In Paris in the 1920's, there was a goldsmith named Louis Antoine. He wanted to make his wife a necklace. He was truly devoted to her, but he was also quite miserly. So, he began by melting down all his gold and making a large solid ring... much too heavy for a necklace... and then began carving it down, as shown here.
First he carved away gold until he had a chain of 8 linked rings - as shown in blue here.
But then he thought: "I could save even more
gold if I carved away gold from each of these
rings, just as I did before! Besides, it would be more beautiful - and most unusual! Nobody has ever seen a necklace of 8 linked chains, each made of 8 rings. My wife would love it."
So that's what he did.
But then he realized he could save even more
gold, and make an even more
beautiful necklace, if he did the same process again!
And so he went on all night, carving each ring into 8 smaller linked rings, over and over. As he became more excited, he worked faster and faster. He did each round of carving twice as fast as the previous one.
And so, when sunrise came, he was done. He had carved away all
the gold from the original solid ring, leaving a wonderfully elegant and fine necklace made of 8 rings made of 8 rings made of 8 rings... on forever!
Holding it up to admire it, he could just barely see it, sparkling in the sunlight. He gave it to his wife, and she was very happy: she knew he was a cheapskate, but she forgave him for that.
This necklace became famous: it's called Antoine's necklace
It is, in fact, yet another version of the famous Cantor space
. We can build this space by taking an interval and repeatedly removing the middle third... or by taking a cube and repeatedly removing middle-third slices in all three directions... and in many other ways.
But the interesting thing about Antoine's necklace is that it gives a wild embedding
of the Cantor space in 3-dimensional space. In other words, no way of warping and deforming 3d space will carry Antoine's necklace to the more usual
ways of sticking the Cantor space in 3d space - like either of the ways I just described.
There are, in fact, uncountably many different wild embeddings of the Cantor space in 3d space. You can get uncountably many by carving versions of Antoine's necklace where you vary the number of rings at each stage!
I thank +Boris Borcic
for kindling my interest in this necklace.
Here's a movie of an approximation to Antoine's necklace:Antoine's Necklace -- 160,000 Tori
This version shows a ring made of 20 rings made of 20 rings made of 20 rings made of 20 rings! #fractals