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
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 are the Pythagorean Theorem
Furthermore, because of design considerations, a tile uses the
b+b=2b variables rather than the
For instance, a tile size of 4:2 ratio continuously tiles the plane with rotation using
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.
c usually doesn’t share a “measure” in common with
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:
chypotenuses as radii angled in relation to the
yaxis and the other
chypotenuse as a radius along the
-yaxis 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.
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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