"The geometry we studied in school, known as Euclidean geometry, was formalized over 2,000 years ago in The Elements. Although, from today's perspective, this work contains formal errors from the very first page (which have been corrected over time, so there's no need to worry), it changed the way mathematics is done. Its author, Euclid, assumed five statements (or postulates) to be true in order to, simply by applying logic, obtain all the theoretical results of plane geometry known at the time,"
"Euclid's first four postulates were very simple and obviousfor example, that a unique straight line passes through any two distinct pointsand were accepted without question. However, the fifth was more complex. It stated, in a rather convoluted way, that through a point outside a given line, only one line parallel to it can be drawn. This prompted the question of whether it could be deduced from the first four, which would therefore be the only necessary postulates."
Hyperbolic geometry provides a consistent alternative to Euclidean geometry in which familiar facts, such as triangle angle sums, can differ. Virtual reality enables immersive experiences that reveal geometric properties beyond symbolic formulas. Euclidean geometry was formalized over two thousand years ago in The Elements and rested on five postulates. The first four postulates were simple and widely accepted, but the fifth dealt with parallel lines and resisted derivation from the others. Centuries of attempts failed to deduce the parallel postulate, prompting the recognition of coherent non-Euclidean geometries and expanding the concept of geometric universes.
Read at english.elpais.com
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