Animations of an alternative dissection of a triangle to a square
The following animations, by Greg Frederickson,
are of a dissection of an equilateral triangle to a square that is an alternative to those 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.
After I won a
Pólya Award for my article,
I pointed people who had trouble visualizing the hinged dissection to
the webpage with my animations.
That caused me to think once again about the dissection.
I wondered about the placement of the hinges and the `little hooks',
noting that the placement was not particularly symmetrical.
Contrast that with the underlying 6-piece dissection in Figure 5,
which is topologically symmetrical, in the sense that
the adjacencies of the pieces are symmetric.
Of course, the introduction of piece 7 breaks that symmetry.
Now suppose that we broke the symmetry the `other way',
by carving piece 7 out of the righthand side of piece 2?
Would we still obtain a swing-hingeable dissection?
That would be neat if it were to work out.
And indeed, it does!
In fact, we can hinge the pieces in the opposite way, topologically speaking, to
the hinging in Figure 7.
Moreover, when we consider adding `little hooks'
to lock the pieces together,
we can introduce once again just three hooks for the alternative dissection.
In this new alternative,
these little hooks will be oppositely placed, topologically speaking,
with respect to the position of the hooks in Figure 9.
We see illustrations for the new dissection,
namely the analogues of Figures 5, 6, and 9,
in this accompanying
set of figures.
What a lovely happenstance! And all the more so because in the one
year between the publication of the article and the presentation of the award,
no one, including myself, had thought to check for such a `mirror image' dissection.
Let's enjoy the new swing-hinged dissection,
which is the alternative to the dissection illustrated in Figures 6 and 7:
of the new swing-hinged dissection of an equilateral triangle to a square.
This version is just as suitable for the top of a pedestal table.
Let's also examine the `little hooks' that can be used in the our new dissection
that is the alternative to what is
illustrated in Figure 9:
of the swing-hinged dissection, with hooks, of the alternative equilateral triangle to a square.
Pretty darn snappy!
Finally, let's have a side-by-side comparison of the two dissections:
of both swing-hinged dissections, with the one from the article on the left,
and the new alternative one on the right.
Now all we need is a real carpenter to craft a pair of actual tables based on the dissections!
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 2009 by Greg Frederickson
and may not be copied, electronically or otherwise,
without his express written permission.
Last updated September 11, 2009.