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:
“We live between two worlds; we soar in the atmosphere; we creep upon the soil; we have the aspirations of creators and the propensities of quadrupeds. There can be but one explanation of this fact. We are passing from the animal into a higher form, and the drama of this planet is in its second act.”
—W. Winwood Reade (1838–1875)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)