Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0905.4128

To Get Overall Shapes and New Data of the 120-Cell and the 600-Cell

Kaida Shi
View original source ↗Read source document ↗Export metadata ↓

Abstract

This research will be helpful for people to display the 2-dimensiona projective models of 4-variable actual problems in many fields, in order to investigate deeply those actual problems. By using the theory of N-dimensional finite rotation group of the regular polytopes, the author established the 2-dimensional projective model of 4-dimensional rectangular coordinate system, and deduced a transformation matrix, and adopt it to display successfully the 2-dimensional overall shapes of two most complicated regular polytopes 120-Cell and 600-Cell. In the meantime, the author calculated all the vertex coordinates and determine the joint relationships between adjacent vertices of the regular polytopes 120-Cell and 600-Cell. Also, this provided a pattern for displaying the 2-dimensional projective model of 4-variable actual problem. (The tables of vertex coordinates of the 120-Cell and the 600-Cell and the table of joint relationship between adjacent vertices of the 120-Cell and the 600-Cell are listed below)

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kaida Shi. 2009-06-13. To Get Overall Shapes and New Data of the 120-Cell and the 600-Cell. https://arxiv.org/abs/0905.4128

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Hilbert's 8th Problem

This note takes the probabilistic half of the Riemann Xi story on its own terms. Every object in the subject is a first passage time: Riemann's kernel is the law of the logarithm of a sum of two hitting times of a three dimensional Bessel process, Polya's approximation is the first passage of a Brownian motion with drift, and the reciprocal Xi function, under the Riemann hypothesis, is the Laplace transform of an infinite convolution of exponentials whose rates are the squared zeros. That reciprocal is written as $F_α(s)=ξ(α)/ξ(α+\sqrt s)$, and complete monotonicity, unconditional for $α\ge1$, is conjectured to persist to the critical basepoint $α=1/2$. Kent's eigenvalue expansion says which laws can arise this way, namely those whose Thorin measure is a Dirichlet spectrum with unit atoms, and Krein's inverse spectral theory turns the hypothesis into the existence of a string. The passage from Riemann to Polya is a flow, not a jump: the Cauchy semigroup on Thorin measures, each step an Esscher tilt followed by a Brownian subordination along a curvature family of hyperbolic Bessel processes, with the arithmetic surviving as Fourier modes damped like $e^{-2πk\varepsilon}$. The arithmetic lives in the atoms and nowhere else. Approximations rank by what they keep: Polya keeps neither atoms nor tempering and is off by a factor of three, a fitted Bessel dimension reaches one per cent, and a few atoms with an erfc tempering stay better than one part in a thousand across four decades. And the flow runs backwards: the Thorin measure of the reciprocal is evaluated from a prime sieve with no reference to any zero, and nonnegative deconvolution of it returns the first ten zero ordinates with unit masses, nine of them to four decimals and one to three. All identities are verified with mpmath, and the scripts are included.

math.GM↗
arXiv2026

Dynamical Geometry of Principal Bundle Constrained Systems: Compatible Pairs, Contact Degeneration, and Adapted Metrics

We study invariant codimension-one constraints on principal bundles through compatible pairs: a constraint distribution and a nonzero coadjoint field parallel for a principal connection. Pairing the field with the connection gives an invariant one-form whose Levi form separates a horizontal curvature contribution from a vertical coadjoint-orbit contribution. This decomposition yields criteria for integrability, contactness, and characteristic reduction; holonomy and stabilizer reductions describe global existence. For evolving compatible pairs, we identify the mixed-curvature obstruction to compatibility and prove that connection transport preserves the Levi geometry. For initially contact data over a closed base of dimension $2n$, we establish matching bounds for the quadratic $H^{n+1}$ cost of contact degeneration: making the paired curvature vanish on a Darboux ball of radius $r$ in time $T$ costs an amount comparable to $[T\log(R_*/r)]^{-1}$. The constructed paths remain contact before $T$, preserve the curvature class, and force the $L^\infty$ norm of every transporting velocity gradient to grow at least as $1/[2(T-t)]$ on the collapsing region in Darboux coordinates. On a closed three-manifold, a contact form preserved by a locally free circle action admits an invariant adapted metric with any prescribed positive curl eigenvalue and unit circle generator. We parametrize all such metrics and prove that their space is contractible. Along the circle-bundle degeneration paths, every continuous tensor limit of normalized adapted metrics is degenerate above the collapsing region.

math.GM↗