Surfing Through Hyperspace by Clifford A. Pickover

Author:Clifford A. Pickover
Language: eng
Format: epub
ISBN: 9780199923816
Publisher: Oxford University Press, Inc.

Figure 5.3 A 4-D being would be the ultimate Houdini and could knot or unknot a string by temporarily lifting it into the fourth dimension. On the left is a string before it has been knotted. (Zöllner tried to have the left string transformed to the right without breaking the wax circle at top.)

Sally traces a squiggle on some dirt using the tip of her shoe (Fig. 5.4). “I don’t think a string can be knotted in 2-D space.”

“You’re right. In Flatland, there’s no way a line can cross over itself. In fact, a string or a line can only be knotted in 3-D space. And any knot you tied in 3-D space will not stay tied in 4-D space because the additional degree of freedom will cause a knot to slip through itself.” You pause. “By analogy, in 4-D space, a creature can knot a plane (surface), but this plane won’t stay knotted in 5-D space. And the knotted plane cannot be formed in 3-D space.”

Sally tugs your hand so that you start walking. “How in the world could you knot a plane?”

“Take a knotted line and then move it upsilon in the fourth dimension. The trail it traces will be a knotted plane. It never intersects itself. Of course, if we simply leave a trail in three-space as we move a knot, it will intersect itself, but since upsilon is perpendicular to all directions in our space, the 4-D knotted plane will not intersect itself.”

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