Rounded Polyhexes


Polyhexes are side-connected hexagons in the hexagonal grid. For each hexagon there are six adjacent neighbors. You can expand the set of polyhexes, if you allow connections from one corner of an hexagon to one corner of another one along a unit line segment of the grid. In the picture below the six adjacent neighbors are marked green and the further ones are marked red. This type of connection is called a pseudo connection because usually there isn't enough space for a physical one.

In April 2013 Jared McComb asked me to have a look at these pieces, but I didn't feel like writing a program. After another e-mail in September 2014 I began to work with these pieces.
A triangle made from three hexagons and three pseudo connections is rather useless for constructions because of the hole in the center, which can't be filled. This applies to some more pieces with four hexagons. Therefore loops with pseudo connections should be forbidden. In accordance with the rounded polyominoes (see OEIS A056840) I worked with components of polyhexes, where the additional pseudo connections form a spanning tree and called them Rounded Polyhexes or Bridged Polyhexes. The rounded n-hexes must contain a total of n hexagons.

Let's have a look at a special rounded tetrahex. Due to different choices of the pseudo connections you get three two-sided and five one-sided pieces with same positions of the four hexagons.

How can we get physical pieces? I reduced the hexagons in size to get some space for a thin pseudo connections. Furthermore I used circles instead of hexagons and cut small cyliders from a bar. Because the circles are reduced in size even originally adjacent ones must be connected by short sticks whereas the pseudo connections need a long stick.
For the one-sided pieces another approach is possible. I used chips for shoppping carts instead of hexagons and halves of rings as bridges to skip adjacent chips on the ground. This justifies the name Bridged Polyhexes.

As usual you can try to construct symmetric figures with sets of rounded polyhexes. Hexagons and hexagonal rings are examples of rotation-symmetric figures for an angle of 60 degree. You get rotational symmetry for an angle of 120 degree, if you make triangles or triangles with truncated corners, which are semi-regular hexagons as shown above. At least parallelograms with rotational symmetry for an angle of 180 degree are also possible. Kind of rectangles can be constructed in the hexagonal grid, if N rows of size L are connected by (N-1) rows of size(L-1) or (L+1) getting two axes of symmetry. A trapezium is an example for a figure with one axis of symmetry.

For two-sided and one-sided rounded polyhexes with a given number of hexagons the number of different pieces and some possible constructions are listed in the following table. Click the different sets to see the sets and constructions made with them.

Number of Hexagons Number of Pieces Total Area Constructions
Two-sided Pieces 3 8 24 rectangle (3*4+4*3);
4 52 208 hexagon with rhomb hole;
hexagon with sides 12-6-12-6-12-6;
parallelogram 16x13;
parallelogram 26x8;
pairs of congruent rectangles;
four congruent berries;
four congruent rectangles(4*7+3*8)
5 433 2165 rectangle (59*19+58*18);
rectangle (19*59+18*58);
rectangle (31*35+30*36);
rectangle (36*30+35*31);
rectangle (167x7+166x6);
rectangle (7x167+6x166);
3 or 4 60 232 pair of a size 9 hexagon and a size 5 triangle
2..4 62 236 pair of hexagons of size 3 and 9
One-sided Pieces 3 10 30 rectangles with holes;
hexagon with sides 3-4-4-3-4-4
4 89 356 hexagon with sides 20-4-13-20-13-4;
hexagon with sides 43-5-4-43-5-4;
lots of rectangles;
set of two rhombs of size 16 and 10;
5 816 4080 four hexagonal rings (19-2);
set of hexagons with sizes of 20,20,25;
rectangle (100*21+99*20);
rectangle (21*100+20*99)
3 or 4 99 386 pair of hexagons of size 8 and 9
2..4 101 390 hexagonal rings (16-11)
hexagonal rings (12-2)