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The notion of a spectral geometry on a compact metric space X is introduced. This notion serves as a discrete approximation of X motivated by the notion of a spectral triple from non-commutative geometry. A set of axioms charaterising…
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point,…
The explicit relationship among perception, communication, and design is being discussed in some detail, in order to relate it to characteristic details of the modeling of the world which defines the scientific and artistic activities of…
Traditional geometry employs idealized concepts like that of a point or a curve, the operational definition of which relies on the availability of classical point particles as probes. Real, physical objects are quantum in nature though,…
Human cognition spans perception, memory, intuitive judgment, deliberative reasoning, action selection, and social inference, yet these capacities are often explained through distinct computational theories. Here we present a unified…
User's perception of product, by essence subjective, is a major topic in marketing and industrial design. Many methods, based on users' tests, are used so as to characterise this perception. We are interested in three main methods:…
Crofton's formula of integral geometry evaluates the total motion invariant measure of the set of $k$-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs…
In this paper, we will study the following pattern recognition problem: Every pattern is a 3-dimensional graph, its surface can be split up into some regions, every region is composed of the pixels with the approximately same colour value…
Videos inherently represent 2D projections of a dynamic 3D world. However, our analysis suggests that video diffusion models trained solely on raw video data often fail to capture meaningful geometric-aware structure in their learned…
This paper makes the case that a powerful new discipline, which we term perception engineering, is steadily emerging. It follows from a progression of ideas that involve creating illusions, from historical paintings and film, to video games…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
Human motion analysis and understanding has been, and is still, the focus of attention of many disciplines which is considered an obvious indicator of the wide and massive importance of the subject. The purpose of this article is to shed…
We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which…
This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…
In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We will argue that this approach provides a geometric semantics for such…
Gestalt theory has provided perceptual science with a conceptual framework which has inspired researchers ever since, taking the field of perceptual organization into the 21st century. This opinion article discusses the importance of rules…
The human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in…
Recognizing spatial relations and reasoning about them is essential in multiple applications including navigation, direction giving and human-computer interaction in general. Spatial relations between objects can either be explicit --…
Humans intuitively recognize objects' physical properties and predict their motion, even when the objects are engaged in complicated interactions. The abilities to perform physical reasoning and to adapt to new environments, while intrinsic…
There is physiological evidence that our ability to interpret human pose and action from 2D visual imagery (binocular or monocular) engages the circuitry of the motor cortices as well as the visual areas of the brain. This implies that the…