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We discuss tiles with the property that two copies of the tile can surround one or two other copies.

Voderberg tiles (V-tiles) of type 1a may admit periodic, rotational and spiral tilings of the plane.
More interesting is that two copies of a tile can surround one or two other copies. But this surrounding is not complete because two points of the internal tile(s) belong to the boundary of the tiling patch.
Adjacent tiles can be transformed into each other by a rotation around a common point of the tiles.

Voderberg tiles of type 1b do not admit rotational and spiral tilings of the plane.
Two copies of a tile can incompletely surround one or two other copies.
Opposite to type 1a, for certain two adjacent tiles the center of rotation is not a point of these tiles.

The principles of this type are described in Heinz Voderberg's paper (see reference below).
A point reflection is applied for type 1, whereas a glide reflection shall be used for type 2.
The author of this web page was not able to construct such a tile yet.
He never has seen a tile of type 2 and does not know if it can be constructed at all.
Can anybody help?

Heinz Voderberg, Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente. Jahresbericht der Deutschen Mathematiker-Vereinigung (1936), Vol. 46, pp. 229-231

See English translation Voderberg-English of Voderberg's article, or a German copy Voderberg-Abschrift.

This tile was found by Casey Mann in 2002:
One tile can be completely surrounded by two other copies of itself, with the enclosed tile having no contact with the boundary of the overall tiling (patch).
The tiles cannot tile the plane.

Two copies of a V-Dragon can surround another copy completely. That means:
There exists a patch of 3 congruent VT-Dragons that is topologically equivalent to a disk and
one Dragon has no point in common with the boundary of the patch.
Additionally each VT-Dragon admits a tiling of the plane.
Rotational and spiral tilings of the plane are not possible.

Only a few tiles are known that admit a tiling of the plane, with some tiles having only 3 neighbors.
Only Voderberg tiles allow monohedral (all tiles are congruent) tilings of the plane, where tiles with only 2 adjacents occur.
And only Voderberg tiles would be able to tile the plane in a way that at least one tile has only 2 neighbors.