Douglas G. Burkholder
"Pentagons in Motion" (first view)
Sculpture in Wood and Brass, 20" x 16" x 12", 2006.
"Pentagons in Motion" (second view)
Douglas G. Burkholder
Associate Professor of Mathematics, Lenoir-Rhyne College, Hickory, NC 28603
" If the five vertices of any irregular non-planar pentagon each slide towards the midpoint of their respective adjacent secant line at speeds proportional to the length of these line segments then at a certain moment in time these five points will flatten out to form a planar affine regular pentagon. As these points continue sliding they fold themselves up and then flatten out once again to form a stellar planar affine regular pentagon. Affine regular pentagons appear regular when viewed from a certain direction.
This sculpture consist of 14 nested pentagons which capture this sense of motion allowing the observer to see these pentagons flattening, folding, and flattening again forming these planar affine regular pentagons. The outermost pentagon is the original arbitrary non-planar pentagon. The original pentagon and the two affine regular planar pentagons are stained dark brown. The observer is challenged to find the viewpoint from which the stellar affine regular pentagon appears to be perfectly regular, and also to find the viewpoint from which the non-stellar affine regular pentagon appears to be perfectly regular.
Details of this, and other constructions, can be found in "Symmetric Linear Constructions in Motion", Bridges Conference Proceedings, pp. 403-410, 2005. "
BurkholderD@LRC.edu
http://www.lrc.edu/mat/burkholder/