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.
|
|
| 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
|
|
|
|
|
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
|
|
|
|
|
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.)
|
|
|
|
|
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
|
|
|
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.
|
|
|
|
This is 6.3.6.3 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 6.4.3.4 but have
differing levels of recursion. On the Shadowfolds site these are
described as "Watering Fujimoto's Garden".
|
|
|
This is the base case. There is only one twist-fold per cell.
|
|
|
|
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 4.4.4.4). The triangles and the central hexagon contain
6.3.6.3 (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.)
|
|
|
|
6.4.3.4 at recursion level 2: each edge in the original design has
been divided into 5 equal segments.
|
Sunburst
|
|
|
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)
|
|
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 4.4.4.4 (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
