I have found a considerably earlier reference for the 4-piece dissection of a parallelogram to a parallelogram in Figure 11.6. It was given by Adolph Göpel in:"Ueber Theilung und Verwandlung einiger ebenen Figuren," Archiv der Mathematik und Physik, vol. 4 (1844), pages 237-239 and Tafel IV.Gustav Adolph Göpel was born in Rostock, Germany in 1812. He was awarded a Ph.D. at Berlin in 1835. He was a teacher at the Werderschen Gymnasium and at the königliche Realschule in Berlin. Finally, he was a civil servant with the königliche Bibliothek (Royal Library). He published two papers in Crelle's Journal and seven more in Archiv der Mathematik und Physik. Adolph Göpel died in 1847.
I was sloppy in my classification of the dissection of a parallelogram to a parallelogram in Figure 11.6. Gavin Theobald alertly pointed out:I've just spotted a mistake in your second book! See figure 11.4. You describe this as a TT1 crossposition. It's not! It's a PP crossposition. If it were a TT crossposition then it would work for trapeziums, but if you think about it, it doesn't work - you've only dissected one half of each trapezium. (In diagram 11.4, if this is a TT1 dissection, then you have dissected one end of each parellogram twice). No, this is, as far as I know, a completely new form of TT dissection that has not been adequately described before. It's a form of TT2 dissection since the area of overlap is twice the area of each shape, but it's not the same since two crosspositions are required.Thanks, Gavin!
There is an extra hinge in Figure 11.31. It's easy to notice, because there is only one piece adjacent to the hinge. Deleting that extra hinge gives a correct description.
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Last updated November 7, 2005.