Animations of dissections influenced by Hugo Hadwiger

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

"Hugo Hadwiger's influence on geometric dissections with special properties," by Greg N. Frederickson,
Elemente der Mathematik, volume 65, number 4 (2010), pp. 154-164.

Swing-hinged dissections

Let's start with the swing-hinged dissection of an equilateral triangle to a square, illustrated in Figure 5 of the article. This is the version that Henry E. Dudeney described in his book, The Canterbury Puzzles. I would have loved to see Dudeney demonstrate his model in polished mahogany with brass hinges!

A simple animation of a swing-hinged dissection of an equilateral triangle to a square.

Twist-hinged dissections

Let's now view the twist-hinged dissection of an equilateral triangle to a square, illustrated in Figures 8 and 9 of the article. All three triangles in the dissection are right triangles.

A simple animation of an equilateral triangle to a square, in which I twist each hinge one at a time.

A more interesting animation of that same twist-hinged dissection of an equilateral triangle to a square, in which I twist a pair of hinges simultaneously. Each such pair of twist hinges corresponds to a swing hinge, so that this animation more closely simulates an animation of the original swing-hinged dissection.

Next up is the symmetrical dissection of a regular hexagon to an equilateral triangle, illustrated in Figure 10 of the article. An analogous dissection exists for every pair of regular p- and 2p-polygons, and it possesses p-fold rotational symmetry!

A simple animation of a twist-hinged dissection of a regular hexagon to an equilateral triangle. Sorry it's so quick, but it seems appropriate to emphasize the 3-fold rotational symmetry by performing isomorphic motions in parallel.

Now let's have some fun with the twist-hinged dissection of a regular heptagon to a square, presented in Figure 11 of the article.

A simple animation of a regular heptagon to a square. The two right triangles result from converting swing hinges to twist hinges. Decompose each right triangle into its two constituent isosceles triangles and glue them onto the adjacent pieces to form the original hinge-snug pieces. Identify the isosceles trapezoid and verify how it repositions the three pieces to which it is hinged.

Stackfolding dissections

Here is a stackfolding dissection of a 1-level square to an 8-level square, as illustrated in Figure 15 of the article.

A simple animation of a stackfolding dissection of a 1-level square to an 8-level square.

Here is a stackfolding dissection of a 4-level triangle to a 3-level triangle, as illustrated in Figure 16 of the article.

A simple animation of a stackfolding dissection of a 4-level triangle to a 3-level triangle. Watch for the cap-cyclic hinging, which involves both of the double-thickness pieces.