Symmetry
Symmetry is Same Measure is Measure Balance. Measure takes various forms. In Geometry, Measure is the Metric. Metric is a scale. Unlike usual scales which have only one \(0\) and either side of it are negative (\(-\)), positive (\(+\)), in Nodal geometry there are many \(0\)s which are called nodes. Nodal geometry is a scale balance with such a new scale. In the symbols of metric, Nodes are represented by commas (,) and the positive or negative on each side is represented by signs \(-\), \(+\). For example, the Real Number Line may be represented as \((-,+)\) where (,) is the \(0\), \(+\) are the positive numbers and \(-\) are the negative numbers.
Nodal Geometry
Uniform Metric is the most symmetric metric. Every point in space is treated the same: symmetrically (uniformly). In such a space, everything is identified to be one. Hence, there is no dynamics (no high or low for flow to take place). For example, the Sphere is an ultimate symmetry (uniform metric) for which all points are equidistant from the centre. A form of this is the Uniformizaiton Conjecture which was proved by Grigori Perelman in his paper Ricci Flow with Surgery on 3 Manifolds.
Laplacian as Identity Cycle
In Calculus, there are two fundamental flows: The Gradient and the Divergence. Gradient and Divergence are Cyclic duals of each other. Composed together in the order: Gradient followed by Divergence which is Divergence after Gradient (Divergence of Gradient), they give the Identity which is named Laplacian. Usually the symbols used to denote these are as follows: Gradient is denoted by \(\nabla\), Divergence is denoted by \(\nabla \cdot\) and their composition in the order given above is the Laplacian denoted by \(\Delta\).
I would like to propose a slight change in the symbols used to denote these. First, I would like to smoothen (mollify) the edges of the Traingles used. I would like to denote Gradient by a lower smooth traingle and the Divergence by an upper smooth triangle. Their composition in the order: Gradient followed by Divergence which is (Divergence of Gradient) gives Identity which is named the Laplacian. I would like to use the placing of lower smooth traingle (Gradient) and upper smooth traingle (Divergence) to meet at their vertex and form \(8\) as the Symbol for the Laplacian (Identity). This is identified as an Equilibrium Cone (reminds of an hourglass). It is rotated \(\infty\) (reflection of horzontal about equiaxis between vertical and horizontal) so as to Identify with Elliptic Regularity as a Fundamental Law arising from the Cyclic Identification of Gradient and Divergence. The meeting point of the two smooth triangles is the node (Fundamental Origin). Both \(\sin\) and \(\cos\) emerge as solutions to the Laplace Eigenvalue Equation Letting the
Nodal Domains
- Courant Nodal Domain Theorem.
Fourier Transform
Gaussian is Identity of Fourier Transform.