Ride the Tide

All in One Geometry.

Compass and straightedge construction to reach 3-4-5-6 fold geometry & many other very interesting ratios like the Golden ratio and the Kepler triangle.

In ancient Mesoamerica, the quatrefoil is frequently portrayed on Olmec and Mayan monuments, such as at La Blanca, Guatemala where it dates to approximately 850 BC. The quatrefoil depicts the opening of the cosmic central axis at the crossroads of the four cardinal directions, representing the passageway between the celestial and the underworld.^[6] Wikipedia

This is the transformation with added circles from simple swastika through to the 137 swastika and then at Master builders grid. The centers of the circles are at each 90 degree corner of every angle.

We see how it starts as a thin quatrefoil standing as a cross and how it evolves to the standing quatrefoil which is very common in the symbology of christianity as well as many other parts of the world.

Transformation with added Pythagoras square, Fractal.

The same transformation as the previous fractal posts. But here with added pythagoras squares.

Pulsating fractal Nr. 2

Transformation of one cubic unit. Shown from two perspectives.

Pulsating fractal.

This is one cubic unit showing the transformation as the angles move. Adding more and more angles makes it to a pulsating hexagon.
It is a visualisation of how the angles are moving from their resting position (swastika) at the center towards the point of maximum compression (Masters builders grid).

Representation of the vectors of movement

2D representation showing the vectors of movement. This is the transformation of swastika to MBG highlighting the path of the angles. As the angles move they pixelate space.

Squaring the circle

Is it possible to do the impossible? Let’s draw a square where the corners is attached to the long leg of each angle, and then draw a circle which is attached to the 90 degree corner of each angle. From the beginning the circle surrounds the square, but as the transformation occurs and the angles start separating the relationship between the square and the circle changes. They are passing through each other, as the circle diminishes in size and the square grows, though still sharing the same center point. It is like two people meet on the same step as one is going up the stairs and the other one is going down, they are bound to meet somewhere, no? Meaning that at one point their circumference and area should be exactly equal. They meet in infinity.

Shifting swastika, to account for all different ratios of the swastika that is found all over the world.

Swastika inside a box to show how the thickness of the swastika is growing as the angles move outward. The added hypotenuse is there to show how it inclines as the structures changes.

A shifting mathematical structure to derive numbers using Pythagoras theorem.

All of these drawings are completely mathematical. Even though some seems artistic, every point in these drawings are mathematical points.
Here is the swastika with a bounding box and with added hypotenuses to use the Pythagorea theorem. The theorem describe the effect when these angles move.
Also note the square and the circle in the middle and see their corresponding numbers you will see that they have equal circumference and area down to 6 decimals at one point as they are growing through each other. That was all that was possible at this moment in this program.

Shifting proof of Pythagoras Theorem

If you draw a diamond square where each corner of the square is attached to the end of the long leg of each angle. What you will get when you start separating the angles is a proof of Pythagoras theorem for every possible triangle with a hypotenuse from root square 2 to root square 5 ,regardless of what unit you use. That means it is a numberless proof for the pythagorean theorem.
You also get a very beautiful square spiral.