Related papers: Abbott Dimension, Mathematics Inspired by Flatland

In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the…

The concepts of spread and spread dimension of a metric space were introduced by Willerton in the context of quantifying biodiversity of ecosystems. This paper develops practical applications of spread dimension in the context of machine…

To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for…

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain…

Since antiquity, from Euclid of Alexandria to Galileo Galilei to Immanuel Kant to Hermann Minkowksi to Albert Einstein, the question of the nature of space and time has occupied scientists and philosophers. In the four-dimensional…

Accessibility is essential for designing inclusive urban systems. However, the attempt to capture the complexity of accessibility in a single universal metric has often limited its effective use in design, measurement, and governance across…

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Several approaches to quantum gravity (including the model of superplastic vacuum; Diakonov tetrads emerging as the bilinear combinations of the fermionis fields; $BF$-theories of gravity; and effective acoustic metric) suggest that in…

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic…

The field of functional recognition or affordance estimation from images has seen a revival in recent years. As originally proposed by Gibson, the affordances of a scene were directly perceived from the ambient light: in other words,…

Some superstring theories have more than one effective low-energy limit, corresponding to classical spacetimes with different dimensionalities. We argue that all but the 3+1-dimensional one might correspond to ``dead worlds'', devoid of…

We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.

In this paper, the intuitive idea of tilt is formalised into the rigorous concept of tilt rotations. This is motivated by the high relevance that pure tilt rotations have in the analysis of balancing bodies in 3D, and their applicability to…

A physical applicability of normed split-algebras, such as hyperbolic numbers, split-quaternions and split-octonions is considered. We argue that the observable geometry can be described by the algebra of split-octonions. In such a picture…

In five dimensional cosmological models, the convention is to include the fifth dimension in a way similar to the other space dimensions. In this work we attempt to introduce the fifth dimension in a way that a time dimension would be…

Heisenberg's uncertainty principle is often cited as an example of a "purely quantum" relation with no analogue in the classical limit where $\hbar \to 0$. However, this formulation of the classical limit is problematic for many reasons,…

We introduce a notion of dimension for the solution set of a system of algebraic difference equations that measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but, as we…

Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This…

During many years since the birth of quantum mechanics, instrumentalist interpretations prevailed: the meaning of the theory was expressed in terms of measurements results. But in the last decades, several attempts to interpret it from a…

In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has…