Twistor theory

PN is the space of real null rays in Minkowski space.


PT is the projective twistor space.


Z is a twistor, and Z2=Z3=0





You see Penrose tiling.

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The Sylvester-Gallai Theorem

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The deformation is the mathematical concept. The Sylvester-Gallai Theorem is that a finite set of points in the plane have the proper line through any two of them, and it passes through a third point of the set. Moreover, it must be on the same line. P is the the noncollinear finite set. S(P) is the set of connecting lines in P. p∈P, s∈S(P) p and s are perpendicular, so they are not the ordinary line. (s*,p*) is the smallest distance, and s* is in the ordinary line. l∩P={p} l must be the ordinary line. This is the contradiction. l∩S(P) There is the cyclically intersection points (p,x1). x1,・・・xk is on l. Moreover it crosses S. This must be ordinary. You also see the extra dimensions. i lines are determined by P.
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Montgomery's pair correlation conjecture

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Hybrid orbitals are the explanation of molecular geometry and atomic bonding. This is Montgomery's pair correlation conjecture, which is the pair correlation between pairs of zeros of the Riemann zeta function . γ and　γ' are imaginary. This is like piling prime numbers . Methane and hydrogen are gasses. C=C is carbon bonding which is in surface. Gasses move like ↑ ↑ ↓. This is called π bond.
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Hybrid orbital

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P^2+Σ2P is the pattern of prime numbers which move like atom. This is Carbon. There are 6 electrons. Σ is hybrid, and atomic orbitals are mixtures. Then, you see sp^3. You move one electron in 2S to 2P. This is called promotion. 2S lose one electron. There are 4 hydrogen atoms. This is tetrahedron and sp^3. Methane is CH4. In sp^2 hybridisation, 2S move to 2P. This is triangle and sp^2.　Ethylen is C2H4 which is the double bond. C=C is connected tightly.
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