Re: Infested Spiral Infinity...

On 08/10/2024 07:40 PM, Ross Finlayson wrote:

On 08/10/2024 07:06 PM, Ross Finlayson wrote:

On 08/10/2024 06:36 PM, Chris M. Thomasson wrote:

Spirals in spirals forever...
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https://paulbourke.org/fractals/septagon
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Paul was kind enough to write about my formula.

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The spiral's the oldest symbol for "mathematics"
in the world, since before recorded history.
Anantha, Thoth, and so on.
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The, "spiral space-filling curve", is as simple as
drawing a line from zero to one, which it is,
in all the dimensions at once. Of course it's,
"non-standard", but anybody who puts pencil to
paper does it each time. It's really about the
first sort of "standard extra-ordinary super-standard",
function, being that it has real analytical character and all.
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DesCartes of course has a theory where vortices
are ubiquitous in the space. Einstein has what's
called his bridge into the centrally symmetric.
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When we were in junior me and my lab partner
made Turbo Pascal draw evolving spirals all
over the screen, but somehow nobody ever had
an epileptic fit, that our goal was a sort of
optical overload, all based on some simple formulas
and the power of the computer.
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We got a good grade for it, ....
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Anyways "infested" isn't the right word,
more like "infectious".
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It seems the idea that "there are no, _non-trivial_,
solutions singularity-free to the field equations",
makes for a particular, _non-trivial_, sort of approach
what's otherwise the trivial, or vanishing, while,
this is already a most usual sort of approach when
there are simply interacting terms, as of what the
linear or non-linear predominate, the idea is to make
a sort of super-standard least-non-trivial to help explain
what all gets involved making a right angle, of a straight line.
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Here that's being called "Zeno's swath".
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 From a spiral-space-filling-curve, for example of a disc,
lines are pretty simple as going out and establishing
what's called "shortest distance is a straight line",
and vice-versa, then simply orthogonal lines are
the most of those, shells, in the least of those, dimensions.
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If you've never heard of the Archimedean spiral,
it's more a usual thing that basically is a spiral,
that, increases distance one in each revolution.
Anyways, it's not constructible with edge and compass.
However, given an Archimedean spiral to accompany
edge and compass, makes for solving angle-trisection,
circle-squaring, and, cube-doubling.
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In case you never heard of it, they already have one.
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So, if you can only bring one more tool after edge and compass,
an Archimedean spiral might be a nice thing to have.
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Less baggage.
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Of course, an Archimedean spiral is sort of
like a spiral-space-filling curve, amplified by infinity.
Or, you know, attenuated in reverse.
It happens to look exactly the same as scaling
the whole integers to the unit interval,
or vice-versa, that being one-dimensional of sorts,
the same sort of thing, the very, very general case.
 From all angles, ....