What's a loop? Scaletiled spirals [ discrete, continuous, vanishing point ]
0x0D89
February 12th, 2022

This ~8 minute read explains important fundamentals of “spacetimescale” logic.

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.

A Cartesian method, with a unit circle superimposed.
A Cartesian method, with a unit circle superimposed.

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

A Golden method, with a 4-color theorem non-sequential order.
A Golden method, with a 4-color theorem non-sequential order.

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.
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.
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.
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.
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."
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.
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.
A Golden method, with a unit circle superimposed.
A Golden method, with a unit circle superimposed.
An "irrational" (c-b)/a method, with a unit circle superimposed.
An "irrational" (c-b)/a method, with a unit circle superimposed.
A "rational" (c-b)/a method, with a unit circle superimposed.
A "rational" (c-b)/a method, with a unit circle superimposed.

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.
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.

To support ape.mirror.xyz , please consider buying an NFT of this “cyberpoetic” document:

.

Arweave TX
kLez1mx8WfRfbrZcL4-pJsomisRsdVnQA9__noWWA_U
Ethereum Address
0x0D89421D6eec0A4385F95f410732186A2Ab45077
Content Digest
7X4RmO_wsuBTJsHnXiEC7IrCYkK3Ji0ULpHA9z5fSxQ