If a regular polygon has an odd number of sides, then its internal structure cannot be described in terms of rhombuses, and we must be satisfied by describing its structure in terms of isosceles triangles, as in Figure 2.2. For instance, the structure of the regular heptagon consists of five isosceles triangles with two angles of 180/7 degrees, three isosceles triangles with two angles of 360/7 degrees, and one isosceles triangle with two angles of 540/7 degrees. A diagram reveals this structure on page 142 in volume 73, number 3 (2018) of Elemente der Mathematik. Sorry that the accompanying text on pages 142 and 143 are in German!)
I did not include the name of the 11-sided polygon. John Conway and Antreas Hatzipolakis worked out a complete system of naming polygons up to the millions. The webpage explaining it has vanished, but here is another list. (I don't know how long this particular webpage will be available.) An 11-sided polygon is called a hendecagon. Now that we know what to call it, we should be ready to dissect it. Shall we start with the two mentioned at the end of Chapter 17: 11 hendecagons to one, and 3 hendecagons to one? As it turned out, it was a hendecagon to a square.
In August 2011, Kevin Jardine, a software developer based in Leiden, the Netherlands, alerted me to a very general tiling result that he has obtained:Decomposition theorem:Check out his remarkable webpages:
If a regular polygon has j = mn sides where j is even with m > 1 and n > 2, then it can be decomposed into rhombs and:
0,1,2,3,...,m n-sided regular polygons if n is even or
0,2,4,...,m n-sided regular polygons if n is odd.
If a regular polygon has j = mn sides where j is odd with m > 1 and n > 2, then it can be decomposed into rhombs and m n-sided regular polygons.http://gruze.org/tilings/universal
http://gruze.org/tilings/decomposition
http://gruze.org/tilings/decomposition_odd
Knowledge of the structure of several stars reaches back at least to 79 C.E., when the Roman city of Pompeii was buried by a volcanic eruption. Paul Calter displays a photograph (Slide 5-23) of a design at Pompeii based on a decomposition of the {12/3} into a central hexagon surrounded by six squares, six 60-degree rhombuses, and six equilateral triangles. His color photograph can be seen in Geometry in Art & Architecture Unit 5.
A second photograph by Calter (Slide 7-20) in Geometry in Art & Architecture Unit 7 is even more impressive. It is of a repeating floor pattern at Pompeii that illustrates the structure for not only an {8/3} but also an {8/2}.
In writing my book I ignored one small point: When one makes a cut, how does one describe the resulting two edges precisely? Does one of the resulting pieces get the points on the cut line, making the other piece look like an open set along the corresponding edge?
At the time, it seemed appropriate to stonewall such nitpicking for a book on math recreations. But the issue has popped up in other contexts, so let's try to find a reasonable model. First, consider physical reality: If you cut a piece of wood, the portion of the wood along the cut line gets "consumed" (converted to sawdust). Even using a precision cutting device such as a laser will consume some portion of material. Although I haven't looked under the microscope at what happens to paper when it's cut by scissors, I suspect that the cut line also gets consumed.
So here is a (mathematical) model for cutting: A figure (a polygon, a star, etc.) is assumed to be an open set. That means that its boundary is not part of the figure itself. When the figure is cut along a sequence of line segments, all points on the segments are removed, resulting in pieces that are open sets. When two pieces are assembled, they are "glued together" by adding their common boundary, minus the endpoints of the common boundary, to the assemblage as "glue". The assembled figure is thus an open set too.
Copyright 1997-2018, Greg N. Frederickson.
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Last updated October 25, 2018.