Animations of swing-hinged dissections of squares

The following animations, by Greg Frederickson, are of swing-hinged dissections of squares that appear in the article:

"Polishing some Visual Gems," by Greg N. Frederickson,
Math Horizons, September 2009, pp. 21-25.

We start with the so-called "Pythagorean runs" of Michael Boardman.
They derive from the following family of identities where n = 1, 2, ... .

(x – n)² + ... + (x – 1)² + x² = (x + 1)² + ... + (x + n)²

The first identity in this family is

3² + 4² = 5²

for which there is already a 4-piece hinged dissection, found in Figure 7.1 of my book Hinged Dissections.
Here is the animation for my dissection, which appears in Figure 2 of the article.

The second identity in the family is

10² + 11² +12² = 13² + 14²

for which there is an 8-piece hinged dissection given in Figures 3 and 4. Here is its animation.

I also have two animations for the second family of identities, where n = 1, 2, ... .

(x – n)² + ... + (x – 1)² + x² + x² = (x + 1)² + ... + (x + n)²

The first identity in this family is

1² + 2² +2² = 3²

for which Sam Loyd knew a 4-piece hingeable dissection, although he was not aware that it was hingeable.
I showed the hinged dissection in Figure 5, and now give its animation.

The second identity in the family is

4² + 5² +6² +6² = 7² +8²

for which I gave an 8-piece hinged dissection in Figure 7. Here we see its animation.

For additional background material on hinged dissections, see:

Hinged Dissections: Swinging & Twisting, by Greg N. Frederickson,
Cambridge University Press, 2002.