When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.
The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.
In algebraic topology, a free abstract mathematical object is generated by elements in a "free manner" ("freely"), i.e., such that the elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic gadget is freely generated by a subset if, for any function where is any other algebraic gadget, there exists a unique homomorphism (which has different meanings depending on what kind of gadgets you're dealing with) such that restricted to is .
If the algebraic gadgets are vector spaces, then freely generates iff is a basis for . If the algebraic gadgets are Abelian groups, then freely generates iff is a direct sum of the integers, with consisting of the standard basis.
See alsoFixed,
Free Group,
Free Variable,
Freely,
Gadget,
Mirror Image,
Rank Explore with Wolfram|Alpha Cite this as:Weisstein, Eric W. "Free." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Free.html
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