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[Using mathematics in everyday situations takes place mainly through the so-called mathematical models; that is, allegories that adapt the real problem to the world of ideas, creating the possibility of dealing with problems scientifically.]
[From Plato to Kepler, some famous philosophers, scientists and alchemists using a remarkable blend of mathematics and faith try to explain the creation of the universe. They make geometric descriptions of allegedly fundamental ingredients of a harmonious cosmos, sometimes scientifically, others...
[The axiomatic Euclidean geometry was unique for 2000 years. Then, in the nineteenth century a certain modernity was established with the flourishing of non-Euclidean geometries. In 1872, Felix Klein presented a way to define geometries without axioms, organizing the space in congruence classes,...
[The twentieth century started with a great deal of applications of non-Euclidean geometries. Topology, as an applied topic, appeared primarily in the study of Einstein’s general relativity and, by the end of the century, material science Nobel prize winners benefited from the topological...
[The romance Flatland by Edwin Abbott is about life in a two-dimensional world. Its inhabitants, the flatlanders, can only imagine a three-dimensional universe. Here we describe a four-dimensional place; that is, a portion of a four-dimensional space enclosed by a hypercube. Although we cannot...
[Closed non-orientable surfaces are connected sum of projective planes. Here we construct the classical models of the projective plane in three-dimensional space; namely, the sphere with cross-cap, the Steiner Roman surface and the Boy surface.]
[Models of three-dimensional objects with boundary abound in our three-dimensional physical world. The boundaries of these objects are surfaces, two-dimensional objects that we can see or touch. For three-dimensional objects finite in size and without boundary, called hypersurfaces here, we have...
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