The Disappearing Hyperbolic Squares

The disk is the Poincaré model of the hyperbolic plane. Each red tile is a hyperbolic square. If either of the two largest red tiles is deleted, the resulting set of red tiles is congruent to the original set. This Demonstration shows the two congruences: the last image on the left is the original set minus the tile on top, while the last image on the right is the original set minus the tile on the bottom. Such a set that is invariant under two deletions does not exist in the Euclidean plane.


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When working with infinite sets, it is easy to make one item disappear. For example, the set is congruent to by translation by one unit. Sierpiński asked if a set can contain two distinct points and so that is congruent to the set obtained by deleting either or from . E. G. Straus proved that such a set cannot exist in the Euclidean line or plane. But in the hyperbolic plane, one can construct such a set and visualize it.
This Demonstration illustrates such a "weak Sierpiński set" by using tiles instead of single points. The term weak is used because the full definition would have being congruent to each set obtained by deleting a finite subset.
The idea is to first work in the group of hyperbolic isometries generated by and (these formulas are for the upper-half plane model of ). The two isometries generate a free group and it is easy to construct a weak Sierpiński set in the context of this group: just let be all of the words ending in or . Then and .
There is a tiling of the hyperbolic plane into squares based on this free group (see diagram). So one can easily lift the Sierpiński set from the group to the disk model of by matching up group elements to the square to which they correspond.
The first snapshot shows the initial configuration: the red tiles. The last snapshot shows the final configuration in the two cases: at left, the upper large red square has disappeared and is shown in light pink; at the right it is the lower large square that has disappeared.
Much more is true. In the free group, and therefore in the hyperbolic plane, one can find, in a constructive manner, a set that is unchanged after the deletion of any finite subset [3; 2, theorem 6.1.6]. And in Euclidean 3-space, making use of the axiom of choice, Jan Mycielski [1] proved in 1958 that there is a set that is invariant under the addition or deletion of any countable set of points.
[1] J. Mycielski, "About Sets Invariant with Respect to Denumerable Changes," Fundamenta Mathematicae, 45, 1958 pp. 296–305.
[2] S. Wagon, The Banach-Tarski Paradox, New York: Cambridge University Press, 1985.
[3] E. G. Straus, "On a Problem of W. Sierpiński on the Congruence of Sets," Fundamenta Mathematicae, 44, 1957 pp. 75–81.
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