- #1
Goldbeetle
- 210
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Dear all,
I've always wondered where the name "manifold" comes from?
Any idea?
Thanks,
Goldbeetle
I've always wondered where the name "manifold" comes from?
Any idea?
Thanks,
Goldbeetle
FredericGos said:
g_edgar said:So the English usage in mathematics is as a translation of the German "Mannigfaltigkeit"
… and illustrates this with German quotations (and his own English translations) from Riemann's http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/"In (I) Riemann defines a "Mannigfaltigkeit" as consisting of the "Bestimmungsweisen" (ways of determination) of a "Größenbegriff" (concept of quantity), with "Bestimmungsweisen" being the points of a "stetige Mannigfaltigkeit" (continuous manifold). The "stetige Mannigfaltigkeit" is not described as being composed from smaller pieces, nor does he mention local flattness. In another, not translated, part he mentions colors and the locations of "Sinngegenstände" (objects of perception) as the only simple concepts giving rise to "stetige Mannigfaltigkeiten". At another point he calls the possible shapes of a spatial figure a "Mannigfaltigkeit".
A manifold is a mathematical concept that describes a space that is locally similar to Euclidean space, meaning that it looks flat when viewed up close, but may have a curved overall shape.
The term "manifold" comes from the Latin word "manifolds", which means "many folds". This describes the idea that a manifold can have many different local shapes, but still have an overall coherent structure.
Some common examples of manifolds include spheres, tori, and cylinders. In general, any surface in three-dimensional space that can be smoothly deformed into a flat plane without tearing or stretching can be considered a manifold.
Manifolds are used in a variety of scientific fields, including physics, engineering, and computer science. They provide a useful way to describe and understand complex systems and spaces, and are particularly important in the study of curved spaces and the theory of relativity.
Yes, manifolds can exist in any number of dimensions. In fact, manifolds are often used to describe spaces with more than three dimensions, such as in string theory or multidimensional calculus.