Folding instructions:
Windmill Fringed Gentian

A photo guide to folding the tessellation, Windmill Fringed Gentian. The tessellation uses closed square twists on both sides of the paper. The "windmill" comes from pinwheeling the central square twist of the molecule, creating a floral ("gentian") structure when backlit.

The crease pattern for 3x3 molecules arrangement.

A 48-square grid will allow for a 3x3 molecules arrangement with a little extra space around the edges.

This guide uses 24cm purple kami (white back side) precreased in a 32-square grid, which allows for a 2x2 molecule arrangement with some extra space on the edges.

Step 1: Initial molecule

This molecule is folded in one quadrant of the paper. For a 48-square grid, it would be folded in the center.

1.1: Front square twist 1

Start with a closed square twist 9 grid spaces away from the top and left edges. This will be the central square twist of the molecule.

Completed: the molecule's central square twist.

1.2: Back square twist 1

Flip the paper to the back side and fold a square twist on the pleat extending to the left of the front side's square twist, following the orientation of the existing pleat. This will be the upper left back square twist.

In progress: Folding the square twist.

Completed: Back square twist 1 (upper left).

1.3: Front square twist 2

Flip the paper to the front side and fold a square twist above the central square twist. This will create an extra pleat to be resolved in the next step.

In progress: Folding the square twist above the previous one.

1.4: Back square twist 2

Flip the paper to the back side and fold a square twist below the existing one. This will resolve the extra pleat from the previous step.

In progress: Folding the square twist below the previous one.

1.5: Front square twist 3

Flip the paper to the front side and fold a square twist to the right of the central square twist. This will create an extra pleat to be resolved in the next step.

In progress: Folding the square twist to the right of the central twist.

1.6: Back square twist 3

Flip the paper to the back side and fold a square twist to the right of the previous one. This will resolve the extra pleat from the previous step.

In progress: Folding the square twist to the right of the previous one.

1.7: Front square twist 4

Flip the paper to the front side and fold a square twist to the bottom of the central twist. This will create two extra pleats to be resolved in the next two steps.

In progress: Folding the square twist below the central twist.

1.8: Back square twist 4

Flip the paper to the back side and fold a square twist above the previous one, completing the grouping. This will resolve one of the extra pleats from the previous step.

In progress: Folding the square twist above the previous one, which will reveal the top left twist in the grouping.

1.9: Front square twist 5

Flip the paper to the front side and fold a square twist to the left of the central twist to resolve the second extra pleat (from 1.7) and complete the molecule.

In progress: Folding the square twist to the left of the central twist.

The initial molecule is complete!

Step 2: Connecting a molecule

Folding a second molecule is slightly different from the initial molecule. First, connected molecules will have a flipped (mirror) symmetry to the original molecule. Second, in order to connect molecules, we start at one of the front side's edge twists instead of with the connected molecule's central twist.

2.1: Connecting twist (top twist)

From the bottom twist of the original molecule, fold a connecting square twist with a single grid space between the pleats.

In progress: Folding the square twist connecting to the previous molecule.

Completed: Top twist of the connected molecule.

2.2: Back square twist 1

Flip the paper to the back. From the bottom right twist of the original molecule, fold a connecting square twist, using the existing pleat lines from 2.1.

In progress: Folding the square twist connecting to the previous molecule.

Completed: Bottom left back twist of the connected molecule.

2.3: Front square twist 2

Flip the paper to the front. Using pleat lines from 2.1 and 2.2, fold a square twist to the bottom right of the first (the right square twist of the molecule.)

In progress: Folding the right square twist of the molecule.

Completed: Right twist of the connected molecule.

2.4: Back square twist 2

Flip the paper to the back. Fold a square twist above the existing one.

In progress: Folding the square twist above the previous twist.

Completed: Top left back twist of the connected molecule.

2.5: Front square twist 3

Flip the paper to the front. Fold the molecule's central square twist. This will create an extra pleat to be resolved in the next step.

In progress: Folding the central square twist of the molecule.

Completed: Central twist of the connected molecule.

2.6: Back square twist 3

Flip the paper to the back side and fold a square twist in the bottom right of the grouping. This will resolve the extra pleat from the previous step.

In progress: Folding the square twist in the bottom right of the grouping.

Completed: Bottom right back twist of the connected molecule.

2.7: Front square twist 4

Flip the paper to the front. Fold the molecule's left square twist. This will create two extra pleats to be resolved in the next steps.

In progress: Folding the left square twist of the molecule.

Completed: Central twist of the connected molecule.

2.8: Back square twist 4

Flip the paper to the back side and fold a square twist in the top right to complete the grouping. This will resolve the extra pleat from the previous step.

In progress: Folding the square twist in the bottom right of the grouping.

Completed: Top right back twist of the connected molecule.

2.9: Front square twist 5

Flip the paper to the front. Fold the molecule's bottom square twist to resolve the extra pleat from 2.7 and to complete the molecule.

In progress: Folding the left square twist of the molecule.

Completed: The connected molecule.

The connected molecule is complete!

Step 3: Additional connections

Connecting additional molecules should be easier now. In fact, if you fold a complete row first, you can then repeat the row below by folding all of those molecules in parallel, which will make the connections easier. Steps will not be explicitly noted but here are some in-progress photos for reference.

Top twists of the second row of connected molecules.

First back twists are the top left / top right for the connected molecules.

Front twists to the left / right of the connected molecules.

Back twists complete the top row of the connected molecules' groupings.

Front central twists are completed.

Bottom left / bottom right back twists.

Front right / left twists, which connect to each other.

Bottom right / bottom left back twists complete the groupings.

The bottom twists complete the connected molecules.

Step 4: Finishing

Although you can fold the pinwheel as the final step of each individual molecule, I find it more efficient to fold all the pinwheels at the end, once all molecules have been completed. Remember that the rotations of connected molecules are mirrored, so the orientation of the pinwheels will alternate between molecules.

First pinwheel fold.

Second pinwheel fold.

Third pinwheel fold.

It helps to partially unfold the first pinwheel fold in order to tuck in the fourth fold.

Notes

A 32-square grid allows for a 2x2 arrangement with some extra spacing.
A 48-square grid allows for a 3x3 arrangement with some extra spacing.
A 64-square grid allows for a 5x5 arrangement with no extra spacing.
The pattern does not need grid rotation, as it aligns perfectly with the sides of the paper.
You can change which square twists you pinwheel, or none at all - different arrangements alter the look of the final pattern.