Cohomology fractals

We introduce cohomology fractals; these are certain images associated to a cohomology class on a hyperbolic three-manifold. They include images made entirely from circles, and also images with no geometrically simple features. They are closely related to limit sets of kleinian groups, but have some key differences. As a consequence, we can zoom in almost any direction to arbitrary depth in real time. We present an implementation in the setting of ideal triangulations using ray-casting.Comment: 8 pages, 30 figures and subfigure

Entanglement entropy, scale-dependent dimensions and the origin of gravity

We argue that the requirement of a finite entanglement entropy of quantum degrees of freedom across a boundary surface is closely related to the phenomenon of running spectral dimension, universal in approaches to quantum gravity. If quantum geometry hinders diffusion, for instance when its structure at some given scale is discrete or too rough, then the spectral dimension of spacetime vanishes at that scale and the entropy density blows up. A finite entanglement entropy is a key ingredient in deriving Einstein gravity in a semi-classical regime of a quantum-gravitational theory and, thus, our arguments strengthen the role of running dimensionality as an imprint of quantum geometry with potentially observable consequences.Comment: 8 pages, 1 figure. Received an Honorable Mention in the 2017 Essay Competition of the Gravity Research Foundatio

Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology

We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the Packed Swiss Cheese Cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of 3-dimensional spheres inside a 4-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of a divergent sum which involves the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of the points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.Comment: 38 pages LaTe

Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns

In this paper, we generalize the idea of star-shaped set inversion fractals using iterations known from fixed point theory. We also extend the iterations from real parameters to so-called q-system numbers and proposed the use of switching processes. All the proposed generalizations allowed us to obtain new and diverse fractal patterns that can be used, e.g., as textile and ceramics patterns. Moreover, we show that in the chaos game for iterated function systems - which is similar to the inversion fractals generation algorithm - the proposed generalizations do not give interesting results

Twisted superfluids in moving frame

{Helical waveguide filled by superfluid is shown to transform the linear displacements of reference frame $\vec V$ into rotations of atomic ensemble and vise versa rotations of reference frame $\vec \Omega_{\oplus}$ are the cause of linear displacements of ensemble. In mean-field Gross-Pitaevskii equation with a weakly modulated gravitation the exact solutions for macroscopic wavefunction $\Psi$ demonstrate the emergence of $redshift$ phase factor being proportional to the population of condensate $N_a$. is shown to have}Comment: 6 pages, 2 figures , submitted to referred journa

Electrostatics in Fractal Geometry: Fractional Calculus Approach

The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume $V_d\sim r^d$ with the fractal dimension $d$ and a complementary host volume $V_h=V_3-V_d$. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation

Star-shaped Set Inversion Fractals

In the paper, we generalized the idea of circle inversion to star-shaped sets and used the generalized inversion to replace the circle inversion transformation in the algorithm for the generation of the circle inversion fractals. In this way, we obtained the star-shaped set inversion fractals. The examples that we have presented show that we were able to obtain very diverse fractal patterns by using the proposed extension and that these patterns are different from those obtained with the circle inversion method. Moreover, because circles are star-shaped sets, the proposed generalization allows us to deform the circle inversion fractals in a very easy and intuitive way