We explain what geometry is, its history and its object of study. Also, the characteristics of each type of geometry.
What is geometry?
Geometry (from the Greek geo“land”, and metric“measurement”) is one of the oldest branches of mathematics, dedicated to study of the shape of individual objects, the spatial relationship between them, and the properties of the space around them.
Although in its beginnings this discipline obeyed, as its name indicates, measurement in its most practical sense, with the passage of time humanity understood that even the most complex abstractions and representations can be expressed in geometric terms.
This is how its numerous branches emerged, hand in hand with mathematical analysis and other forms of calculation, especially those that link geometric representation with numerical and algebraic mathematical expressions.
The geometry It is a fundamental branch of mathematics on which numerous disciplines are based (such as technical drawing or architecture itself) and serves as a complement to many others (such as physics, mechanics, astronomy, etc.). In addition, it has given rise to numerous artifacts, from the compass and pantograph, to the global positioning system (GPS).
History of geometry
Geometry has its origins practically in the first human civilizations. The ancient Babylonians were the inventors of the wheel and therefore of the geometry of circumferences. For this reason, they were probably the first to recognize the infinite potential of geometric study, which they soon applied to astronomy.
The ancient Egyptians did the same, who cultivated it enough to apply it in their majestic architectural works, given that at that time geometry and arithmetic were eminently practical sciences.
Numerous Greek historians, such as Herodotus (c. 484-c. 425 BC), Diodorus (c. 90 BC – c. 30 BC), and Strabo (c. 63 BC – c . 24 AD) recognized the importance of the Egyptian geometric legacy, and were considered the creators of the discipline. However, It was the ancient Greeks who gave geometry its formal aspect thanks to its advanced philosophical model.
Of particular importance was the mathematician and geometer Euclid (c. 325 – c. 265 BC), recognized as “father of geometry”, who proposed the first geometric system for verifying results, through his celebrated work The elementscomposed around the year 300 BC. C. in Alexandria. There the differences between the plane (two-dimensional) and space (three-dimensional) are stated for the first time.
Other important contributions to the geometry of the period were those of Archimedes (c. 287 – c. 212 BC) and Apollonius of Perga (c. 262 – c. 190 BC). However, In subsequent centuries the development of mathematics moved to the East (India, specifically, and the Muslim world), where geometry was developed along with algebra and trigonometry, linking them with astrology and astronomy.
So, interest in the discipline returned to the West only in the Renaissance European, in which many new names were added to its study, thus giving rise to projective geometry and especially Cartesian geometry or analytical geometry, fruit of the work of the French philosopher René Descartes (1596-1650), bearer of a new geometric research method that revolutionized and modernized this field of knowledge.
From then on, modern geometry took place by the hand of great scholars such as the German Carl Friedrich Gauss (1777-1855), the Russian Nikolai Lobachevski (1792-1856), the Hungarian János Bolyai (1802-1860), among many others, who managed to deviate from Euclid's classical axioms and found a new field of discipline: non-Euclidean geometry.
Object of study of geometry
The geometry deals with the properties of space and in particular with the forms and figures that inhabit it whether two-dimensional (plane) or three-dimensional (space), such as points, lines, planes, polygons, polyhedra, etc. These types of objects are understood in terms of idealizations, that is, mental projections of space, from there to transfer (or not) their conclusions to the concrete world.
Geometry Types
Geometry has numerous different branches, and its classification generally responds to the relationship it establishes with Euclid's five basic postulates, of which only four have been widely demonstrated since ancient times. The fifth, on the other hand, had to be modified to give rise to different families of geometries.
Thus, we must distinguish between:
absolute geometry one that is governed by the first four postulates of Euclid.
Euclidean geometry which also accepts the fifth Euclidean postulate as an axiom, giving rise to two variants: the geometry of the plane (two-dimensional) and the geometry of space (three-dimensional), according to the classification of the ancient Greeks.
Classic geometry the one in which the results of the Euclidean geometries are compiled.
Non-Euclidean geometry emerged in the 19th century, is one that brings together the different geometric systems that deviate from Euclid's fifth postulate, however accepting the first four or some of them. Among them are:
- Elliptic or Riemannian geometry which obeys the first four postulates of Euclid and presents a model of constant and positive curvature.
- Hyperbolic or Lobachevskian geometry which obeys only the first four postulates of Euclid and presents a model of constant and negative curvature.
- spherical geometry understood as the two-dimensional surface geometry of a sphere (rather than a straight plane), is a simpler model of elliptical geometry.
- Finite geometry whose system obeys a limited number of points (unlike Euclid's infinite geometry), and whose models apply only in a finite plane. There are two types of finite geometries: affine and projective.
Continue with: Formal sciences
References
- “Geometry” on Wikipedia.
- “History of geometry” on Wikipedia.
- “Geometry” at the University of Alcalá (Spain).
- “The geometries” in the Ministry of Education of the Argentine Republic.
- “Geometry (Mathematics)” in The Encyclopaedia Britannica.
