in Dissections: Plane & Fancy, by Greg N. Frederickson

- Nickname for Geoffrey Thomas Bennett
- An earlier octagon-to-square dissection
- An earlier attribution for the octagon-to-square dissection
- More on Ernest I. Freese
- A pretty dissection by Wolfgang Stöcher
- Hexagram to hexagon with no turning over
- Bruce R. Gilson
- Completing the square applied to the {12/2}
- Completing the square applied to {18} to {6}

Robert Macmillan wrote that when he went down to Emmanuel College in 1941, he came to know G.T. Bennett for a short while before the latter's death. Bennett was popularly known then as "Beaver" Bennett, because he had a formidable white beard in his old age. Macmillan explains that in Britain in the 1930's a beard was called a "beaver", although he does not know why. Perhaps this usage comes from an alternate meaning for beaver, deriving from the Middle English "baviere", which came from Middle French, which was a piece of armor protecting the lower part of the face.

When Henry Dudeney posed the problem of dissecting an octagon to a square, in his "Perplexities" column in theStrand Magazine,volume 71 (1926), page 416, he stated that he believed that no such dissection had ever been published. But the problem of dissecting an octagon to a square in as few pieces as possible was posed earlier by James Blaikie inMathematical Questions and Solutions from "The Educational Times."In response to his challenge, Henry Martin Taylor gave an 8-piece dissection in that periodical in volume 16 of the second series (1909), pages 81-82. Thus Henry Dudeney's 7-piece dissection, which he gave in theStrand Magazine,volume 71 (1926), page 522, was an improvement of one piece over the earlier dissection. Both dissections converted the octagon to a rectangle and then performed a P-slide. Dudeney's conversion avoided one additional crossing of cuts.

The lovely 5-piece dissection of an octagon to a square appeared much earlier, in an anonymous Persian manuscript,Interlocks of Similar or Complementary Figures, from approximately 1300 C.E. A more complete discussion of this is included in my second book,Hinged Dissections: Swinging & Twisting.

Harry Lindgren wrote in a 1963 letter to C. Dudley Langford that Ernest I. Freese was a partner in the architectural firm Boyle, Meldt & Freese. The firm is almost certainly bogus, its name exploiting a pun on the name Freese. So who was responsible for this bit of fun? Take another look at Freese's biography and see if you can guess!

Also, check out Ernest Irving Freese's photograph and "autobiography" (approx 46K). FromPencil Points,vol. 11 (1930) page 224.

An earlier example of his humor appeared in "A Bungle-Ode," by Ern. Freese, inThe Architect and Engineer of California,March 1918.

Freese's 5-part series on perspective projection was also published as a 43-page book,Perspective Projection; a Simple and Exact Method of Making Perspective Drawings,Pencil Points Press, New York, 1930.

Using the completing the tessellation technique, Wolfgang Stöcher has designed a very pretty dissection of a regular hexagon to an equilateral triangle. Since it has 7 pieces, it is not minimal. But it has 3-fold rotational symmetry and thus is really attractive. Of especial interest, the two small triangles that are added each have sidelength one half of that of the large equilateral triangle. I know of no other example of completing the tessellation for which the side of an added polygon is one half the side of one of the given polygons.

In March 2007, Gavin Theobald discovered a way to modify my 6-piece hexagram-to-hexagon dissection so that no pieces need be turned over. He enlarged the two triangles that are turned over so that each resulting piece has mirror symmetry and thus need not be turned over. He did this by introducing curved cuts, so that one piece becomes a "pie slice" and the other is the portion of a circle cut off by a chord. The circular cuts have a radius equal to the length of the edge of the hexagram.The "chord slice" is also an improvement in that it is a considerable enlargement of the tiniest piece, which had been somewhat of a blemish on the dissection. As far as I know, this is the first essential use of circular cuts in a dissection whose figures do not have curved boundaries. Really nifty!

Martin Gardner wrote a column on geometric dissections inScientific Americanin 1962, and gave a table that listed the number of pieces that suffice for various dissections. Noting that there was no entry for the hexagram to hexagon, then-20-year-old Bruce R. Gilson found the 7-piece dissection that I mention on page 154. After seeing my books, Bruce got in touch with me, and I found out what he had been doing during the intervening four decades:

Bruce R. Gilson was born in 1942 in the Bronx, in New York City. He earned B.S., M.S., and Ph.D. degrees in chemistry from City College (1962), Yale University (1964), and the University of Virginia (1969). Since then, he has worked primarily in the computer programming field. Bruce has maintained interests in natural and constructed languages, paper money from foreign countries, public transportation, unusual music scales, and music from his boyhood years. Bruce now lives in the Washington, DC area.

Gavin Theobald found a lovely 9-piece dissection of a {12/2} to a square, using the completing the tessellation technique. Although it has one more piece than another dissection that Gavin found (on page 135), this one turns no pieces over. However, Gavin then went on to find an 8-piece dissection with no pieces turned over! Gavin illustrates the new dissection on his dodecagram webpage.

In March 2007, Gavin Theobald found a lovely 12-piece dissection of an {18} to a hexagon, using the completing the tessellation technique. The dissection is based on Wolfgang Stöcher's technique which he used to dissect any {3n} to an {n}. (See Chapter 3 of Piano-Hinged Dissections: Time to Fold!) Gavin's dissection involves a considerable reduction in the number of pieces from Stöcher's for the particular case ofn = 6, and furthermore turns no pieces over.

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