 What's a loop? Scaletiled spirals [ discrete, continuous, vanishing point ]
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February 12th, 2022

## What’s a scaletile?

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.”

# what’s a scale?

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 Golden method 1/2 scale attempt, with a gap in the plane.

# what’s a tile?

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. A Golden method using tiles of 4:2 ratio, with a 4-color theorem non-sequential order.

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. A Golden method using tiles of 4:2 ratio, attempting to rotate into 2:4.

# what’s a rotation scale?

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. A (c-b)/a method using tiles of 4:2 ratio, with a 4-color theorem sequential order.

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. A (c-b)/a method "child" using tiles of a preceding (c-b)/a method "parent."

Or reorganizing the `(c-b)/a` method “child” and “parent” along a bottom right corner “limit,” rather than tending along a spiral pattern. The previous (c-b)/a method "child" and "parent" nesting, with a different spatial order.

# what’s a (c-b)/a method?

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;
• a third `c` hypotenuse as a radius along the `-y` axis,
• with `c-b` along the `-y` axis as well.

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. A "rational" (c-b)/a method, with Pythagorean measures and a unit circle superimposed.

# conclusion

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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