Antione’s Necklace

“dans une telle étude, les yeux de l’esprit et l’habitude de la concentration remplaceront la vision perdue"

“in such a study the eyes of the spirit and the habit of concentration will replace the lost vision” (​Lebesgue to a sightless Antione​)

Louis Antoine, géomètre aveugle
Louis Antoine, géomètre aveugle

Had Antione’s attuned fingers
Felt necklaces trickle through? Coiled chains weaved between digits, Falling into cupped palm.

Cold.
But alive.
Vibrating with, Imbued with, Embedded​ with, Infinitely many strings.

Did their metallic bite excite him Into countering,
Canter himself?
With the unequivocal idea,

Of uncountably many tori, Constructing a necklace,
(Which contains none)
A Cantor Set,
Homeomorphic to no standard, ​C​.

He used his mind to see,
There exists disconnected spaces, With no connected compliment.

Incredible.

“Postulation”
(Kaisa 2019)

I created a 3rd iteration Antione’s Necklace. It is composed of 4096 single chain links representing tori hereafter known as T​0.​ Each iteration is ​n​ = 16 of the previous.

Physical Construction:

  1. Begin with T​0,​ a solid torus, represented by a single chain link.
  2. Chain together n = 16 T​0.​
  3. Link the first and last T​0​ in the chain together to form T​1.​ This is the first iteration for Antione’s Necklace.
  4. Chain together n = 16 T​1.​
  5. Link the first and last T​1​ in the chain together to form T​2.​ This the second iteration for Antione’s Necklace.
  6. Chain together n = 16 T​2.​
  7. Link the first and last T​2​ in the chain together to form T​3.​ This is the third iteration for Antione’s Necklace.

Summary:

  • T​0​ = 1 solid torus
  • T​1​ = 16 T​0
  • T​2​ = 16T​1​ = 256 T​0
  • T​3​ = 16 T​2​ = 256 T​1​ = 4096 T​0

A true Antione’s Necklace begins similarly. (1) Start with a single, solid torus. (2) Replace with a chain of length ​n​ of smaller, solid tori. (3) Now, replace each of those smaller, solid tori (the first iteration) with a smaller chain of length ​n​ of even smaller, solid tori. (4) Now replace those even smaller, solid tori with an even smaller chain of length ​n ​of even smaller, solid tori... (5) Infinitely repeat step 4. Each step produces an iteration.

Perhaps the most perplexing (and interesting) property of Antione’s Necklace is that one constructs it with infinitely many tori. But, as the tori’s diameters become infinitely small... there is not a single torus in Antione’s Necklace. One can always replace a single torus with a chain of chains of chains of... of tori.

Antione’s Necklace is totally disconnected. It is often explained with the analogy:

"a string of beads, but without the string, forming a necklace that cannot fall apart"

Antione's Necklace or How to Keep a Necklace From Falling Apart​

(Beverly L. Brechner, John C. Mayer.)

However, the complement of Antione’s Necklace not connected. He uses this property amongst others to prove that, while Antione’s Necklace is a Cantor Set by definition, it is not homeomorphic to the standard Cantor Set, ​C.​ This was in direct opposition with the belief that all Cantor Spaces are ambiently equivalent.

I hope this art piece brings a smile to one of the most notable and influential blind mathematicians, Louis Antione.