Antoine's Necklace and Menger's Sponge

Antoine's Necklace (1920) is an interesting Topological shape, in that it is shaped like infinite tori without a single solid torus in the bunch. How can Infinity equal zero? Is it permissible for our brains to explode while contemplating this apparent contradiction? Inevitable even?

Well calm down, inevitability boys and girls, infinity doesn't really exist except in the mind of a Mathematician, but it's usefulness is uncontested. I've suffered enough brain explodery thinking about this stuff so let's see if I can make it simple.

To construct an Antoine's Necklace, think of a torus, a doughnut if you will, and replace it with a linked chain. How many links you may ask? It could be as little as two if you bend them but let's not do that. Say a dozen, or however many you want. The point is that while each link appears to be a torus, we can continue the thought and construct the necklace by imagining each link is a chain of smaller links themselves. And so on with those links, and their links, and their links an so on forever, the diameters of links eventually decreasing to zero..

Time for a picture:

Antoine's Necklace

Mathematically this is referred to as "homeomorphic with a Cantor set". A Cantor set is a special set of points with infinitely many gaps between them. The Necklace is "totally disconnected ... because for any 2 different points, there is some stage of construction such that the 2 points will lie on different tori." (Brechner and Meyer).

Louis Antoine (1888-1971) came up with this idea. He was blinded in World War I at age 29 and told by Mathematician Henri Lebesgue to study two and three-dimensional topology, because "in such a study, the eyes of the spirit and the habit of concentration will replace the lost vision."



Menger's Sponge

A Menger's sponge is a fractal object with an infinite number of cavities. This image  is from a fascinating website of fractals and polyhedra, "Of a Fractal Nature",  copyright Paul Bourke and Gayla  Chandler, at http://fractalnature.com

Menger sponges have been around since 1926 thanks to Karl Menger (1902-1985).

They are essentially objects with zero area but infinite perimeter!

??!!  What the heck ?

You can see how this is true. Think of  a cube like a Rubik's cube built of  27 smaller cubes
cubes in a a 3 x 3 matrix. Now remove the center cube and the six cubes that face it. You have 20 cubes left. Keep doing this for each smaller cube cube and so on down to infinity. Infinite perimeter, zero area.

Does it have mass?

Well according to Clifford Pickover in The MaTH bOOK (the source of this page's knowledge), "Dr. Jeannine Mosely has constructed a Menger sponge model from more than 65,000 business cards that weighs about 150 pounds (70 kilograms)."