2023/03/13

Given a hexagonal grid with unit length 1 you can connect hexagons side by side getting ordinary polyhexes. Additional connections with distances of 3 between the centers of the hexagons leads to the rounded or bridged polyhexes already discussed earlier. At last you may think of connections with distances of 2√ 3 between the centers of the hexagons, which skip over a whole hexagon. For the connections you need short bridges of length 1 and long bridges of length √ 3, respectively. The picture bewlow shows the three possible kinds of connections from the yellow hexagon. (sided by side with the green hexagons, short bridges to the red hexagons, long bridges to the blue hexagons)

How can we count the number of possible pieces? Since closed loops of short or long bridges prevent constructions without holes, the bridges should produce a spanning tree with ordinary polyhexes as components. Let's have a look at some pieces with four single hexagons and three long bridges,where the positions of the hexagons are all the same. We have 3 two-sided and 5 one-sided different pieces due to the selection of possible bridges.

How can we get physical pieces? For the one-sided pieces A-shaped bridges of different height and width can be added as shown in the title. But these pieces can't be turned upside down. Therefore I constructed hexagons of slightly smaller size to allow short bridges to pass bewtween the hexagons. Furthmore they get kind of grooves on the front and backside to create space for the long bridges. If you don't reduce the hexagons in size and add only additional grooves at the edges, you get problems combining some pieces as the following example shows. Take the piece with a 120° angle. The edges of the full size hexagons must go under the bridges of the I-piece but their own bridges must go above on the grooves of the I-piece.

For sets of pieces with three and four hexagons I counted the number of pieces and made some symmetric constructions. Since the number of pieces grows very fast, we have already 2537 two-sided pieces with five hexagons. Click the sets in the table to see some examples for the smaller sets. The obj-files for the two-sided trihexes are here .

Number of Hexagons | Number of Pieces | Total Area | |
---|---|---|---|

Two-sided | 3 | 16 | 48 |

4 | 180 | 720 | |

One-sidded | 3 | 22 | 66 |

4 | 323 | 1292 |

For each kind of symmetry one construction with the printed pieces is shown and some other figures are added.

You can also modify a triangle of size 9 or you can combine three rhombs of size 4x4. Chains of rhombs may also be possible but in some cases you might need crossing bridges or bridges skipping empty places.

Kind of rectangles are made from y rows with x hexagons and y-1 rows of x-1 hexagons. The picture shows two rectangles of this type.

Among the constructions without holes are parallelograms of size 16x3, 12x4 and 8x6.

Here are a few examples for figures with one axis of symmetry. Other chains of trapezia even with two axes of symmetry seem to be possible.

The printed pieces are rather fragile, because the bridges are too thin and they don't stick perfectly to the hexagons.

You can also take a hexagon with a side length of 4 and remove a single hexagon or seven hexagons from the center. Remaining hexagons must be distributed symmetrically.

The shown tristars can also be constructed using three congruent parts and the triangle pieces. Removing congruent parts from the corners of larger triangles yields some other symmmetric figures.

Since 66=2*3*11 the only possible parallelograms are 22x3 and 11x6. Cutting triangles of size 2 from opposite corners of larger parallelograms leads to other symmetric constructions.

There are only two trapezia possible. Cutting congruent triangles from the bottom corners of larger trapezia or triangles allows to make kind of hexagons or one pentagon with one axis of symmetry.

Rectangle 3x258+2x259

The figure is divided into six same sized parts and two smaller ones. At first parts 1 to 5 were solved, then the smaller parts and at last part 6. A first approach to solve the smaller parts at last didn't work.

Rectangle 24x27+23x28

The whole figure was divided into stripes of eight rows. I think you can see the separating lines.

Rectangle 28x23+27x24

Stripes of eight rows can be easily combined to fill a large part of the whole rectangle.

Rectangle 259x2+258x3

Such a construction isn't possible. The straight piece with three long bridges doesn't fit into the very small pattern.