Draft:Niblett's Dimensional Distribution Method

• Comment: Badly formatted, poorly sourced, and hard for those unfamiliar with the subject to understand. I dream of horses (Hoofprints) (Neigh at me) 01:37, 23 September 2024 (UTC)

• Comment: This appears to be WP:ORIGINAL RESEARCH. Unless we have sources on this, reliably published by people unconnected with its inventor, we cannot include it. —David Eppstein (talk) 21:39, 23 September 2024 (UTC)

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• • Niblett’s Dimensional Distribution Method** is a mathematical approach developed by Oscar Alexander Niblett for the distribution of points (or atoms) along the vertices, edges, faces, and, in higher dimensions, volumes and cells of geometric shapes in 2D, 3D, and 4D spaces. This method provides a systematic way to calculate how discrete points can be arranged on standard geometric figures, including polygons, polyhedra, and polychora, by accounting for their dimensional components.

1. 1. 1. Overview

The method is a generalized extension of simple edge-distribution principles in 2D shapes and applies them to more complex 3D and 4D geometries. The core of the method involves subtracting corner points (vertices) and evenly distributing the remaining points across the edges, faces, and cells of these shapes.

1. 1. 1. Equation

Niblett’s Dimensional Distribution Method is governed by the following general equation:

\[ P = \frac{n - V}{d} \]

Where: - \(P\) represents the number of points or atoms distributed per unit (such as edge, face, or cell). - \(n\) is the total number of points or atoms. - \(V\) is the number of vertices (corners or points where edges meet). - \(d\) is the dimension-dependent factor (edges for 2D, edges and faces for 3D, and edges, faces, and cells for 4D).

1. 1. 1. Application in 2D

In 2D, this method can be applied to polygons. For example, in a square with \(n = 28\) atoms and \(V = 4\) vertices, the number of points distributed along each edge can be calculated as:

\[ P = \frac{28 - 4}{4} = 6 \]

This means each edge of the square will have 6 points, including the vertices at the endpoints.

1. 1. 1. Application in 3D

In 3D, Niblett’s method applies to polyhedra such as cubes, tetrahedrons, and other 3D shapes. The distribution of points involves accounting for vertices, edges, and faces. For example, for a cube with \(n = 38\) atoms, \(V = 8\) vertices, and \(E = 12\) edges, the equation is:

\[ P = \frac{38 - 8}{12} = 2.5 \]

In this case, 2.5 atoms are distributed along each edge, meaning some edges may have 2 atoms and others may have 3.

1. 1. 1. Extension to 4D

In 4D, Niblett’s Dimensional Distribution Method extends to **polychora**, which are the 4D analogs of 3D polyhedra. A common example is the **tesseract** (or hypercube), which has vertices, edges, faces, and 3-dimensional cells.

The modified equation for 4D becomes:

\[ P = \frac{n - V}{E + F + C} \]

Where: - \(E\) is the number of edges, - \(F\) is the number of 2D faces, - \(C\) is the number of 3D cells (such as the cubic cells of a tesseract).

For instance, for a tesseract with \(n = 80\) atoms, \(V = 16\) vertices, \(E = 32\) edges, \(F = 24\) faces, and \(C = 8\) cells, the equation is:

\[ P = \frac{80 - 16}{32 + 24 + 8} = \frac{64}{64} = 1 \]

This means one atom is placed on each edge, face, and cell of the tesseract.

1. 1. 1. Generalization and Use Cases

Niblett’s Dimensional Distribution Method provides a generalized approach to distributing points across geometric shapes in higher dimensions, potentially finding applications in: - **Molecular modeling**: Arranging atoms in molecules or crystals. - **Material science**: Optimizing lattice structures in 3D and 4D. - **Computer graphics**: Efficiently placing vertices and edges in modeling and animation.

The method's versatility allows it to be extended to other shapes beyond standard polygons and polyhedra, including irregular geometries.

1. 1. 1. References

- Combinatorial Geometry and Discrete Geometry. - Lattice Structures in Crystallography.