Regular and Semi-Regular Tessellations in Paper

This is my favorite method of sanity maintenance. (Allegations that this doesn't make one particularly sane are cheerfully acknowledged.) Each of these was folded from one sheet of paper... no cutting and no drawing allowed. Just folding. I haven't yet tracked down the origins of this particular branch of origami, but I know that some of the early work was done by Kawasaki and Fujimoto and most of the recent innovation has been by Chris Palmer. All of the designs on this page were derived from pictures he's put up on his Shadowfolds site. (Warning: Shadowfolds is visually beautiful but is all Macromedia Flash. If you're running recent versions of Netscape or IE you probably already have this installed.) He designed them. I just folded them. If you're interested in the math behind these patterns, I recommend the What is a Tessellation? page at the Math Forum.

The translucent pieces were all folded from glassine. The two red pieces began from pieces 500mm square; all the others began as 700mm squares. (The one exception is the brown triangles-and-hexagons strip, which began as a piece roughly 1 meter by 500 mm.)

No diagrams exist for most of these. A good source to get you started is Chris's Hira-Ori tape. (Origamido lists this as out of stock. If you're interested, call them and ask if this is still true. The tape is worth it.) If you're looking for other web pages, I know of the following three:

Without further ado, here are the images! Everything you see here is a thumbnail... you can click on it for a higher-resolution version.

Opaque Stuff

Neither of these patterns have much at all to do with regular tessellations, although it's possible to get the octagon collapse to work as part of a tiling of squares.

Gold flower
tower Bird
base octagon collapse
A flower tower is a recursive pattern based on a closed-back twist fold. Tom Hull posted some instructions for making a flower tower to the origami mailing list a while back. The crease pattern for this piece consists of a ring of eight bird bases surrounding a central octagon.

Translucent Stuff

A Minor Variation

octagons (open back 2) Red
octagons (open back 2, iso-area)
octagons (open back 2) detail Red octagons
(iso-area) detail
These two pieces are essentially the same pattern. On the left are regular twist octagons (open back II) where all the pleats emerge from the same side of the paper. On the right, the pattern has been rotated 22.5 degrees, and alternating pleats emerge from different sides of the paper. The result is that the pattern on the right looks the same on both sides of the paper, a technique known as iso-area folding.

Twist Octagons

twist octagons (open back I) Orange twist
octagons (darker version)
Detail of orange octagons (center) Detail of orange octagons (corner)
This piece is all variations on the open-back (I) octagon twist. The four stars in each corner of the paper also incorporate a closed-back octagonal twist in the center of the pattern. (Yes, the detail of the center is out of focus. I'll re-take the picture when I get a chance.)

twist octagons
twist octagons (variant 1) Green
twist octagons (variant 2)
This is what happens when you fold an open-back I octagon on top of a closed-back octagon twist. The variations come from things you can do to the closed-back twist before the final collapse.

Triangles and Hexagons Detail of
All of the tessellations so far have been based on a square grid except the red iso-area octagons, which are based on a pattern of octagons and squares. This piece was folded on a grid of equilateral triangles instead. This is actually fairly straightforward: if you start with a square, it's easy to construct a crease pattern with folds at 30- and 60-degree angles to the center lines, and once you have a few of those it becomes a simple matter of subdivision. The piece of paper for this began as a long rectangle. take 2 take 2
(detail of center)
This is filling a hexagon instead of a long rectangle. This pattern was originally due to Shuzo Fujimoto.

Levels of Recursion

The following three pieces are what Chris Palmer calls whirl spools. They are each based on the Archimedean tessellation but have differing levels of recursion. On the Shadowfolds site these are described as "Watering Fujimoto's Garden".,
base case,
base case, detail of center
This is the base case. There is only one twist-fold per cell.,
recursion level 1
(recursion level 1) detail
Recursion level 1. Note how the edge of each cell has been divided into 3 segments. The square segments contain 3x3 twist folds (the pattern The triangles and the central hexagon contain (as above). Each recursion level in this pattern increases the number of divisions by 2 in order to get the handedness of the pattern to match up. (I should probably explain this in more detail.)
(recursion level 2)
(recursion level 2) (detail) at recursion level 2: each edge in the original design has been divided into 5 equal segments.


star progression 12-fold star progression (detail of center)
This is one of those designs that I could sit and stare at for hours. The center is straightforward enough: a 12-fold closed-back twist surrounded by closed-back triangles and squares. However, the pleats that radiate outward from the center have been folded into a series of ever-expanding 12-pointed stars instead of simply proceeding straight. The detail of the center is a bit messy because I haven't learned how to collapse that part cleanly just yet. I find myself dragging the rest of the paper around to try to get enough slack to fold the pleats over. Perhaps a bigger version would be easier.

Purple Octagons (for Merlin and Uriel)

Purple octagons
This was made as a wedding gift for Merlin and Uriel. Like most of the others, it's based on a floor tiling from the Alhambra. If you stare at it long enough, it's based on 8.8.4 (like the red twist octagons above), but it's easier to treat it as (a regular grid of squares) and deal with the vertices of the grid separately.

Green Hexagons (for tyee and OJ)

This was a wedding gift for tyee and OJ. It's another tiling pattern from the Alhambra. Unlike most of the tessellations I've made, this one incorporates different pleat widths: if you look at the insides of the stars on top and bottom, they're the same patterns as in the middle, only smaller. It's possible to fold this so you can't see the grid of triangles in all the light areas, but I rather like the effect. I'll put in a better picture of this once tyee and OJ get a chance to take one.

I will be updating this page as I fold more of these and obtain better pictures. Many thanks to Tanner Lovelace for his time and effort taking pictures one semi-cloudy afternoon. There are more pictures in the origami folder of my gallery.

Page created by Andy Wilson
Last updated 3 July 2002