Updates to Chapter 9, "Watch Your Step!",
in Dissections: Plane & Fancy, by Greg N. Frederickson

Showing that PP-minus is contained in Pythagoras-extended

I should have given more explanation for the last sentence of the first paragraph on page 93: The membership of PP-minus in Pythagoras-extended isn't obvious. To show this, recall that PP-minus has m greater than 1, n=1, p=1, and q=m+2. Thus x, y, z, and w are respectively m+q, mq-1, mq+1, and 2. Double each of the x, y, z, and w values, and interchange x and y. Then determine the corresponding new values of m, n, p, and q. It follows that m'=m+1, n'=m-1, p'=q-1, and q'=q+1. Thus the new value of m is 2 more than the new value of n.

Plato-extended versus Pythagoras-extended

Actually there is a more serious problem dealing with the Plato-extended class, namely that it is contained in the Pythagoras-extended class! Consider any instance of the Plato-extended class with n=1. I claim that the same solution, but with x and y interchanged, is generated by m'=m+1, n'=m-1, p'=q-p, and q'=q+p. This is included in Pythagoras-extended, since m'=n'+2. So either I should not have defined the Plato-extended class at all. Or perhaps, more reasonably, I should have defined the Pythagoras-extended class to include only instances for which m=n+1.

More information on James Schmerl

James H. Schmerl was born in 1940 in Storrs, Connecticut. He earned an A.B. and a Ph.D. in mathematics at the University of California, Berkeley, in 1962 and 1970, respectively.

An earlier dissection of Greek crosses

Figure 9.24 presents Robert Reid's 7-piece dissection of Greek crosses for 32 + 42 = 52. Unbeknownst to Robert and me, the problem of dissecting Greek crosses for 32 + 42 = 52 had earlier been posed by L. P. Mochalov, who gave a 9-piece dissection in Vladimir Belov's 1992 book. (See the updates to chapter 1.)


A typo on page 94: On the last line, I should have m = 4.
(This is corrected in the paperback edition.)
Two typos on page 95: In Method 9, the last line in step 2 should have the word "odd" rather than the word "even". In step 3, the w should be replaced by mp - nq. If this quantity is negative, then the move starts in the space to the left of the step piece, with the cut beginning when the piece is reached.
(These are corrected in the paperback edition.)
In correcting these last two typos in the paperback edition, the compositor introduced two more: The second line of step 2 of Method 9 should read
[ leftfloor (m-2)/2 rightfloor times ]:
Obviously, the symbols for leftfloor and rightfloor didn't print properly.

Zigging or zagging?

Koji Miyazaki, Hirohisa Hioki, and Naoki Odaka pointed out that there was a problem with Method 13 on page 104. The simplest way to correct it seems to be to replace the body of the repeat loop with:
Wobbly step starting at the left vertex:
[n times]:
{[n times]: {Move/cut dn-right 1; Move/cut right 1.} [n+1 times]: {Move/cut up-right 1; Move/cut right 1.}}
Mark the position 1 to the left of the current position.
[n times]: {Move/cut dn-right 1; Move/cut right 1.}
Move/cut from the last mark to the lower left vertex.
Rotate 120 degrees clockwise around the center.

Tri-root class extended to octagons, decagons, etc.

A surprising feature of the Tri-root class of solutions for 7-piece dissections of hexagons is that the class can be generalized to handle any regular polygon that has an even number of sides, with the exception of the square. In September 1998 I discovered (2m+1)-piece dissections of {2m}s for x = 1 and y = 2n sin (pi / m) for every whole number n. Two weeks later I extended this result to regular polygons with an odd number of sides, with the exception of the triangle. The dissection for a {2m+1} uses 2m+2 pieces. These are described in a manuscript that I am preparing. I have found a similar generalization for stars, which I mention in my updates to Chapter 18, in Star class extended.

Copyright 1997-2003, Greg N. Frederickson.
Permission is granted to any purchaser of Dissections: Plane & Fancy to print out a copy of this page for his or her own personal use.

Last updated October 4, 2003.