draconic, adj. /drəˈkɒnɪk/ pertaining to, or of the nature of, a dragon. [Latin *draco*, *-ōnem*, < Greek δράκων dragon] — *The Oxford English Dictionary*

In Curvous Energy, I looked at the strange, beautiful and complex fractal known as the dragon curve and showed how it can be created from a staid and sedentary square:

A dragon curve

Here are the stages whereby the dragon curve is created from a square. Note how each square at one stage generates a pair of further squares at the next stage:

Dragon curve from squares #1

Dragon curve from squares #2

Dragon curve from squares #3

Dragon curve from squares #4

Dragon curve from squares #5

Dragon curve from squares #6

Dragon curve from squares #7

Dragon curve from squares #8

Dragon curve from squares #9

Dragon curve from squares #10

Dragon curve from squares #11

Dragon curve from squares #12

Dragon curve from squares #13

Dragon curve from squares #14

Dragon curve from squares (animated)

The construction is very easy and there’s no tricky trigonometry, because you can use the vertices and sides of each old square to generate the vertices of the two new squares. But what happens if you use lines rather than squares to generate the dragon curve? You’ll discover that less is more:

Dragon curve from lines #1

Dragon curve from lines #2

Dragon curve from lines #3

Dragon curve from lines #4

Dragon curve from lines #5

Each line at one stage generates a pair of further lines at the next stage, but there’s no simple way to use the original line to generate the new ones. You have to use trigonometry and set the new lines at 45° to the old one. You also have to shrink the new lines by a fixed amount, 1/√2 = 0·70710678118654752… Here are further stages:

Dragon curve from lines #6

Dragon curve from lines #7

Dragon curve from lines #8

Dragon curve from lines #9

Dragon curve from lines #10

Dragon curve from lines #11

Dragon curve from lines #12

Dragon curve from lines #13

Dragon curve from lines #14

Dragon curve from lines (animated)

But once you have a program that can adjust the new lines, you can experiment with new angles. Here’s a dragon curve in which one new line is at an angle of 10°, while the other remains at 45° (after which the full shape is rotated by 180° because it looks better that way):

Dragon curve 10° and 45°

Dragon curve 10° and 45° (animated)

Dragon curve 10° and 45° (coloured)

Here are more examples of dragon curves generated with one line at 45° and the other line at a different angle:

Dragon curve 65°

Dragon curve 65° (anim)

Dragon curve 65° (col)

Dragon curve 80°

Dragon curve 80° (anim)

Dragon curve 80° (col)

Dragon curve 135°

Dragon curve 135° (anim)

Dragon curve 250°

Dragon curve 250° (anim)

Dragon curve 250° (col)

Dragon curve 260°

Dragon curve 260° (anim)

Dragon curve 260° (col)

Dragon curve 340°

Dragon curve 340° (anim)

Dragon curve 340° (col)

Dragon curve 240° and 20°

Dragon curve 240° and 20° (anim)

Dragon curve 240° and 20° (col)

Dragon curve various angles (anim)

Previously pre-posted:

• Curvous Energy — a first look at dragon curves