Mathematics

II.1 Foreword

"To understand the Universe, you must understand the language in which it's written, the language of Mathematics." Galileo Galilei, The Father Of Science

As a mathematician, I have discovered that the geometry of the universe is 2𝛑 dimensional.

where ζ is the Riemann zeta function ζ(s).

II.2 Negative dimensions

Dans l'univers il y a 3 dimensions d'espace et 1 dimension de temps.

la distance d entre deux événements situés en (x,y,z,t) et en (x’,y’,z’,t’) dans l’espace-temps s’écrit :

d=(x-x’)²+(y-y’)²+(z-z’)² – c².(t-t’)²

Ce n’est pas une distance euclidienne à cause du signe « moins », donc le temps n’est pas une dimension “comme les autres”, même après l’avoir transformé en distance grâce à la vitesse de la lumière. L’espace-temps n’est donc pas un espace euclidien, mais un espace de Minkowski.

Ainsi je peux dire que les 3 dimensions d'espace sont des dimensions "positives" parce que la mesure d'une distance dans ces dimensions est positive selon la mesure de Minkowski alors que la dimension de temps est "négative" puisque la mesure d'une distance dans cette dimension est négative selon la mesure de Minkowski .

II.3 Probabilities

We may associate an event occurring in the future to a probability.

P(X=a) = 1/3

P(X=b) = 1/3

P(X=c) = 1/3

Somehow, one may consider each of this probability as a unique time-dimension.

We can then consider a space-time with multiple time dimensions.

II.4 Quaternions

P.R. Girard's 1984 essay The quaternion group and modern physics^[21] discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely SO(3), the Lorentz group, the general theory of relativity group, the Clifford algebra SU(2) and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics of rigid body motion. Next he used complex quaternions (biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions (split-biquaternions) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.^[22]

The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "Plate trick").

Wikipedia