Blinded in the First World War at age 29, Louis Antoine was a French mathematician who struggled to cope with his newly darkened world. Sensing his friend’s need for new purpose, Henri Lebesgue suggested that Antoine begin studying two- and three-dimensional mathematical objects. What resulted was the discovery of Antoine’s necklace.
Antoine’s necklace is a theoretical object in three-dimensional space that modern mathematicians continue to study. David Wright, a faculty member in the Department of Mathematics, has undertaken a study that builds upon the existing known properties of this object.
An easy way to visualize this necklace is to imagine a rock that has been chiseled away to create a chain of unbroken links. Then, each of these links is chiseled into a miniature necklace, whose links are then chiseled into other necklaces, and so forth. This pattern continues indefinitely.
Though it seems fragile, a physical replica of Antoine’s necklace wouldn’t fall apart.
“If you put your fingers around its outermost link and tried to pick [Antoine’s necklace] up,” Wright explained, “it might shift a little bit just like some beads would, but it would not fall through your fingers.”
Similar to Antoine’s necklace, the Bing-Whitehead necklace is the focus of Wright’s research project. However, the Bing-Whitehead pattern is not formed by chiseling, but by intertwining Bing and Whitehead links, which are large and delicate. Imagine twisting two rubber bands together, or one rubber band around itself, to make different structures.
Another difference between these two necklaces is their stability. The beginning Bing-Whitehead links easily disentangle themselves and fall apart when gently shaken. However, if these graceful links are repeatedly woven together, the first few links may fall apart, but it is impossible to untangle all of them.
Wright studied the effect that different combinations of Bing and Whitehead links would have on a finished product. He sought to discover whether wrapping these links in different patterns, or codes, would create different necklaces.
This problem also interested three other mathematicians, one from Oregon and two from Slovenia. In collaboration with these sharp minds, Wright was able to solve the problem and determined that constructing Bing-Whitehead necklaces according to different codes really do produce different necklaces. These findings were published in the Transactions of the American Mathematical Society.