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:
“For my part, I have no hesitation in saying that although the American woman never leaves her domestic sphere and is in some respects very dependent within it, nowhere does she enjoy a higher station . . . if anyone asks me what I think the chief cause of the extraordinary prosperity and growing power of this nation, I should answer that it is due to the superiority of their woman.”
—Alexis de Tocqueville (1805–1859)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)