TITLE: Are there locally jammed arrangements of spheres of zero density?
QUESTION [16 upvotes]: I know of a remarkable result from a paper of 
Matthew Kahle (PDF download), that there are arbitrarily low-density
jammed packings of congruent disks in $\mathbb{R}^2$:

In 1964 Böröczky used
  a subtle disk packing construction to disprove a conjecture of Fejes Tóth, that a
  locally jammed arrangement of disks in the plane, one where each disk is held in
  place by its neighbors, must have positive density. 
Böröczky, K. "Über stabile Kreis- und Kugelsysteme." Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 7, 79-82, 1964.

Surprisingly (to me), this natural candidate is not the lowest density:
     

My question is whether or not Böröczky's construction extends to $\mathbb{R}^3$
and higher dimensions. (I do not yet have access to Böröczky's 1964 paper.)
Addendum. (I add this prompted by user j.c.)
My question is answered: Yes. First, through the help of
Wlodek Kuperberg and other commenters, we learned what exactly is Böröczky's
1964 zero-density packing. Indeed it is "subtle" as Matthew Kahle said.
Second, simply layering identical copies of that packing of spheres in $\mathbb{R}^3$
results in a jammed packing of zero density, as confirmed by Benoît Kloeckner.
What remains is to correct Wikipedia's misleading entries on the topic, as detailed by 
Gerry Myerson.

REPLY [12 votes]: Wlodek Kuperberg sent me this
'diagram from the "Lagerungen"' 
of Böröczky's packing, over which I overlaid several congruent disks:
 
(Yes, small cracks or overlays are visible between disks that should be tangent.
The diagram is not 100% metrically accurate.)

Added: The vertical red lines suggest a slight concavity.