What's a loop? Scaletiled spirals [ discrete, continuous, vanishing point ]

0x0D89

February 12th, 2022

There are a variety of ways to discretely tile objects to create a continuous, flat and finite plane.

For instance, this is a Cartesian method that does it.

And this is a “Golden” method that does it.

That Cartesian method uses 1 tile size whereas this Golden method uses Infinite tile sizes.

Further, this Golden method uses 1 scale for its Infinite tile sizes; as such, its tiles are “scaletiles.”

Like a tile’s size, scale is a magnitude; but scale is the magnitude that modifies the size of a tile.

In other words, size is the variant magnitude and scale is the invariant magnitude.

This Golden method must use the `((5^(1/2))-1)/2`

scale in order to fixate Infinite tile sizes on a continuous, flat and finite plane.

For instance, if the Golden method attempts `1/2`

scale, there’s a gap and the plane isn’t continuous.

A tile’s size doesn’t need to have a 2:2 ratio.

For instance, a size of 4:2 ratio continuously tiles the plane with `((5^(1/2))-1)/2`

scale.

But ratios other than 2:2 have asymmetry and, as such, include a rotation property.

For instance, the 4:2 ratio can rotate into the 2:4 ratio.

If a size of 4:2 ratio attempts `((5^(1/2))-1)/2`

scale while rotating for each size change, there’s a gap and the plane isn’t continuous.

Using observational evidence, the formula that determines scale for the simplest form of tile rotation is the `(c-b)/a`

method.

`a`

, `b`

and `c`

are the Pythagorean Theorem `(a^2)+(b^2)=(c^2)`

variables.

Furthermore, because of design considerations, a tile uses the `a`

and `b+b=2b`

variables rather than the `a`

and `b`

variables.

For instance, a tile size of 4:2 ratio continuously tiles the plane with rotation using `((17^(1/2))-1)/4`

scale.

And that plane is also a tile size of `2+(4*(((17^(1/2))-1)/4))`

:4 ratio, which continuously tiles the plane with rotation using `(sqrt(2(11 + sqrt(17)))-2)/(1+sqrt(17))`

scale.

Or reorganizing the `(c-b)/a`

method “child” and “parent” along a bottom right corner “limit,” rather than tending along a spiral pattern.

The philosophical concept is that the `c`

“irrational” length of the hypotenuse in a right triangle embodies an Infinite energy within a “field of view.”

And at the heart of this perpetual nature of `c`

is the square root of natural numbers.

The fact `c`

usually doesn’t share a “measure” in common with `a`

or `b`

is an inherently recursive aspect of Nature with powerful utility value.

This “nonhalting” aspect of `c`

can guide Infinite expression outside the hypotenuse length.

The “vanishing point” perspective drawing pioneered by Brunelleschi is one of many example use cases.

`(c-b)/a`

is another example use case.

The following are some instances that show how `(c-b)/a`

applies to rotation scale.

Notice the radius measure, in the context of imagining:

- two different
`c`

hypotenuses as radii angled in relation to the`x`

axis,- with
`a`

along the`x`

axis, one`b`

along the`y`

axis and the other`b`

along the`-y`

axis;

- with
- a third
`c`

hypotenuse as a radius along the`-y`

axis,- with
`c-b`

along the`-y`

axis as well.

- with

This image displays `a`

as a yellow line, `b`

as a green line, `c`

as a pink line and `c-b`

as an aqua line.

This has gone over the basics of what a “scaletile” is, and the “diagonal” methods behind it, but has not explained the philosophical “spacetimescale” context.

A follow-up will look at the square roots behind these methods, and explore different ways to geometrically index square roots.

Another follow-up will look at the exponent “power law” index behind these methods, and explore different ways to record this geometric index.

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