In particular, the centroid of a parallelogram is the meeting point of its two diagonals. The centroid of many figures ( regular polygon, regular polyhedron, cylinder, rectangle, rhombus, circle, sphere, ellipse, ellipsoid, superellipse, superellipsoid, etc.) can be determined by this principle alone. In particular, the geometric centroid of an object lies in the intersection of all its hyperplanes of symmetry. If the centroid is defined, it is a fixed point of all isometries in its symmetry group. The centroid of a ring or a bowl, for example, lies in the object's central void. A non-convex object might have a centroid that is outside the figure itself. The geometric centroid of a convex object always lies in the object. It may be added, in passing, that the proposition did not become common in the textbooks on plane geometry until the nineteenth century. The first explicit statement of this proposition is due to Heron of Alexandria (perhaps the first century CE) and occurs in his Mechanics. Thus Archimedes could not have learned the theorem that the medians of a triangle meet in a point-the center of gravity of the triangle-directly from Euclid, as this proposition is not in Euclid's Elements. While it is possible Euclid was still active in Alexandria during the childhood of Archimedes (287–212 BCE), it is certain that when Archimedes visited Alexandria, Euclid was no longer there. Bossut credits Archimedes with having found the centroid of plane figures, but has nothing to say about solids. This book was highly esteemed by his contemporaries, judging from the fact that within two years after its publication it was already available in translation in Italian (1802–03), English (1803), and German (1804). In 1802 Charles Bossut (1730–1813) published a two-volume Essai sur l'histoire générale des mathématiques. I, p. 463) that the center of gravity of solids is a subject Archimedes did not touch. However, Jean-Étienne Montucla (1725–1799), the author of the first history of mathematics (1758), declares categorically (vol. While Archimedes does not state that proposition explicitly, he makes indirect references to it, suggesting he was familiar with it. When, where, and by whom it was invented is not known, as it is a concept that likely occurred to many people individually with minor differences. The center of gravity, as the name indicates, is a notion that arose in mechanics, most likely in connection with building activities. The French use " centre de gravité" on most occasions, and others use terms of similar meaning. The term is peculiar to the English language. It is used as a substitute for the older terms " center of gravity," and " center of mass", when the purely geometrical aspects of that point are to be emphasized. The term "centroid" is of recent coinage (1814). 4.12 Of a tetrahedron and n-dimensional simplex.
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