Agnes Scott College
Larry Riddle, Agnes Scott College

Sierpinski Pedal Triangle

Description

Let T = S(0) be an acute triangle with vertices A, B, and C. The pedal triangle of T is the triangle formed by the three points A1, B1, and C1 that lie at the feet of the three altitudes of T, i.e. from each vertex drop a perpendicular to the opposite side until it intersects with that side.

pedalTriangle

The point E where there three altitudes intersect is the orthocenter of triangle T. The pedal triangle divides the original triangle into four smaller triangles. Remove the interior of the pedal triangle to get S(1).

pedalTriangle2

Each of the three remaining triangles is similar to the original triangle T [Proof]. Now repeat this procedure and remove the pedal triangle for each of these three remaining triangles to obtainS(2).

iter0
S(0)
iter1
S(1)
iter2
S(2)
iter3
S(3)

Continue to repeat the construction to obtain a decreasing sequence of sets

$$ S(0) \supset S(1) \supset S(2) \supset S(3) \supset \cdots $$

The Sierpinski pedal triangle is the intersection of all the sets in this sequence, that is, the set of points that remain after this construction is repeated infinitely often.

 

Construction
Animation

iterall
Sierpinski pedal triangle with angles 65°, 50°, and 65°


Iterated
Function
System

The three triangles left after the pedal triangle is removed can be obtained from the original triangle by a sequence of scaling, reflection, rotation, and translation. The transformations are completely determined by the three angles of the original triangle.

pedalTriangle2

If A, B, and C represent the angles at the corresponding vertices in the figure above, then the IFS is given by the following three functions [Proof].

\({f_B}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {{{\cos }^2}(B)} & {\cos (B)\sin (B)} \\ {\cos (B)\sin (B)} & { - {{\cos }^2}(B)} \\ \end{array}} \right]{\bf{x}}\)
 
scale by cos(B)
vertical reflection
rotate by B
 
\({f_C}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {{{\cos }^2}(C)} & { - \cos (C)\sin (C)} \\ { - \cos (C)\sin (C)} & { - {{\cos }^2}(C)} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {{{\sin }^2}(C)} \\ {\cos (C)\sin (C)} \\ \end{array}} \right]\)
 
scale by cos(C)
vertical reflection
rotate by −C
 
\({f_A}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - \cos (A)\cos (C - B)} & {\cos (A)\sin (C - B)} \\ {\cos (A)\sin (C - B)} & {\cos (A)\cos (C - B)} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {{{\sin }^2}(C)} \\ {\cos (C)\sin (C)} \\ \end{array}} \right]\)
 
scale by cos(A)
horizontal reflection
rotate by −(C−B)
 

The Sierpinski Pedal Triangle consists of three self-similar pieces corresponding to the three functions in the iterated function system.

IFSpedal
70°-60°-50° Sierpinski Pedal Triangle
[Enlarge]

Similarity
Dimension

The Sierpinski pedal triangle is self-similar with 3 non-overlapping copies of itself, each scaled by the factors cos(A), cos(B), and cos(C), respectively. Therefore the similarity dimension, d, of the attractor of the IFS is the unique solution to the Moran equation \[\cos {(A)^d} + \cos {(B)^d} + \cos {(C)^d} = 1\]

(There is a unique solution because the function \(h(d) = \cos {(A)^d} + \cos {(B)^d} + \cos {(C)^d}\) has h(0)=3 and is strictly decreasing to 0 as d increases.)

The 65°-50°-65° Sierpinski pedal triangle shown above has dimension 1.59572.

The 70°-60°-60° Sierpinski pedal triangle has dimension 1.60410.

The figures below show a 30°-75°-75° Sierpinski pedal triangle with dimension 1.65931 and a 55°-85°-40° triangle with dimension 1.79750.

30-75-75pedaltriangle  55-85-40triangle


Right Triangle
Animation
Notice that as the original triangle ΔABC approaches a right triangle, the dimension of the corresponding Sierpinski pedal triangle approaches the value 2. If, in fact, ΔABC is a right triangle, then the pedal triangle is actually just a line segment that divides ΔABC into two similar right triangles. So there is nothing to "remove" at each step in the iterative construction and the Sierpinski pedal triangle is just the original ΔABC with dimension 2. You can see the "pedal triangle" line segments filling up the original triangle in the animation.

Zhang, Hitt, Wang, and Ding conjectured that the ordinary Sierpinski gasket (where all angles equal 60°) has the smallest fractal dimension among all Sierpinski pedal triangles.

 

References

  1. Xin-Min Zhang, Richard Hitt, Bin Wang,and Jiu Ding. "Sierpinski Pedal Triangles," Fractals, Vol. 16, No. 2 (2008), 141-150 [Available at http://www.rhitt.com/research/hittzhang.pdf]