A Parade of Algorithmic Mathematical Art, by Greg N. Frederickson
A geometric dissection is a cutting of a geometric figure
into pieces that we can rearrange to form another figure.
Such visual demonstrations of the equivalence of area
(or volume) span from the ancient Greeks
through current-day postings on the world-wide web.
During the past century,
the emphasis has been on minimizing the number of
pieces for any given dissection, which resulted in
some remarkably beautiful mathematical creations.
That was the subject of my first book (in 1997).
As dissection methods have become more sophisticated,
attention has also focused on special properties,
such as having all the pieces of a dissection be connected by hinges.
Swing hinges allow the pieces to swing from one figure
to the other figure.
My second book (2002) surveys swing-hinged dissections,
and also introduces twist-hinged dissections.
A twist hinge allows one piece to be twisted with respect to another.
My third book (2006) explores hinges
that allow pieces to be folded one on top of the other.
When giving talks based on my second book,
I found it useful to demonstrate actual physical models.
I included in my third book a CD containing video clips
in which I demonstrated various folding models.
That stimulated me to produce computer-generated animations
for talks on discoveries made since my third book.
Here you will see a parade of animations and video clips
that I produced over the last five years.
Just as Alexander Calder had his circus, I have my parade!
What makes this art?
We see the application of elegant dissection methods
to symmetrical figures such as regular polygons and stars.
We also see the design of objects such as a table whose top
swings from a triangle to a square, and garden benches that
twist to ring around a tree.
We see individual motions smoothly executed
to reveal symmetry and other lovely properties.
I still cannot resist
watching these animations over and over again.
What makes this algorithmic?
Algorithms can be seen both in the creation of smooth motion
and in the use of dissection methods on certain
infinite families of figures.
While perhaps more hidden, hints of the algorithms are detectable
if you pay close attention.
So relax and enjoy the mesmerizing motion!
Sources (all by Greg N. Frederickson)
Articles in journals
- Unexpected twists in geometric dissections,
Graphs and Combinatorics, 23[Suppl] (2007), pp. 245-258.
- The heptagon to the square, and other wild twists,
Mathematical Intelligencer 29,
4 (2007), pp. 23-33.
- Designing a table both swinging and stable,
College Mathematics Journal 39,
4 (September 2008), pp. 258-266.
— won the 2009 MAA George Pólya Award.
- Polishing some visual gems,
Math Horizons , September 2009, pp. 21-25.
-
Casting light on cube dissections,
Mathematics Magazine 82, 5 (December 2009), pp. 323-331.
- Hugo Hadwiger's influence on geometric dissections with special properties,
Elemente der Mathematik 65, 4 (2010), pp. 154-164.
- Folding polyominoes from one level to two,
College Mathematics Journal 42, 4 (September 2011), pp. 265-274.
Papers presented in conferences and workshops
- Symmetry and structure in twist-hinged dissections of polygonal rings
and polygonal anti-rings,
Proc. Bridges Donostia: Mathematics, Music,
Art, Architecture, Culture,
San Sebastian, Spain (July 2007), pp. 21-28.
- Dissecting and folding stacked geometric figures,
DIMACS Workshop on Algorithmic Mathematical Art: Special Cases and Their Applications,
Rutgers University, Piscataway, NJ
(May 11-13, 2009).
Books
- Dissections: Plane & Fancy,
Cambridge University Press,
1997.
- Hinged Dissections: Swinging and Twisting,
Cambridge University Press,
2002.
- Piano-hinged Dissections: Time to Fold!,
A K Peters Ltd,
2006.