## Thoughts on animations of swing-hinged dissections of a triangle to a square

On the webpage Animations of swing-hinged dissections of a triangle to a square I presented several animations of swing-hinged dissections of an equilateral triangle to a square that appear in the article:
"Designing a Table Both Swinging and Stable", by Greg N. Frederickson,
College Mathematics Journal, volume 39, number 4 (September 2008), pages 258-266.
Those of you who are interested in animations may wonder why I "chopped up" my second and third animations. Why didn't I start rotating all of the pieces at the same time, and stop rotating them all at the same time? There are two reasons. First, I wanted people to be able to focus on various aspects of the motion one at a time, rather than completely overwhelming them. Second, and more importantly, if all pieces moved simultaneously, then the motion wouldn't be correct for a physical object, since some of the pieces would interpenetrate at times during the motion. Here's the explanation why:
The problem is a result of having all of the pieces complete their rotations during precisely the same interval of time. This means that N5 and N7 will rotate twice as fast as N1, N3, N4, and N6, since they must rotate through 360° rather than 180°, because each has two hinges separating them from the immovable piece N2.
Let's see how the overlap comes about. Consider the square in Figure 6. For simplicity in the description, rotate the whole square 30° counterclockwise, so that the edge of piece N4 that touches N5 and N2 is vertical. Let R be the length of the edge of N4 that is now vertical, and let r be the length of the edge of N5 that is now vertical. Note that r is clearly greater than R/2. Now rotate pieces N1, N3, N4, and N6 by θ, and at the same time rotate N5 and N7 by 2θ.
Let's describe the position of the apex of N5 for θ in the range 0 ≤ θ ≤ 45°: The x-value of the apex will change by –R sin(θ) + r sin(2θ), and its y-value will change by RR cos(θ) – r + r cos(2θ). Taking derivatives of each value as a function of θ, we get
dx/dθ = –R cos(θ) + 2r cos(2θ)
dy/dθ = R sin(θ) – 2r sin(2θ)
Taking the limit as θ decreases to zero, we get
lim{θ→0} dx/dθ = –R + 2r
lim{θ→0} dy/dθ = 0
Since r is greater than R/2, we see that the apex of N5 starts off moving horizontally to the right, with it slowly and continuously then shifting direction down and to the right. Unfortunately, immediately after the rotation of pieces begins, N5 will collide with N2, because the angle between their adjacent edges in the neighborhood of the apex is 30° below the horizontal, and there cannot be an abrupt shift in direction of 30° as θ continuously increases.
For additional background material on hinged dissections, see:
Hinged Dissections: Swinging & Twisting, by Greg N. Frederickson,
Cambridge University Press, 2002.

Text is copyright 2008 by Greg Frederickson
and may not be copied, electronically or otherwise,
without his express written permission.

Last updated September 1, 2008.