Heptiamond 2

Tessellations with Tileable Rotational Pairs

Rotational Pair 1

Each ‘cell’ can be mirrored individually. The three diagrams show the most basic variations.

The first example is identical with the Basic Strip of Pair 6 (blue fill).

The second example is identical with the Basic Zigzag of Pair 6 (blue fill).

Rotational Pair 6

Basic Zigzag.

Rotational Pair 7

Basic Strip, Basic Zigzag, Skewed Zigzag, Endless Zigzag:

The first two examples are identical with the first two of Pair 1 (blue fill). The other two can be regarded as variations of Pair 1 as well.

The first diagram can be regarded as a tessellation of a Flower (red line) and also as a tessellation of six-piece ‘Windmills’ (blue fill bottom left).

The second diagram shows a tessellation of a pair of Flowers — the pink one is a mirrored version of the yellow one.

Other Tessellations

Strips to Stars
The following Themes form Strips with smooth edges. Create variations by mirroring or rotating individual Strips, by sliding them on or off the grid, and exchanging the two types between diagrams.

Strip 1, Strip 1 & rotated, Strip 2, Strip 2 & mirrored:

Two more Themes — they can be combined into a new tessellating pattern (below left). The smooth edges become apparent again when either of the Themes is multiplied horizontally (middle and right).

The two Themes can be widened locally (blue fills) and then tessellated again. The continuous smooth edges are no longer available from this point.

Among the wealth of combinations, the six below are of particular interest because they also occur —

— in Endless Stars like these. The coloring highlights their complexity.

As the previous diagrams make clear, a single yellow or red Theme with partial widening can function as a Dam between the original smooth-edged Strips. the ultimate length of the Strips is determined by the placement, or omission, of similar Dams chosen from the above examples.

Twisted Twins
The Themes have equal halves, one of which has been rotated 180 degrees.

Some Themes built around this core↓

Cropped diamond — can be mirrored.

Double Twin — one of the above plus Pair 7, separated by green lines. Both rows can be multiplied.

Beyond Double Twin — a Theme becomes wrapped in single-piece Strips (blue):

Themes and Strips can be multiplied in various ways. Strips can have other Themes between them (see further above). In the last diagram, the original Theme serves as a Dam in a sea of Strips.

Twisted Twins Growing — some examples of what might be called Metamorphosis in tessellation: in every next stage of the series, more Heptiamonds are added to the original Theme.

First example (2 stages):

Second example (3 stages):

(The second diagram is a little excursion; it shows the first Theme paired with its mirrored and rotated counterpart in blue.)

Third example (stage 1, 2 and 5 shown):

Stage 5 is the most regular with ten Heptiamonds per ‘tile’.

Double Twin and Endless Twin (last diagram):

In the Double Twin, tiers of the yellow/pink group and of the blue group can be copied individually. The Endless Twin has a Dam from which the blue Strips start out in opposite directions.

Arches
This mirroring Twin-like form has infinite variations, as this sampler illustrates. The first diagram contains the Basic Zigzag with Rotational Pair 6 (blue), with the current Theme showing the mechanism. In the other diagrams, pink groups are yellow groups mirrored.

Endless Blocks
Coloring is used here to highlight sub-groups within each of the three sectors.

This Block has the maximum number of Pairs 1.

Some complex Blocks. In the first Block, the boundaries between some sub-groups keep changing before finally settling into repetition. The green arrow in the bottom right corner announces yet another growing spur.

This Block becomes regular outward after a ‘false start’ around the center:

Block around a central form that can also be filled with Heptiamonds 5:

Block around a central group of three cropped diamonds:

Endless Stars around the Flower

Star 1.

Star 2.

Star 3.

Many more Blocks and Stars remain to be found with Heptiamond 2, especially around center groups.