Affine Regular Pentagon Sculptures

Year: 2006 Authors: Douglas G. Burkholder

Core claim

A carefully chosen symmetric linear construction from a non-planar pentagon always yields planar affine regular stellar pentagons, generalizing Jesse Douglas’s procedure.

Topics

symmetric linear construction, nested pentagons, stellar pentagon, computer-generated sculpture

Domains

affine geometry, polygon construction, weighted averages, golden ratio, sculpture, geometric art, computer-generated art

Methods

iterative point construction, midpoint construction, ratio division on line segments, symmetry-based averaging

Media

pentagon diagram, computer-generated sculptures

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Douglas G. Burkholder School of Computing Science & Mathematics Lenoir-Rhyne College 625 7th Avenue NE Hickory, North Carolina, 28603 E-mail: burkholderd@lrc.edu

Abstract

In this paper we shall describe how to apply symmetric linear constructions to a random non-planar pentagon to construct mathematically and artistically interesting sculptures, such as in Figure A. This process will always produce a nested set of affine regular stellar pentagons. This generalizes a procedure created by Jesse Douglas.

Figure A

Starting from a random non-planar pentagon, we shall describe a sequence of steps consisting of constructing line segments between pairs of known points followed by locating a new point on this line at a specified ratio of this length. At the completion of these steps, label the last point constructed ${\nu }_l$ . We repeat the same process from the vantage point of each successive vertex of the original pentagon and label the last point of each sequence ${\nu }_i$ respectively. By connecting the ${\nu }_i$ in order we form a new pentagon. Although the final sculpture is not symmetric, since the process is symmetrical, this is called a symmetric linear construction. If this process is carefully selected, then the resulting pentagon will be planar and affine regular. By including variations on the theme, we can construct a collection of nested

affine regular pentagons within the original pentagon. Affine regular pentagons, if oriented just right, will cast a shadow which is regular.

Given a random pentagon with vertices $p_1,p_2,\dots ,p_5$ , any finite sequence of steps of the type listed above can be written as a weighted average $w_1{p}_1+w_2{p}_2+\dots +w_5{p}_5$ of these 5 points. By symmetry, all of the vertices of the final pentagon constructed can be written as $v_i=w_1{p}_i+w_2{p}_{i+1}+\dots +w_5{p}_{i+4}$ , where subscripts are modulo 5. If these weights $w_i$ measure the distances from successive vertices of a regular stellar pentagon to an arbitrary line, with distances below the line measured as negative distances, then the pentagon constructed from these weighted averages will be affine regular stellar [1]. For our sculpture we shall place the line parallel to and a distance of $a$ units

Figure 1: Calculating the weights

above the base, as shown in Figure 1. By letting $a$ vary, we shall be able to construct a nested set of affine regular stellar pentagons within our sculpture. If the distance between the base $p_4{p}_2$ and the horizontal line $p_1{p}_5$ is 1 unit then the height of the pentagon is the golden ratio $\phi =(1+\sqrt{5})/2$ . Thus, the desired weights, in order, are $(1-a),-a,(\phi -a),-a,(1-a)$ . While these weights will always generate affine regular pentagons, we must normalize these by dividing by $(2+\phi -5a)$ before we can convert them into a sequence of linear steps. This gives us the following formula for calculating the location of the new vertices:

To convert these weights into a sequence of steps consisting of locating points on line segments we first rewrite the above in groupings of pairs of the form $(1-c)x+cy$ . The above is equivalent to:

For any pair of points $x$ and $y$ , the point $(1-c)x+cy$ is located on the line $xy$ at ratio $c$ of the distance between $x$ and $y$ as measured from $x$ towards $y$ . For example, if $c=1/2$ then we find the midpoint; if $c=1.5$ then we go from $x$ to $y$ and then go an extra half the distance of $xy$ past $y$ ; and if $c=-1/2$ then we travel half

the distance of $xy$ from $x$ away from $y$ . We start from any non-planar pentagon, such as the one shown in Figure 2. We can interpret the above formula, reading from the inside out, as: 1) Locate the midpoint $m_{15}=(p_1+p_3)/2$ as shown in Figure 3 below. 2) Construct the line segment $p_2{p}_4$ and locate the midpoint $m_{24}=(p_2+p_4)/2$ . 3) Construct the line $m_{15}{m}_{24}$ between these midpoints and locate the point which is $-a/(1-2a)$ of the distance from the $m_{15}$ towards $m_{24}$ . Call this point $q$ . 4) Construct the line from $q$ to $p_3$ and locate the point which is $(\phi -a)/(2+\phi -5a)$ of the distance from $q$ towards $p_3$ . This last point is the desired point ${\nu }_1$ . Repeating this in a symmetric manner for the other four points will construct points ${\nu }_2,\dots ,{\nu }_5$ . By connecting these in order, we obtain an affine regular stellar pentagon. Note that by letting $a=0$ we get the stellar construction first created by Jesse Douglas in [2]. To create the following sculptures, we repeat the above process for different values of $a$ . Notice that steps 1), 2) and half of 3) do not need to be repeated for different values of $a$ . Figures 4 and 5 consist of computer generated sculptures starting from the pentagon in Figure 2. Figure 6 consist of both the sculpture in Figure 4 and the sculpture in Figure 5.

Figure 2: Non-planar pentagon

Figure 3: Construction $a=-0.1$

We finish with several observations. As the line in Figure 1 moves closer to the middle, $(2+\phi -5a)$ gets smaller causing the resulting pentagons to get larger. This explains why values between 0.3 and 1.0 were ignored. As can be seen, the affine regular pentagons all lie upon the same plane, however, none of the lines constructed, other than the pentagons, lie on this plane. One wonders, what makes this plane special?

Figure 4: $1.0\le a\le 1.4$ Figure 5: $-0.1\le a\le 0.3$ Figure 6: Combined Sculpture

References

[1] Douglas Burkholder, “Parallelogons and Weighted Averages,” Revista de Física y Matemática (FISMAT) de la Facultad de Ciencias, Escuela Politécnica Nacional (Quito-Ecuador), XII (2004). [2] Jesse Douglas, A Theorem on Skew Pentagons, Scripta Mathematica, Vol. 25, 1960.