St. Andrews, Scotland)

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Hmm.

--

I'm the guy who's rarely seen hanging out with anyone; who's not comfortable around people, and less so with intimacy.

But I am seen handing out mathematics in the square.

People sometimes ask me why...

that's a tough one.

--

I take to math and find forms, some of which I call Aphrodite's Curves; these seem distinctly feminine.

As much as anything, it is the first-degree coefficient which determines whether they're generic, androgynous or female;

so in time I made this variable. .The result was an analog for the most intimate curve of all.

Though I wouldn't quite say it's human, the intent is achingly there.

There's little that's going to do much in that channel, without rubbing against that bulb. .Cheers.

--

One can be amused at the irony, of the marginally social academic finding an object of desire.

But I ask you to take seriously... that it exists at all, that it's there to be found.

Mathematics seems to approach being a living thing on one side of the mirror, as it were.

And we living things seem to approach being unalive -- complicated, but unalive -- on the other.

The material and spiritual worlds are at a boundry where either can be described as both,

when that cold clinical thing called "mathematics"

that students push their way through

in so many briary books

is in heat.

--

--

--

About the characters. .They are adapted from Japanese Kanji.

And from left to right they would read:."Flower, Bird, Wind, Moon".

Taken together, (according to Kodansha's Guide), these are "beauties of nature", or "elegant pursuits".

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I hope you agree.

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