organiker'at'lycos.com

Revised May 2006

Overview of theory

Software for calculating quantitative parity divergence

Petitjean, Michel, J. Math. Phys. 40(9) 4587 (1999)

J. Math. Phys. 43(8) 4147 (2002)

Compt. Rend. Acad. Sci. (Paris), serie IIc, 4(5) 331 (2001)

J. Math. Chem. 22(2-4) 185 (1997)

J. Math. Chem. 23 429 (1998)

(*S*)-[6.6]Chiralane optimized by
Hartree-Fock 6-31G(d) has overall CHI=0.982423 and CHI=1 exactly
for its carbon skeleton alone. Here is a different view of
[6.6]chiralane.

[6.6]Chiralane is a condensed alkane, C_{27}H_{28},
built of fused homochiral twist-boat cyclohexane rings. Its skeletal
graph is non-planar by Kuratowski's theorem. (A graph is planar if
and only if it has no subgraph homeomorphic to K_{5} or
K_{3,3}). It has three C_{2}-axes and four
C_{3}-axes of rotation present overall - point group
**T** (not **T _{d}** or

Many other point group **T** C_{30}-C_{50} skeletons
give CHIs below 0.06. We believe large CHI generation requiries a
convex mass distribution. Several ring size variations of the chiralane
skeleton generate CHI=1. Flat graphs in point group **T** or **I** can
also deliver CHI=1 (hollow chirolanes and fullerenes).

[6.6]Chiralane in 3-D manipulable VRML (e.g., Cortona viewer) here (wrl), and here (wrl) with more display options.

Molecule | Overall CHI | Graphics | Molecule | Overall CHI | Graphics | |
---|---|---|---|---|---|---|

[5.5]Chiralane | 1.000000 | CHIME | C44 Fullerene point group T |
1.000000 | CHIME | |

[5.7]Chiralane | 0.985962 | CHIME | C52 Fullerene point group T |
1.000000 | CHIME | |

[6.6]Chiralane | 0.982423 | CHIME | C92 Fullerene point group T |
1.000000 | CHIME | |

[6.7]Chiralane | 0.979747 | CHIME | C100 Fullerene point group T |
1.000000 | CHIME | |

[7.7]Chiralane | 0.984151 | CHIME | C140 Fullerene point group I |
1.000000 | CHIME | |

[8.8]Chiralane | 0.989588 | CHIME | C260 Fullerene point group I |
1.000000 | CHIME | |

[7]Chirolane | 0.986908 | CHIME | ||||

[8]Chirolane | 0.989588 | CHIME |

Gravitation theory can be parity-even math: Galileo, Newton, Green's
function; Einstein and metric gravitation overall; non-heterotic
string theory. Gravitation theory can be parity-odd math: teleparallel
Cartan, affine Weitzenböck. Parity-even gravitation *postulates*
the Equivalence Principle (EP): all local centers of mass vacuum free fall
identically regardless of chemical composition or mass distribution;
inertial and gravitational masses are fundamentally indistinguishable.
Parity-odd gravitation contains a chiral pseudoscalar vacuum background
that diastereotopically interacts with opposite parity mass distributions.
Only one class of theory can be correct. All other predictions are
identical within testable limits.

The proper challenge of spacetime geometry is test mass geometry. Extremal
parity divergence of otherwise chemically and macroscopically identical
paired (sets of) test masses, a parity Eötvös experiment, is an
important and untried
challenge (pdf) to
the Equivalence Principle. Practical considerations would run a
parity Eötvös experiment with single crystal solid spheres of
enantiomorphic space group P3_{1}21 (right-handed screw axes) and
P3_{2}21 (left-handed screw axes)
quartz sparse,
quartz dense,
Point scatter in these plots of log(1-CHI) vs. radius is not noise.
It arises from variation of moments of inertia of increasing radius spherical
samples of anisiotropic lattice. Placement of the sphere's center is
arbitrary. Each unique origin will give a different CHI value versus
a given radius. The overall graph will remain unchanged.

Dr. Penelope Smith at Lehigh University notes that CHI is a connection between eigenvalues, special functions, and their representation theory with solid angles and exponentials of fractions of pi. The graph intercept is the solid angle subtended by the smallest vertex angle of a polyhedron (the supplement of its dihedral angle) defined by the c-axis helix,

log(1-CHI)= -2[log(radius)] + [(180-) ()/60] -

( is the smallest vertex angle in the helix. The slightly distorted tetrahedral O-Si-O helix angle is 110.56° vs. 109.47° undistorted)log(1-CHI) = -2[log(radius)] + 0.494277

3-cm quartz test mass has CHI = 1 - 1.387·10^{-16}, theory

3-cm quartz test mass has CHI = 1 - 1.535·10^{-16}, graph fit

Return to Uncle Al, click here.