Synchronization in the Time-Space of the Present

Archimedes Theorem on the Area of the Sphere: The Equiareal Projection of the Sphere on the Embracing Cylinder and the Area-Preserving Ontophainetic Screen

Radial Projection on the Embracing Cylinder Preserving the Area of the Sphere

Key to Archimedean Ontophainetic Synchronization: Symplectic Interpretation of the Pythagorean Theorem

(1) Invariant Enclosure of Area through Rotation and Translation

(2) Area of Annulus on the Left Equals the Area of the Disk on the Right

Complex and Symplectic Physical Phase Space Rendering

Harmonic Parallel Transport at the Centre of Synchronization in the Present

and the Measure-Preserving Transformation on the Equiareal Disk

The disk of all synchronized tangents records synchronically and invariantly the area of the annulus with respect to a center that is not explicit in advance - to be identified with the present with respect to an ontophainetic screen. Since the tangential component in the direction of motion following the ordering aspect of time does not have any impact on the rate of change of area, it all depends mechanically upon rotation with respect to this center, as well as upon the rate of this rotation, that is, how fast the winding takes place.

Mechanical Method of Archimedes - Spiral as a Bridge for the Rectification of the Synchronization Disk

The spiral upon reaching the periphery of the disk, that is upon completion of a full rotation unrolls the perimeter of the disk into the linear length magnitude of the subtangent rooted at the center of synchronization.

The linear length magnitude of the subtangent is determined by the intersection of the tangent of the spiral at the point where it meets the disk with the vertical line extending from the center.

The spiral of Archimedes is polystrophic. The unrolling continues in countable discrete steps upon completion of each integer number of completed rotations with respect to the center of synchronization.

The unrolled linear length magnitude is quantized in integer units of the fundamental length equal to the perimeter of the disk.

The area of the disk is equal to the area of the orthogonal triangle upon completion of a full rotation in synchronization with the corresponding translation.

Area of the Sphere Equals the Area of 4 Disks of the Same Radius

Area of each disk equals the area of a corresponding orthogonal triangle via the theorem of rectification of the disk by the spiral. 4 orthogonal triangles, 2 above the equator, and 2 below the equator, generate the spectral partition spectrum on the screen of the unrolled cylinder preserving the area of the sphere through the radial projection. This articulates the measure-preserving/equiareal transformation from the sphere to the unrolled ontophainetic cylinder obtained through synchronization in the present and rectification.

Double Helix and Helical Standing Wave of Moment Resolution in Prime Harmonics