User:David Eric Ffell

FFELLONIC GEOMETRY
Ffellonic geometry encompasses a collection of geometric configurations based on the symmetry arising from attaching spheres of equal size. This series evolves as spheres gain increased capability to connect. Ffellonic geometry is structured as a finite progression composed of twelve Levels, organized in a hierarchical order. Each Level in Ffellonic geometry comprises of a pair of isomorphic structures: a Ffellonic Form and its corresponding Canalicchio Dual.

Origin of Name
The term "Ffellonic" derives its name from David Fell, the individual who originally coined this geometrical concept. The inclusion of the double 'ff' in 'Ffellonic' reflects his connection to the Welsh-speaking community in North Wales, where 'f' is pronounced as 'v,' while 'ff' is pronounced as 'f'.

Construction
The construction of the Ffellonic geometry arises from the parameters used to construct the polyhedral Platonic Solids. The parameters are responsible for the highest form of symmetry in which each vertex has the same number of symmetrically arranged edges of equal lengths.

While the traditional description restricts the Platonic Solids to polyhedral shapes, Ffellonic geometry expands the application of these symmetry parameters beyond polyhedra. Specifically, Ffellonic geometry emerges from extending the parameters of symmetry and uniformity inherent to the Platonic Solids. This expanded framework preserves the pure symmetrical essence of the Platonic Solids.

Dual forms of the Platonic Solids
To gain a comprehensive understanding of this geometry across all geometrical forms, it is imperative to acknowledge that the Platonic Solids are not composed of merely five distinct shapes; rather, they consist of two separate, progressing series, each comprising of three polyhedra.

The five Platonic Solids are divided into two series of duals

One utilizes equilateral triangles as the fundamental geometric unit. In these, the number of converging triangles at each vertex increases sequentially. The other construction type uses regular polygon faces as its basic building block. In these forms, the number of edges in each face increases progressively, with three faces converging at each vertex.

Sphere Arrangements
Extending and preserving the innate geometric purity of the Platonic Solids proves unattainable by solely increasing the number of edges or faces converging at each vertex. To overcome this challenge, a new set of principles must be introduced. These principles are not derived from the geometry of the Platonic Solids themselves but rather from their source, specifically from three distinct arrangements of four, six, and twelve equal-sized connecting spheres. These are the arrangements created when spheres of similar binding capacity are freely allowed to attach.



Symmetry created by attaching spheres
The self-assembly of equal spheres, such as bubbles or droplets, often leads to highly symmetric structures due to the uniform interactions between the spheres. This phenomenon is governed by the minimization of surface free energy, leading the system to "choose" the geometry with the minimum surface area for a given volume. Common symmetric structures formed by equal spheres include hexagonally packed cylinders, bicontinuous structures with P, D, and G surfaces, and icosahedra. These structures exhibit various symmetries such as rotational, reflectional, and translational symmetries. For instance, hexagonally packed cylinders have six-fold rotational symmetry along the cylinder axis as well as reflectional symmetry perpendicular to it. Such symmetries emerge spontaneously from the simple physical interactions between identical building blocks.

Construction of the Platonic Solids
When constructing the Platonic Solids, there are two crucial points that define these arrangements of spheres: one point located internally within each sphere and the other situated externally.

The internal point is the centre of each sphere.



When the points of adjacent spheres are connected by lines, the tetrahedron, octahedron and icosahedron are created.

The external point is centrally located between the spheres and is formed by the tangent intersections of the sphere's touching points.



by connecting lines between these points, the tetrahedron, cube and dodecahedron are created.

The quantity of spheres attached to each other defines the three distinct arrangements that constitute the Platonic Solids

The following are the sphere arrangements which form the vertex of each Platonic Solid and are responsible for the polyhedral shape.



Ffellonic geometry builds upon the Platonic solids by systematically increasing the number of equal-sized spheres that meet at each vertex. This method generates geometric forms that maintain the symmetry and stability of the Platonic solids as the sequence expands. The core principle is using vertex-sphere congruency, the property that defines the Platonic solids, as a framework to construct an amplified set of highly symmetric polyhedral structures. By adhering to this rule of congruent vertices and incrementing the number of spheres, Ffellonic geometry progresses through iterative expansion beyond the five Platonic solids.

The symmetry within Ffellonic geometry is attainable only when attaching spheres conform to one of the following twelve inherent natural arrangements. These particular spherical arrangements provide the foundation for symmetric order within Ffellonic geometry.



Ffellonic Forms
By connecting the centres of each sphere the Ffellonic Forms are created.



Canalicchio Duals
While the duals of the Canalicchio Duals are created by connecting the tangent transections of the sphere's touching points.