## 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} + *x*^{2} = (*x* + 1)^{2} + ... + (*x* + *n*)^{2}

The first identity in this family is
3^{2} + 4^{2} = 5^{2}

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
10^{2} + 11^{2} +12^{2} = 13^{2} + 14^{2}

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} + *x*^{2} + *x*^{2} = (*x* + 1)^{2} + ... + (*x* + *n*)^{2}

The first identity in this family is
1^{2} + 2^{2} +2^{2} = 3^{2}

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
4^{2} + 5^{2} +6^{2} +6^{2} = 7^{2} +8^{2}

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.

Text and animations are copyright 2008 by Greg Frederickson

and may not be copied, electronically or otherwise,

without his express written permission.

*Last updated September 10, 2009.*