Mathematical Concepts Essential for understanding Space
This list of spacial concepts will more likely help the foxes rather than the hedgehogs.
Foxes like horizontal approach to understanding the universe including its mathematical underpinings rather than a vertical "depth" approach.
This "many things" approach to mathematics taken by Terence Tao, Fields medalist, and professor of mathematics among others.
Mathematics has evolved over time. It is "of this world" which means it has evolved in the context of our human sensory limits, and then progressively, through our instrumentation enabled experimentation. Mathematics has been challenged by human problems and the physics of the world and evolved directly as a result of those challenges.
The unreasonable effectiveness of mathematics in the natural sciences" by Eugene Wigner (1960 Communications in Pure and Applied Mathematics) explores the mystifying "why" problem.
The standard 6 parameter cosmological-constant (Lambda - Λ) model of the universe, the Cold Dark Matter (CDM) model has been successfully aligned with large array of cosmological observations. It describes space at very large scales and the infintesmally small scale. A Scalar field is a function that assigns a single scalar value (a number) to each point in a mathematical space. This space can be physical space, a coordinate system, or any other set of points where a scalar value can be defined at each location. Mathematical space refers to a set of points with defined properties, such as Euclidean space (2D or 3D), a coordinate system, or a more abstract space like a function space. A scalar field is a way to describe a property or quantity that varies across a mathematical space. The field itself exists within the structure of that space, and the field values are assigned to specific locations within that space.
Space is central to mathematics. Successive definitions of space have given us new tools for understanding the nature of reality and for computational progress.
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry has a very long and distinguished history".
Here is a 12 min review of modern geometry
Space is not uniform. It's clumpy, bumpy, curved and warped at different scales including the quantum level (see image) Nevertheless "idealized" classic "Euclidean" space has been foundational to the advancement of our mathematics, science, and technology.
Euclidean space is the fundamental space of geometry, intended to represent physical space. It is a idealized two dimensional infinite plane. Euclidean concepts include points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry.
Cartesian Space
Cartesian space, also known as Cartesian coordinate space, is a mathematical construct that uses an ordered set of numbers (coordinates) to specify the position of points in a space. It is typically characterized by a coordinate system with perpendicular axes (x, y, z in three dimensions) and a defined origin.
Minkowski Space
Minkowski space is the main mathematical description of spacetime in the absence of gravitation.
Vector Space (also called a linear space) - Vector spaces are an abstract concept in linear algebra.
In Newtonian physics, space and time had independent identities and gravity.
A Banach space is a complete vector space equipped with a norm. Completeness means that every Cauchy sequence (a sequence where the terms get arbitrarily close to each other) converges to a limit within the space.
A Hilbert space is a Banach space where the norm is derived from an inner product. The inner product allows for the definition of concepts like orthogonality (perpendicularity) and angles, which are not directly definable in a general Banach space.
Einstein Space Time - Special Relativity
Special relativity is the theory of a fixed, flat space-time, without gravity. In Einstein's theories, the ideas of absolute time and space were superseded by the notion of spacetime in special relativity, and curved spacetime in general relativity.
Minkowski space-time is flat, static and infinite.
Einstein Space Time - General Relativity
General relativity is the theory of dynamic space-time, in which curvature gives rise to gravity.
A metric space is a set where you can measure the distance between any two elements (points) using a specific function called a metric or distance function. This metric must satisfy certain properties, ensuring it behaves like a reasonable notion of distance. The most familiar example is Euclidean space (like the 2D or 3D space we live in) where the distance is calculated using the standard distance formula (Pythagorean theorem).
Metric spaces provide a general framework for studying concepts like continuity, convergence, and other topological properties in various mathematical fields. They are used in areas like analysis, geometry, and even in computer science for tasks like image processing and machine learning.
The French mathematician Maurice Frechet
initiated the study of metric spaces in 1905.
Black Holes - Where "the rules of space and time" breakdown
Microscopic Scale - between the macroscopic scale and the quantum scale.
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of function
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