Higher Dimensions
A knot in three dimensions can be untied when placed in four dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.
In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.
Read more about this topic: Knot Theory
Famous quotes containing the words higher and/or dimensions:
“Dreams have a poetic integrity and truth. This limbo and dust- hole of thought is presided over by a certain reason, too. Their extravagance from nature is yet within a higher nature.”
—Ralph Waldo Emerson (1803–1882)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)