For whatever it's worth, there are just two different shapes for all the pieces!
In the first paragraph of page 162, the second sentence starts "According to Langford (1967b)..." The reference should be to Langford (1967a) .
Actually, we can check this directly in my fourth book (Ernest Irving Freese's Geometric Transformations). The dissection that Langford referred to is in Plate 118.
In Ernest Freese's diagrams (from the 1950's), which I got access to late in 2002, there is a diagram of the same 12-piece dissection of three hexagrams to one that Harry Lindgren published in 1964. Thus the right two thirds of Figure 17.11 were discovered independently by Ernest Freese and by Harry Lindgren. Apparently, neither man realized that the dissection is hingeable.
In Figure 17.9, I showed a 19-piece hinged dissection of two {12/4}s to one. In November 2019 I designed an 18-piece hinged dissection for that problem. It's interesting because each small star is identically dissected and hinged. (Note that I show the conversion of just one of the two small {12/4}s to half of the larger one, because the other is identical!) It took quite some patience on my part to produce the diagram below, so I hope that readers will bring their own fine measure of patience to bear while working through it. (!)
For whatever it's worth, there are just two different shapes for all the pieces!
Copyright 2003-2019, Greg N. Frederickson.
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Hinged Dissections: Swinging & Twisting
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Last updated November 12, 2019.