Boring, dull, staid, stiff, everyday, ordinary, unimaginative, unexceptional, crashingly conventional — the only interesting thing about squares is the number of ways you can say how uninteresting they are. Unlike triangles, which vary endlessly and entertainingly, squares are square in every sense of the word.

And they don’t get any better if you tilt them, as here:

Sub-squares from gray square (with corner-numbers)

Nothing interesting can emerge from that set of squares. Or can it? As I showed in Curvous Energy, it can. Suppose that the gray square is dividing into the colored squares like a kind of amoeba. And suppose that the colored squares divide in their turn. So square divides into sub-squares and sub-squares divide into sub-sub-squares. And so on. And all the squares keep the same relative orientation.

What happens if the gray square divides into sub-squares sq2 and sq9? And then sq2 and sq9 each divide into their own sq2 and sq9? And so on. Something very unsquare-like happens:

Square-split stage #1

---

Stage #2

---

Square-split #3

---

Square-split #4

---

Square-split #5

---

Square-split #6

---

Square-split #7

---

Square-split #8

---

Square-split #9

---

Square-split #10

---

Square-split #11

---

Square-split #12

---

Square-split #13

---

Square-split #14

---

Square-split #15

---

Square-split #16

---

Square-split (animated)

The square-split creates a beautiful fractal known as a dragon-curve:

Dragon-curve

---

Dragon-curve (red)

And dragon-curves, at various angles and in various sizes, emerge from every other possible pair of sub-squares:

Lots of dragon-curves

And you get other fractals if you manipulate the sub-squares, so that the corners are rotated or reverse-rotated:

Rotation = 1,2 (sub-square #1 unchanged, in sub-square #2 corner 1 becomes corner 2, 2 → 3, 3 → 4, 4 → 1)

---

rot = 1,2 (animated)

---

rot = 1,2 (colored)

---

rot = 1,5 (in sub-square #2 corner 1 stays the same, 4 → 2, 3 stays the same, 2 → 4)

---

rot = 1,5 (anim)

---

rot = 4,7 (sub-square #2 flipped and rotated)

---

rot = 4,7 (anim)

---

rot = 4,7 (col)

---

rot = 4,8

---

rot = 4,8 (anim)

---

rot = 4,8 (col)

---

sub-squares = 2,8; rot = 5,6

---

sub-squares = 2,8; rot = 5,6 (anim)

---

sub-squares = 2,8; rot = 5,6 (col)

---

Another kind of dragon-curve — rot = 3,2

---

rot = 3,2 (anim)

---

rot = 3,2 (col)

---

sub-squares = 4,5; rot = 3,9

---

sub-squares = 4,5; rot = 3,9 (anim)

---

sub-squares = 4,5; rot = 3,9 (col)

Elsewhere other-accessible…

• Curvous Energy — a first look at dragon-curves
• Back to Drac’ — a second look at dragon-curves