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10/4/2020

Most of us started our earliest rudimentary lessons in geometry when we played with shapes and building bricks as infants. Geometry is the study of the relationships between points, lines, surfaces, angles, and shapes, so we encounter geometry daily.

Geometry arose independently in a number of early cultures as a practical way of dealing with lengths, areas, and volumes. Geometry began in Greek mathematics as early as the 6th century BC and by the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid’s Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, while Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages.

By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Currently, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are fundamental to it. These include the concepts of angle, curve, distance, line, point, plane, and surface, as well as the more advanced ideas around topology and manifold.

Geometry has applications in many fields, including art (where mathematics meets art, since all the elements of geometry are easier to understand when you draw a diagram), architecture, physics, as well as other branches of mathematics. In geometry, you'll have to develop proofs. So, having a starting point (what you’re given) and an endpoint (what you want to know) will be a huge help in the process — your goal is to map a route between the two. Once you have everything written down and your diagrams drawn, you can start the proof process. Mathematical proofs are not easy tasks, so if you get stuck, go back a step and see if there is a different property or theorem that you can apply.

Properties and theorems

These are your most useful tools for drawing diagrams, forming relationships and developing proofs. It will make your life so much easier if you’re able to recall properties and theorems for various shapes, angles and lines; properties of lines, parallelograms and angles; theorems of lines, triangles and angles. If you make flashcards for all the properties and theorems you need to know, you can go through them every day.

Here are some of the most important theorems for triangles:

**Side-side-side (SSS):** If two triangles have all three side measures congruent (in other words, if the three sides of one triangle are all the same length as the three sides of the other triangle), then the two triangles are congruent. Congruent lines are often marked by short lines.

**Side-angle-side (SAS):** If two triangles have two sides congruent and the angles in between the two sides are also congruent, then the two triangles are congruent.

**Angle-side-angle (ASA):** If two triangles have a congruent side that touches two congruent angles, then the triangles are congruent.

**Hypotenuse-Leg (HL): **This is a special triangle theorem that can only be used for right triangles. It states that if you have two right triangles and you know that their hypotenuses are congruent and one pair of their sides is also congruent, then the triangles are congruent.

**Angle-angle-angle (AAA): **Triangles with three equal angles are similar but not necessarily congruent.

Euclid’s postulates

Euclid is often referred to as the “Father of Geometry”. Geometry was developed based on his five postulates — as in, if properties are the cement foundation and theorems are the bricks, then Euclid’s five postulates make up the ground on which the cement is poured and the bricks are laid. Without them, there would be no tower of geometry.

Here are Euclid’s five postulates and a brief explanation of each:

- A straight-line segment can be drawn joining any two points (two points define a line).
- Any straight-line segment can be continued in either direction indefinitely in a straight line (a straight line has infinite measure).
- A circle can be drawn around any line segment with one end of the line segment serving as the centre point and the length of the line segment serving as the radius of the circle (any line segment can be spun to create a circle and serve as the circle’s radius).
- All right angles are congruent (they all measure 90 degrees).
- If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is known as the “parallel postulate”.

**Geometry concepts**

The following are some of the most important concepts in geometry.

**Angles**

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry. In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be calculated using the derivative.

**Axioms**

Euclid took an abstract approach to geometry in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines and planes. Euclid’s approach to geometry was its rigor and has come to be known as axiomatic or synthetic geometry.

**Congruence and similarity**

Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Congruence and similarity are generalised in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.

**Compass and straightedge constructions**

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction has to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.

**Curves**

A curve is a one-dimensional object that may be straight (like a line) or not; curves in two-dimensional space are called plane curves and those in three-dimensional space are called space curves. In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.

**Dimension**

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane), and 3 (our ambient world conceived of as three-dimensional space), mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the system’s degrees of freedom. For instance, the configuration of a screw can be described by five coordinates.

In general topology, the concept of dimension has been extended from natural numbers to infinite dimensions and positive real numbers (in fractal geometry). In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.

**Length, area, and volume**

Length, area and volume describe the size or extent of an object in one dimension, two dimension and three dimensions respectively. In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem. Area and volume can be defined as fundamental quantities separate from the length, or they can be described and calculated in terms of lengths in a plane or three-dimensional space. Mathematicians have found many explicit formulas for area and formulas for the volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral or the Lebesgue integral.

**Lines**

Euclid described a line as “breadthless length” which “lies equally with respect to the points on itself”. In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it. In differential geometry, a geodesic is a generalisation of the notion of a line to curved spaces.

**Manifolds**

A manifold is a generalisation of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighbourhood that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighbourhood is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory.

**Metrics and measures**

The Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity. In a different direction, the concepts of length, area and volume are extended by measure theory, which studies methods of assigning a size or measure to sets, where the measures follow rules similar to those of classical area and volume.

**Planes**

A plane is a flat, two-dimensional surface that extends infinitely far. Planes are used in every area of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis; and so on.

**Points**

Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid’s definition as “that which has no part” and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points.

**Surfaces**

A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology, surfaces are described by two-dimensional ‘patches’ (or neighbourhoods) that are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations. A sphere is a surface that can be defined parametrically or implicitly.

**Symmetry**

Symmetric shapes such as the circle, regular polygons, and platonic solids held a deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of da Vinci, M.C. Escher and others. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. A different type of symmetry is the principle of duality in projective geometry, among other fields. This meta-phenomenon can roughly be described thus: in any theorem, exchange point with a plane, join with meet, lies in with contains, and the result is an equally true theorem.

**Triangles**

A scalene triangle has no identical sides and no identical angles. An isosceles triangle has (at least) two identical sides and two identical angles and an equilateral triangle has all three sides and all three angles identical. The longest side of a scalene triangle is the opposite of the largest angle. Similarly, the shortest side of a scalene triangle is the opposite of the smallest angle.

**Geometry today**

**Algebraic geometry**

The field of algebraic geometry developed from the Cartesian geometry of co-ordinates. It underwent occasional periods of growth, accompanied by the creation and study of projective geometry, birational geometry, algebraic varieties and commutative algebra, among other topics. From the late 1950s to the mid-1970s, it had undergone major foundational development, largely due to the work of Jean-Pierre Serre and Alexander Grothendieck. In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials. It has applications in many areas, including cryptography and string theory.

**Complex geometry**

Complex geometry studies the nature of geometric structures modelled on or arising out of, the complex plane. Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables, and it's found applications to string theory and mirror symmetry.

Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces. Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre, who introduced the concept of sheaves to the subject and illuminated the relations between complex geometry and algebraic geometry. The primary objects of study in complex geometry are complex manifolds, complex algebraic varieties, and complex analytic varieties, as well as holomorphic vector bundles and coherent sheaves over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi-Yau manifolds, and these spaces find uses in string theory. In particular, worldsheets of strings are modelled by Riemann surfaces and superstring theory predicts that the extra six dimensions of ten-dimensional spacetime may be modelled by Calabi-Yau manifolds.

**Computational geometry**

Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal and linear programming. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc.

**Convex geometry**

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. It has close connections to convex analysis, optimisation and functional analysis, and important applications in number theory.

Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity. The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings, and lattices.

**Differential geometry**

Differential geometry uses tools from calculus to study problems involving curvature. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics, econometrics, and bioinformatics, among others.

**Discrete geometry**

Discrete geometry is a subject that has close connections with convex geometry. It is concerned mainly with questions of the relative position of simple geometric objects, such as points, lines, and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc. It shares many methods and principles with combinatorics.

**Euclidean geometry**

Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics, astronomy, crystallography, and many technical fields, such as engineering, architecture, geodesy, aerodynamics, and navigation. The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts - points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry.

**Geometric group theory**

Geometric group theory uses large-scale geometric techniques to study finitely generated groups. It is closely connected to low-dimensional topologies, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem.

**Non-Euclidean geometry**

Euclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori. This view was at first somewhat challenged by thinkers such as Saccheri, then finally overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss. They demonstrated that ordinary Euclidean space is only one possibility for the development of geometry. Riemann’s new idea of space proved crucial in Einstein’s general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.

**Topology**

Topology is the field concerned with the properties of continuous mappings and can be considered a generalisation of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness. The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. Subfields of topology include geometric topology, differential topology, algebraic topology, and general topology.