426 research outputs found
Puzzling the 120-cell
We introduce Quintessence: a family of burr puzzles based on the geometry and
combinatorics of the 120-cell. We discuss the regular polytopes, their
symmetries, the dodecahedron as an important special case, the three-sphere,
and the quaternions. We then construct the 120-cell, giving an illustrated
survey of its geometry and combinatorics. This done, we describe the pieces out
of which Quintessence is made. The design of our puzzle pieces uses a drawing
technique of Leonardo da Vinci; the paper ends with a catalogue of new puzzles.Comment: 25 pages, many figures. Exposition and figures improved throughout.
This is the long version of the shorter published versio
Enumerating Gribov copies on the lattice
In the modern formulation of lattice gauge-fixing, the gauge fixing condition
is written in terms of the minima or stationary points (collectively called
solutions) of a gauge-fixing functional. Due to the non-linearity of this
functional, it usually has many solutions called Gribov copies. The dependence
of the number of Gribov copies, n[U] on the different gauge orbits plays an
important role in constructing the Faddeev-Popov procedure and hence in
realising the BRST symmetry on the lattice. Here, we initiate a study of
counting n[U] for different orbits using three complimentary methods: 1.
analytical results in lower dimensions, and some lower bounds on n[U] in higher
dimensions, 2. the numerical polynomial homotopy continuation method, which
numerically finds all Gribov copies for a given orbit for small lattices, and
3. numerical minimisation ("brute force"), which finds many distinct Gribov
copies, but not necessarily all. Because n for the coset SU(N_c)/U(1) of an
SU(N_c) theory is orbit-independent, we concentrate on the residual compact
U(1) case in this article and establish that n is orbit-dependent for the
minimal lattice Landau gauge and orbit-independent for the absolute lattice
Landau gauge. We also observe that contrary to a previous claim, n is not
exponentially suppressed for the recently proposed stereographic lattice Landau
gauge compared to the naive gauge in more than one dimension.Comment: 39 pages, 15 eps figures. Published version: minor changes onl
Euler potentials for the MHD Kamchatnov-Hopf soliton solution
In the MHD description of plasma phenomena the concept of magnetic helicity
turns out to be very useful. We present here an example of introducing Euler
potentials into a topological MHD soliton which has non-trivial helicity. The
MHD soliton solution (Kamchatnov, 1982) is based on the Hopf invariant of the
mapping of a 3D sphere into a 2D sphere; it can have arbitrary helicity
depending on control parameters. It is shown how to define Euler potentials
globally. The singular curve of the Euler potential plays the key role in
computing helicity. With the introduction of Euler potentials, the helicity can
be calculated as an integral over the surface bounded by this singular curve. A
special programme for visualization is worked out. Helicity coordinates are
introduced which can be useful for numerical simulations where helicity control
is needed.Comment: 15 pages, 12 figure
Conformal compactification and cycle-preserving symmetries of spacetimes
The cycle-preserving symmetries for the nine two-dimensional real spaces of
constant curvature are collectively obtained within a Cayley-Klein framework.
This approach affords a unified and global study of the conformal structure of
the three classical Riemannian spaces as well as of the six relativistic and
non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both
Newton-Hooke and Galilean), and gives rise to general expressions holding
simultaneously for all of them. Their metric structure and cycles (lines with
constant geodesic curvature that include geodesics and circles) are explicitly
characterized. The corresponding cyclic (Mobius-like) Lie groups together with
the differential realizations of their algebras are then deduced; this
derivation is new and much simpler than the usual ones and applies to any
homogeneous space in the Cayley-Klein family, whether flat or curved and with
any signature. Laplace and wave-type differential equations with conformal
algebra symmetry are constructed. Furthermore, the conformal groups are
realized as matrix groups acting as globally defined linear transformations in
a four-dimensional "conformal ambient space", which in turn leads to an
explicit description of the "conformal completion" or compactification of the
nine spaces.Comment: 43 pages, LaTe
Transverse Mercator with an accuracy of a few nanometers
Implementations of two algorithms for the transverse Mercator projection are
described; these achieve accuracies close to machine precision. One is based on
the exact equations of Thompson and Lee and the other uses an extension of
Krueger's series for the projection to higher order. The exact method provides
an accuracy of 9 nm over the entire ellipsoid, while the errors in the series
method are less than 5 nm within 3900 km of the central meridian. In each case,
the meridian convergence and scale are also computed with similar accuracy. The
speed of the series method is competitive with other less accurate algorithms
and the exact method is about 5 times slower.Comment: LaTeX, 10 pages, 3 figures. Includes some revisions. Supplementary
material is available at http://geographiclib.sourceforge.net/tm.htm
A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres
We consider Poisson's equation on the -dimensional sphere in the situation
where the inhomogeneous term has zero integral. Using a number of classical and
modern hypergeometric identities, we integrate this equation to produce the
form of the fundamental solutions for any number of dimensions in terms of
generalised hypergeometric functions, with different closed forms for even and
odd-dimensional cases
Modelos, axiomática y geometría del plano hiperbólico
First of all, we need to understand why there are other geometries such as Hyperbolic geometry besides the intuitive Euclidean geometry. Who discovered this geometry? Lobachevski, a young scientist who decided to leave the medical career to devote himself completely to the study of a geometry that he called “imaginary geometry”, is a founder of this geometry. He made progress not only in mathematics but also in physics, such as Einstein’s theory of relativity. How did Lobachevski come up with this geometry? Before giving an answer to this question, let’s see what an axiomatic system is
A model for Hopfions on the space-time S^3 x R
We construct static and time dependent exact soliton solutions for a theory
of scalar fields taking values on a wide class of two dimensional target
spaces, and defined on the four dimensional space-time S^3 x R. The
construction is based on an ansatz built out of special coordinates on S^3. The
requirement for finite energy introduces boundary conditions that determine an
infinite discrete spectrum of frequencies for the oscillating solutions. For
the case where the target space is the sphere S^2, we obtain static soliton
solutions with non-trivial Hopf topological charges. In addition, such hopfions
can oscillate in time, preserving their topological Hopf charge, with any of
the frequencies belonging to that infinite discrete spectrum.Comment: Enlarged version with the time-dependent solutions explicitly given.
One reference and two eps figures added. 14 pages, late
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