Agnes Scott College
Larry Riddle, Agnes Scott College

Terdragon

Description

 

Construction
Animation

Construction via line segments

Begin with a line segment. As the first iteration, replace this segment with three segments in a Z shape, each scaled by a ratio \({\bf{r}} = \frac{1}{{\sqrt 3 }}\). The middle segment makes an angle of 60° with the first and last segments. Following along the original segment, we start the first new segment to the left. For the second iteration, replace each of the three segments with three new segments in the same motif, each scaled by the ratio r. The new segments are placed to the left, then to the right, and then to the left. Continue this construction, always alternating the new segments between left and right along the segments of the previous iteration. This generates the "terdragon curve". The following figure shows the first three iterations for this construction.

LineConstruction

Construction via paper folds

Fold a long narrow sheet in thirds, folding the first third towards you and the last third away from you. Fold three times again in the same directions (Example). Continue folding, always folding in the same directions. After several folds, open the sheet so that every fold is at 60deg; and view the sheet from the edge. In general, n sets of folds produce an order-n dragon. The following picture shows an order-2 dragon.

order2


Iterated
Function
System

Starting with an initial set that is a horizontal unit line segment, the construction shown above requires three copies of the segment, each scaled by a factor r. The first must be rotated by 30°, the second by −90°, and the last by 30°. The second and third segments must then be translated so that the path remains connected.

IFSdetails

We see that

\[r = \frac{{1/2}}{{\cos {{30}^ \circ }}} = \frac{{\frac{1}{2}}}{{\frac{{\sqrt 3 }}{2}}} = \frac{1}{{\sqrt 3 }}\]

 

IFS
Animation

This yields the following IFS.
 
\({f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { - \sqrt 3 / 6} \\ {\sqrt 3 / 6} & {1/2} \\ \end{array}} \right]{\bf{x}}\)
 
   scale by \(1/ \sqrt 3\), rotate by 30°
 
\({f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 0} & { \sqrt 3 /3} \\ {-\sqrt 3 /3} & { 0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ \sqrt 3 / 6 \\ \end{array}} \right]\)
 
   scale by \(1/ \sqrt 3\), rotate by −90°
 
\({f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { - \sqrt 3 / 6} \\ {\sqrt 3 / 6} & {1/2} \\ \end{array}} \right]{\bf{x}}+ \left[ {\begin{array}{*{20}{c}} 1/2 \\ -\sqrt 3 / 6 \\ \end{array}} \right]\)
 
   scale by \(1/ \sqrt 3\), rotate by 30°
 

The terdragon consists of three self-similar pieces corresponding to the three functions in the IFS.

terdragon
[Enlarge]

L-System

Angle 30
Axiom F
F —> +F−−−−F++++F−

Similarity
Dimension

The terdragon is self-similar with 3 non-overlapping copies of itself, each scaled by the factor r < 1. Therefore the similarity dimension, d, of the attractor of the IFS is the solution to

\[\sum\limits_{k = 1}^3 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/3)}}{{\log (1/\sqrt 3 )}} = 2\]

Special
Properties

 
 
 

Three copies of the terdragon tile the plane. In addition to the regular terdragon, one copy must be rotated by 60° and the other by −60°. Click on each of the buttons in succession to see this tiling in the square below.

blank

Six copies of the terdragon can be joined together at a common point at 60° angles.

sixCopies2Level3 sixCopies2

The boundary of the terdragon is a fractal with dimension log(4)/log(3) = 1.26186 [Details].

terdragonBoundary

The area of a terdragon with an initial segment of length b is equal to \(\dfrac{b^2}{2 \sqrt 3}\) [Details].

Variations

Fudgeflake

Three copies of the terdragon form a fudgeflake. The fudgeflake can also be formed by placing copies of the lower boundary of the terdragon around the three sides of an equilateral triangle (similar to the construction of the Koch snowflake). Copies of the fudgeflake will tile the plane (if you look at the tiling picture above, you can see the red/blue/green grouping of three terdragons that make up one fudgeflake). Click on the name for more details.

3terdragons

 

References

  1. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Sringer-Verlag, 1990.
  2. Davis, Chandler and Donald J. Knuth. "Number representations and dragon curves" J. Recreational Math. 3 (1970) 66-81 (Part 1), 133-149 (Part 2). Reprinted with extra addendum in Selected Papers on Fun and Games, Donald Knuth, CSLI Publications, 2011.