Space, Geometry and Physics

      … It seems to me, in addition to this, to discern in Sarsi a firm belief, that in philosophizing it is necessary to lean on the opinions of some celebrated author, so that our mind, when it is not wedded to the discourse of another, should in everything remain sterile and fruitless; and perhaps estimates that philosophy is a book and a man’s fantasy, like the Iliad and ‘Orlando furioso, books in which the least important thing is that what is written in them is true. Mr. Sarsi, the thing is not like that. Philosophy is written in this very great book that continually stands open before our eyes (I say the universe), but it cannot be understood unless we first learn to understand the language, and know the characters, in which it is written. He is written in the mathematical language, and the characters are triangles, circles, and other geometrical figures, without which means it is impossible to humanly understand a word of it; without these it is a vain wandering through a dark labyrinth. Galileo Galilei (The Assayer, ch. VI, 1623)

Questions:

graph LR
a([ What is the concept of Space in Physics today? ])
b([ What is the connection between Geometry and Physics? ])
c([ What geometric methods are used in modern Science and Technology? ])
a --- B --- C

The answers have changed profoundly due to great advances in mathematics, physics, and the sciences in general in the 20th century, such as:

  • the theory of special and general relativity
  • quantum field theories
  • condensed matter physics and theoretical physics of complex systems
  • the application of mathematical methods of complexity physics to other natural and social sciences
  • the computer revolution from numerical computation to artificial intelligence.

Only 2,300-year-old geometry is studied in schools, when there are so many geometries of which the ancient one is only a small part. To make only a very partial list of the most important disciplines in the resumes of specialists:

  • (Multi-)linear algebra
  • Discrete Geometry
  • Computational Geometry
  • Digital Geometry
  • Analytical Geometry
  • Differential Geometry
  • Fractal Geometry
  • Absolute Geometry
  • Hyperbolic non-Euclidean geometry
  • Parabolic or Euclidean Geometry
  • Non-Euclidean Elliptic (and Spherical) Geometry
  • Simplectic geometry
  • Projective Geometry
  • Affine Geometry
  • Descriptive Geometry
  • Epipolar (or Stereoscopic) Geometry
  • Algebraic Geometry

The main objects studied and used in real-world applications are very varied, for example:

  • Vector or affine spaces with 2,3,4,… n dimensions
  • Infinite-dimensional spaces (Banach’s, Hilbert’s)
  • Vectors, Operators, Shapes and Tensors
  • Fractals, non-integer dimensional spaces
  • Varieties that can be differentiated
  • Fibered spaces
  • Curves and surfaces in n dimensions
  • Polytopes in n dimensions
  • Graphs and networks
  • Nodes
  • Regular or imperfect meshes
  • Covering or doweling
  • Packing or filling

The study, because of the substantial unity of the mathematical language, is closely related to:

  • Linear Algebra, Vector Spaces
  • Abstract Algebra, Group Theory
  • Mathematical Analysis

Geometries are in a sense a special case of a much broader and more important field of science, topology, which is its basis and foundation. Topology in turn is distinguished into:

  • General Topology
  • Algebraic Topology
  • Differential Topology
  • Discrete Topology

The very origin of the word geometry is misleading, in fact it comes from the Greek geos (earth) and metria (measurement), thus literally measuring the earth. But geometry also deals with spaces in which the concepts of measurement, length, angle, of measures, and have nothing to do with the globe (one of the many reasons why it is harmful to study Greek, Latin and classical culture in schools). Moreover, the synthetic method of ancient geometry is hardly ever used in applications today.

The most important findings have been discovered in the past 150 years, starting in the late 1800s when alliance between scientific revolution and industrial revolution was fully developed, and recognized the value social of science as an engine of technological development and economic growth. Only in recent decades have we begun to understand the applications to science of topology and new geometries, an extremely vibrant area of contemporary research.