Wednesday, April 27, 2011

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)."

6 comments:

Pat B said...

Wow, someone has business cards that are truly fractal??? .. maybe she jests. My question is can you ever be "inside" a Menger sponge?

Steven Colyer said...

With your question the brain explodery continues! Thank goodness for skulls to keep that stuff in one place or we'd all be dead and The Universe would have to carry on without us!

Well it's tough to be "inside" something that has no area, and if area is zero and volume is area times length then it doesn't even matter which direction of length (usually perpendicular) because 0 times anything is zero.

And yet, the child doesn't know that, so there he is.

Ignorance is bliss?

:-)

Pat B said...

IS that why I'm so happy?

Steven Colyer said...

I don't know WHY you're happy, but let me venture a guess that THE reason is that you're married to a wonderful woman, who dayam man, looks a LOT like my own.

A good life partner cures tons of ills. Yeah yeah, all guys know fidelity is a great sacrifice and no woman understands that because in our culture "marraige" is their goal, but so what?

Yin and yang baby, yin and yang, and IMO a man and a woman fit so well together that they defy even Mathematics, that is to say in human relationships:

1 + 1 > 2

Heh.

Woman + Man = Stability. I think that's something we can all wrap our heads around. Rather than being "boring", as many of the young-uns view it, it's actually quite the opposite. To go down the decades of our lives with one VERY special person, and doubly so if you procreate with them, is THE thing with me.

In short, it frees up time to do what you intellectually WANT to do, which I see you doing, and me too. It's a peaceful, easy feeling, to quote a line from a song by The Eagle.

And as a fringe benefit, you can have sex with them. What's not to love? :-)

Anonymous said...

The Child in Menger Sponge image is copyright Paul Bourke and Gayla Chandler. You may email me at gayla.chandler@gmail.com. You referenced Cliff Pickover's Math Book. This image in included in his book with our permission, and the copyright is referenced there. Please make clear reference to our copyright on your page at the place of the image and REMOVE access to the image in larger form. The child was my then 20-month-old grandchild playing with a squishy menger sponge that I made for her. I caught her throwing it in the air and catching it. As a thought experiment for MY webpage, Paul extracted Sydney Renee and placed her in one of his computer-generated sponges. That is the history of the image. Sydney Renee is now 8-1/2 years old.

Please contact me about this.

Thank you.

Steven Colyer said...

Hi! Have done so. Thank you!