Identity

Identity is Symmetry. Symmetry is Identity.

  • Symmetry is Identification.

  • As a word, Symmetry reads as “same metric”. Same metric is same measure is same amount is same mass.

  • Same amount of what? Same amount of asymmetry on each side. Same amount (measure) of each side. Same amount of the forms of identity.

  • Equilibrium is same amount is same mass. Equilibrium is mass balance.

  • Symmetry is equilibrium.

  • Symmetry is mass balance.

  • Symmetry is Measure balance.

Identifying Reflections

Symmetry expresses its forms through assymetry. Symmetry expresses in different forms through reflections. Reflection is asymmetry. Identifying the asymmetries gives symmetry. Composing asymmetries together gives symmetry. Symmetry is Identity. We may diagrammatically represent

  • In Matter language, one is condensed, the other is fluid. Equilibrium interface between the two identifies both forms condensed and fluid. Equilibrium interface is the symmetry. Equilibrium interface is the identity.

Condensed \(0\) Fluid

  • In Natural language, Identity gets partitioned into two sides across a mirror.

  • In Duality language, The duals are asymmetric. Each is the reflection of the other. Identifying duals gives non-dual identity. Symmetry is non-dual.

  • In English language, one is straight, the other is crooked.

  • In Horizontal Direction language, one is left, the other is right. One is forward, the other is backward. Identity is right here. Self is identity.

  • In Vertical Direction language, one is down, the other is up.

  • In Amount language, one is high, the other is low. Identity is same level.

  • In Sign language, one is \(-\), the other is \(+\).

  • In Binary language, one is \(0\), the other is \(1\).

  • In Rotation language, one is clockwise, the other is anticlockwise. Within Rotation, vertical and horizontal are reflections.

  • In Orthogonality language (as a form of Rotation language), one is along some direction, the other is along the “orthogonal direction”.

  • In Planar Geometry language, one is radius, the other is angle. Identifying the two gives Circle as the identity.

  • In Geometry language, Coordinate axes are the forms of asymmetry. Origin is the identity.

  • In Mathematics language, various theories are the forms of asymmetry. The equivalence of theories gives different Identities.

    • \(0\) is the Prime Identity.

    • \(=\) is the Mirror across which asymmetry (reflections) occurs. \(=\) is the identification.

    • \(1\) is the Identity in Group Theory.

    • In Field Theory, \(0\) is the identity of one group with operation \(+\) and \(1\) is the identity of another group with the operation \(\cdot\). Composing the two groups gives the corresponding Field as the Identity. When both Identities \(0\) and \(1\) are identified, it gives \(0\) as the trivial group, which is the Prime Identity. One may look at this as the reverse: Field arise from \(0\) by an assymetry- a group and further assymetry (another reflection) to form a second group. In Euclidean Geometry, a Field may be seen as two parallel lines with \(0\) as the origin of the first line and 1 as the origin of the other line. In Number theory, a Field is seen as the Rationals. One may keep Repeating one group in different forms to give multiple reflections of itself and form band (imagine splitting of light into a spectrum of colors which is the band). Or a group can be rotated around to form a Spokes of a Wheel. Complete rotation forms a Disc.

    • In Geometry, Circle ( \(0\) ) is the Prime Identity. The \(n\)-Speheres are Identity for dimension \(d\).

    • Geometrization Conjecture is the Identity proved by Grigori Perelman to identify three dimensional topologial structures with the corresponding unique geometric structure.

  • In Group language, one is an element \(a\), the other is its inverse \(a^{-1}\), identified together forms \(a a^{-1}= 1\) with \(1\) as the identity.

  • In Calculus language, one is differentiation, the other is integration, identifying together gives \(1\) as the identity. That is \(I \,\, \cdot \,\, D = 1\). To state in the other way, differentiation and integration are reflections of each other, that is, \(D\,\, =\,\, I\). In Sign language, \(D\) acts as \(-\) and \(I\) acts as \(+\). This translates to \(I \,\, \cdot \,\, I = \,\,D \,\, \cdot \,\, D\,\, = \,\, 1\) or \(\,\,- \,\,-\,\, = \,\,+ \,\, + \,\, = 1\). To clarify, the signs here represent assymetric directions they are not to be confused with the usual operations, although there exists a way to identify them. Formally, we may differentiate the Identity \(1\) into two identites \(-1\) and \(1+\). This converts Calculus from a Group to a Field. That is, \(D \,\, \cdot \,\, D = -\,\,1\) and \(I \,\, \times \,\, I\,\, = \,\,1\,\, +\,\, 1\). In basic calculus the two group operations \(\cdot\) and \(\times\) are identified to \(\cdot\). The two \(I\)’s on either side of \(\times\) may be differentiated to give “roots of \(1+\)”. These are represented by the symbols \(-1\) and \(1+\), which by definition obey \((\,\,-\,\,1 \,\,) \,\, (\,\,-\,\, 1\,\,)\,\,=\,\, (\,\, 1\,\,+\,\,) \,\, (\,\, 1\,\,+\,\,) \,\, =\,\,1\,\,+\) . The two \(D\)’s on either side of the group operation may further be differentiated to give “roots of \(-1\)”. These are represented by the symbols \(-i\) and \(i+\), which by definition obey \(\,\, - \,\,i \,\, i\,\, - \,\, = \,\,i \,\,+ \,\,+\,\, i\,\,= \,\,-\,\,1\,\,\). See that Even always maintains its own Identity on Differentiation whereas Odd splits its own Identity into two new Identities. This has far reaching consequences, one of which is that finite Fields of Even number of elements have characteristic \(2\).

    • In functions, one is \(\exp\), other is \(\log\), identifying gives \(1\) as the Identity.

    • \(-theta\) \,\,,\,\, \(\theta+\) identify to \(0\) as the Identity in the Angle group where \(\theta\) acts as the identifier of angles. Algebraic symbols \(x\) or \(theta\) which identify various quantities are symmetries hence they are the identity. Algebra is symmetry. Here in Angle Group, \(\theta\) is one Identity which gets connected to \(0\) as the prime identity through \(+\) and its inverse \(-\). This is stated as \(\,\,-\,\,\theta\,\, \,\,\theta\,\, + = \,\,0\).

    • The identity ( \(\theta\) ) composed (\,\, \(cos\) \,\,,\,\, \(\sin\) \,\, ) gives the identities \(\)

    • \(0\) and \((\,\, -\,\, (\,\, -\,\, 1\,\, )\,\, \,\,(\,\, 1\,\, +\,\, )\,\, +\,\, ) \,\, (\,\, \pi \,\,)\) are duals which are identified by the angle group to \(0\) as the Identity.

    • In Differential Equations, Gradient is one side and Divergence is the other side, they identify together to form Laplacian as the Identity. \(\nabla \cdot\) is one and \(\nabla \times\) is the other, Identified together they give Helmholtz Theorem as the Identity.

    • In Linear Algebra, the Dot product is one and the cross product is the other, composing together, gives scalar triple product as the Identity. This is the analogue of Determinant. Determinant is the Identity in Square Matrices. Correspondingly, D’Alembertian is the Identity in Wave type Differential Equations. Composing duals in a space fills up “space”. For example, disc can be composed with its axis to form cylinder as the identity. Here disc and the axis are duals which composed together fill up the space of the cylinder. That is, every point on the axis acts as a mirror about which disc on one side may be reflected to the other side and composition of all these discs forms a cylinder. “Field like behaviour”. Surface of a cylinder cn be formed by composing the axis with a circle. The points along the axis forms one group and the angles along a circle forms another group. Composing the two gives the surface of a cylinder (which is the field). Alternatively, radii of multiple circles may be composed with angles of each circle to form a plane (which is another field). Symmetrically, angles along one circle may be composed with angles along another circle to give a sphere. That is a circle (which is itself a rotation) can be rotated about any axis of itself to give a sphere.

  • In Complexity language, one is \(P\), the other is \(NP\). \(P\) and \(NP\) is Identity.

  • In Games Language, Unique Games conjecture is Identity.

  • In Electromagnetic language, one is Electric field \(E\), the other is Magneic field \(B\).

  • In Space-time language, one is Space, the other is Time.