Related papers: Borobudur was Built Algorithmically
Consider a pyramid made out of unit cubes arranged in square horizontal layers, with a ledge of one cube's length around the perimeter of each layer. For any natural number $k$, we can count the number of ways of choosing $k$ unit cubes…
A class of simplified measures is constructed to capture the key features of generic spatio-temporally chaotic systems. A combined analytical and numerical investigation allows us to extablish the scaling beahviour of the fractal dimension…
We present structural properties of two-dimensional polymers as far as they can be described by percolation theory. The percolation threshold, critical exponents and fractal dimensions of clusters are determined by computer simulation and…
Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy…
A fundamental challenge in materials design is linking building block attributes to crystal structure. Addressing this challenge is particularly difficult for systems that exhibit emergent order, such as entropy-stabilized colloidal…
Matt Blum conjectured that the number of tilings of a hexagonal dungeon with side-lengths $a,2a,b,a,2a,b$ (for $b\geq2a$) equals $13^{2a^2}14^{\lfloor a^2/2\rfloor}$. Ciucu and the author of the present paper proved the conjecture by using…
We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad {a,b,c,d} = abcd + dcba as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are…
Jordan algebras were first introduced in an effort to restructure quantum mechanics purely in terms of physical observables. In this paper we explain why, if one attempts to reformulate the internal structure of the standard model of…
It is demonstrated that iterative repeating of some simple geometric construction leads unavoidably in the limit to the golden ratio. The procedure appears to be quickly convergent regardless of a ratio of initial elements sizes. This could…
The eigenvalues of the Hyperspherical billiard are calculated in the semiclassical approximation. The eigenvalues where this approximation fails are identified and found to be related to caustics that approach the wall of the billiard. The…
Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-\lambda$ over $\mathbb{F}_{q^2}$ is given, where $\lambda$ is a unit in $\mathbb{F}_{q^2}$. Based on this…
In this note, we provide equivalent definitions for fractal geometric dimensions through dyadic cube constructions. Given a metric space $X$ with finite Assouad dimension, i.e., satisfying the doubling property, we show that the…
By viewing the covers of a fractal as a statistical mechanical system, the exact capacity of a multifractal is computed. The procedure can be extended to any multifractal described by a scaling function to show why the capacity and…
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal…
We explore the fractal nature of particle showers using Monte-Carlo simulation. We define the fractal dimension of showers measured in a high granularity calorimeter designed for a future lepton collider. The shower fractal dimension…
Robots deployed at orders of magnitude different size scales, and that retain the same desired behavior at any of those scales, would greatly expand the environments in which the robots could operate. However it is currently not known…
The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies.…
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities…
Cuspidal robots are robots with at least two inverse kinematic solutions that can be connected by a singularity-free path. Deciding the cuspidality of generic 3R robots has been studied in the past, but extending the study to…
Some years ago we proposed a new approach to the analysis of galaxy and cluster correlations based on the concepts and methods of modern statistical Physics. This led to the surprising result that galaxy correlations are fractal and not…