Overview of Geometries
What is Geometry?
| Space | Generic set whose elements are called points |
| Topology | General and abstract definition of concepts such as vicinity, continuity, connection, compactness and convergence, and of the properties of forms that do not change by continuous deformations of space, i.e., without cuts, tears, overlaps and gluing |
| Topological space_ | Space with a topology |
| Metrics | Definition of distance between points in space, from which we derive the definition of lengths and angles |
| Measure | Definition of the magnitude of a subset of points in space, from which derive the definition of area, volume, and so on |
| Metric space_ | Topological space_ in which a metric, that is, a distance between two points in space, is defined, and possibly a measure |
| Group of transformations | Set of operations acting on the points of the space, with properties such as: 1. existence of inverse, 2. existence of identity or neutral element, 3. group membership of the combination of two transformations, 4. indifference of the order of evaluation of three transformations (associativity). |
| Symmetry | Property of the space, or objects contained in it, that remains unchanged after the application of a transformation of a given group |
| GEOMETRY | Study of the invariant properties of a metric space with respect to a group of transformations, or symmetries, that preserve the metric |
Also called Topology is the branch of mathematics that studies topological spaces, the most important basic of mathematics along with Abstract Algebra and the unifying language provided by Category Theory (functors, morphisms, structures, classes and sets.) The term comes from the Greek topos, “place,” and logos, “study.” Topology is the most general discipline underlying Geometry.
The concept of group comes from Abstract Algebra, which studies structures defined in a set with one or more algebraic operations. Examples of algebraic structures are field (numerical), vector space, ring, lattice, algebra (algebra over field, algebra over lattice,…) and precisely group.
symmetries have fundamental importance not only in Geometry but also in Theoretical Physics_, are a bridge between the two disciplines. In fact, from the symmetries of the space or physical system considered conservation principles are derived (energy, momentum, angular momentum, …) and the form of the equations of motion in both classical physics and quantum field theory, and in the standard model of fundamental forces and elementary particles. In addition, the concepts of spontaneous symmetry breaking and topological order are the basis of the study of states of matter.
Basic concepts: Space, Topology, Group of transformations, Symmetry, Metrics
tD graph a([GROUP THEORY]) --- S(SYMMETRIES) --- G((GEOMETRIES)) b([TOPOLOGY]) --- T(<br> METRIC SPACES) --- G
The view of Geometry as Geometry of groups of transformations has been outlined in Felix Klein’s Program of Erlangen (1872). Its extension to Topology is the work of Poincaré(1895), Hausdorff(1914) and others in the first half of the twentieth century.
graph LR
t1([TOPOLOGY]) ==> T2(Form without Measure)
g1([GROUPS]) ==> G2(Measurement of Symmetries)
n1([NUMBERS]) ==> N2(Measurement of Dimensions)
a1([REAL NUMBERS]) ==> A2(1D geometry of the line)
b1([COMPLETE NUMBERS]) ==> B2(2D plane geometry)
c1([QUATERNIONS]) ==> C2(3D space geometry)
d1([N-PLE OF REALI]) ==> D2( Geometry of N-Dimensional Spaces)
Different methods and strategies are used in the study of geometric problems, which we can summarize as the following
| Matter | Subject and Tools |
|---|---|
| Vector spaces | Linear (and multilinear) algebra to study vectors, operators, tensors, and as analytic geometry of n-dimensional spaces |
| Group Theory | Abstract Algebra for the general study of groups |
| Graph Theory | Study of discrete structures composed of vertices (or nodes) and links between vertices |
| Algebraic topology (and geometry) | Study of topological spaces and algebraic varieties with the tools of abstract algebra |
| Differential pathology (and geometry) | Study of differentiable varieties, fiber spaces, differential forms, connections, curves and surfaces with the tools of differential and integral calculus |
| Node Theory | Study of the topology of nodes, i.e., closed curves intertwined in low-dimensional spaces |
| Analytic Geometry | Coordinate systems for tracing geometric problems back to algebra and differential and integral calculus |
| Synthetic Geometry__ | Demonstrations of theorems from appropriate postulates and constructions with ruler and compasses |
Today in scientific and technological applications, the most important methods are the former, while the latter, ancient synthetic geometry, which is studied in high schools, is practically never used.
The relationship between different geometries and the transformations studied can be summarized in a table.
| Geometry | Transformations | Description |
|---|---|---|
| General Topology | homomorphisms_ | continuous deformations of space |
| Differential pathology | diffeomorphisms | continuous deformations that are also differentiable functions |
| Projective geometry | projective transformations | projections of a plane that do not preserve distances or angles, but only certain proportions, such as the birapport of four distinct aligned points, including improper points at infinity |
| Affine geometry | affine transformations | projective transformations preserving parallelism between lines |
| - | translations | displacements of a fixed distance in the same direction |
| - | homotetries | conformal and affine transformations that preserve angles but not lengths |
| - | similitudes | composition of homotetries and isometries |
| metric geometry | isometries | transformations that preserve the measure of distances and angles, in Euclidean and non-Euclidean geometries |
| - | rotations | rotational movement around a fixed central point |
| - | reflections | specular transformations around a fixed center/axis/plane |
Metric geometry can be either Euclidean or non-Euclidean.
Euclidean geometry is a metric geometry with a definition of distances and angles such that Euclid’s 5th postulate, or postulate of parallels, in which the Pythagorean theorem, the theorem of the sum of the angles of a triangle, and so on, apply, the geodesic lines are straight lines.
Non-Euclidean geometries are metric geometries with a definition of distances and angles such that they do NOT satisfy Euclid’s 5th postulate, in which the Pythagorean theorem and the theorem of the sum of the angles of a triangle do not apply, and geodesic lines do not are straight lines.
Geometry is thus based on Topology, which studies more general transformations of space. Based on the type of transformations we have a hierarchy of geometries, from the most general to the most specialized. In addition, in the age of Computers and Artificial Intelligence, the following cannot be overlooked the role of discrete and computational geometry:
tD graph
a((TOPOLOGY <br> General - Algebraic)) -------- A2([TOPOLOGY <br> Discrete <br> Combinatorics])
a --- L([ANALYSIS <br> MATH])
a --- B([TOPOLOGY <br> DIFFERENTIAL])
a --- G([METRIC SPACES])
b --- L
b --- D((GEOMETRY <br> DIFFERENTIAL))
g --- D
a ---- P([GEOMETRY <br> PROJECTIVE])
p --- F(GEOMETRY <br> AFFINE)
f --- E{VECTOR SPACES <br> LINEAR ALGEBRA}
e --- H(VECTOR SPACES <br> METRIC with standard<br> and/or scalar product)
aND --- S(GEOMETRY<br>SIMPLECTIC)
d ----- K(GEOMETRY <br> EUCLIDEA <br> Flat Spaces)
d ---- J(GEOMETRY <br> NON-EUCLIDARY <br>CURVED SPACES)
d --- H
d -.- S
p ------ M(GEOMETRY <br> DESCRIPTIVE)
p ------- N(GEOMETRY <br> EPIPOLAR)
a2 --- C2(GRAPH THEORY <br>KNOT THEORY)
a2 ---- B2((GEOMETRY <br> DISCRETE))
b2 --- D2(GEOMETRY <br> COMPUTATIONAL)
b2 ---- D3(GEOMETRY <br> DIGITAL)
b2 --- E2(DOWELS)
b2 ---- F2(PACKINGS)
b2 --- G2(POLYTOPES)