# Ernest Irving Freese's Geometric Transformations

In its simplest form, a geometric dissection refers to subdividing a polygon into a finite number of pieces and to reassemble them to form another polygon. It is proved that this is always possible, but the challenge is then to obtain it with a minimal number of pieces. Proving the minimality for an orthogonal polygon is shown to be NP-hard. If every piece can pivot around a vertex that stays connected to a neighbouring piece during the transformation, it is called a hinged dissection. When unfolded, it forms a string of connected pieces. Piano-hinged means that the connection is not at the vertices, but that edges are connected so that this corresponds to folding a paper version, but these are not considered in the present book.

Frederickson starts this book with a short history of dissections. They were first studied in the 19th century by some mathematicians a.o. Farkas Bolyai, but they became popular in the 20th century when they featured in newspaper puzzle columns by Sam Lloyd and by H. Dudeney, (who collected them also in their collected puzzle books) and later M. Gardner. Harry Lindgren provided a way to derive dissections by overlapping tessellations of the plane around the middle of the 1900s. That is also the time that Freese was preparing his work on the topic.

In his book Dissections: Plane & Fancy (1997) Frederickson mentions some 'over 200 plates' prepared by Freese of which a few loose copies were circulation, but the whole manuscript was not located. In his second book Hinged Dissections: Swinging & Twisting (2002), Freese has disappeared from the reference list, but in Piano-hinged Dissections: Time to Fold! (2006), Freese's work is very prominently present in the form of appendices to the chapters written by Fredericson. Some of the plates are reproduced and Frederickson provides comments. What has happened? Frederickson gives an explanation, which is actually a very remarkable story. That story is retold in this book. Freese, who obviously knew about these dissection problems from the puzzlers columns and from puzzle books that he got as a present from his wife. He had collected his ideas on dissections in the form of 200 plates that he finished a couple of months before he died. He wrote in 1957 to a friend that he had been intensively busy preparing them, that a blueprint could be obtained for $28.00, but that probably nobody would be interested in publishing his drawings. After his death several people tried to obtain the manuscript from his wife Winifred but she had written a letter to Ginsburg (a friend of Freese and editor of a journal) asking him to take care of the manuscript. However Ginsburg had died three weeks before her husband, so he never answered and the manuscript was forgotten. Frederickson obtained Winifred's address only after she passed away, but her son Bill was living in the house now. Frederickson wrote him a letter but the manuscript was not found. After Bill died, a cousin found the letter and the manuscript so that Frederickson finally could make a copy of Freese's plates in 2003. That explains why Freese features in the appendices of his 2006 book.

Because Frederickson also got hold of some other material, he can add a biography of Ernest Irving Freese (1886-1957) as the second chapter of this book. A very wild adventurous life this Los Angeles architect has lived. Mostly self-taught, he first worked for an architectural firm, later as an independent architect, but he also travels the country as a tramp, and later goes on a bicycle world tour, working when he needs money or as a crew member on a ship to pay for his fare. He published articles in cycling magazines and in architectural and construction journals. After an earthquake in 1933 he started a campaign to construct safe schools (he was by then father of three). He was an assertive man with strong opinions.

The main purpose of this book is to finally publish the manuscript by Freese. It was originally not conceived as a commercial product, so it is a notebook that consists of loose hand-made geometric constructions with little text in Freese's elegant slanted handwriting. Frederickson has kept the order of the numbering of the places, subdivided them into chapters and provided an introductory text and explaining notes, references, and new results per chapter (Freese has no references) and this text is followed by the relevant plates. So, while in his 2006 book Frederickson used Freese's results as an appendix to his own work, here it is the other way around, it is Freese's work with Frederickson commenting. The plates are beautifully reproduced after being digitally processed to remove stains. Their original size is 8.5 x 11 inch (21.6 x 27.9 cm) which is also the somewhat unusual format of this book. Moreover to keep the originals intact and in the state that Freese has created them, these pages get no headers or page numbers, they only have the original encircled plate numbers.

Freese had divided the plates into sections corresponding to the geometry of the objects. The subdivision into smaller chapters is a decision of Frederickson. The idea is that chapters will group transformations that are somewhat similar so that a common introduction is possible, although Frederickson is also commenting on all the separate plates within each chapter. The chapters are then about transforming isoscele or equilateral triangles, followed by squares, crosses, rectangles, and n-polygons and n-polygrams up to n = 12 and they conclude with some unclassified miscellaneous figures. All the dissections are two-dimensional, so no 3D generalizations. It is made clear with references that some of the dissections were found later by others, when Freese's work was unavailable. Everyone interested in geometric dissections, and this kind of puzzles, either mathematically or recreationally will embrace this publication. But also the readers interested in the history and certainly those who became curious about this mystery man and his manuscript, after reading Frederickson's 2006 book, will be fully satisfied with this respectful reproduction eventually made available for a general public.

**Submitted by Adhemar Bultheel |

**10 / May / 2018