# Rgdboer

Joined 11 August 2004

When a man writes to the world, he summons up all his reasons and deliberations to assist him; he searches, meditates, is industrious, and likely consults with his judicious friends; after all which done he takes himself to be informed in what he writes as well as any that writ before him.

My great-grandparents lived in the land of Herman Boerhaave and Abel Tasman, most of my grandparents were born there. When I was a boy I took an interest in coin collecting, like the Lincoln cent and the Canadian penny that circulated in Michigan. In a local library I read about meteorology, a perpetual spur to understanding. A schoolmate showed me how chess was played; it captured my attention. As an adolescent I read Hans Kmoch's Pawn Power in Chess. The YMCA offered swimming lessons in its pool, a summer camp, and a room for weekly meetings of the city chess club.

I was amazed that my crystal radio worked without batteries -- the power to drive the earphone was "in the air". At that time there was no internet, but shortwave communications offered a window on the world. I studied Morse code, took up amateur radio practice. Radio Amateur's Handbook served as a textbook. The League magazine QST discussed impedance matching a transmitter to an antenna, opening up the frequency domain. Though I was fascinated with electronics, reading Mathematics and the Imagination exposed me to greater abstraction.

I put some of my physical energy into learning to ride a unicycle. In the final year of junior high school there was a science fair in which I competed with a problem in combinatorics. The creative experience of innovative mathematics was rewarding enough without award. (Such creativity or originality is prohibited in this encyclopedia: Some of my original work has been archived by Wayback Machine: Complementary numerals, Corner flow and Common ground, and Adventures in 9-space.)

After junior high school I started studying through the summers, taking a courses in speech and typewriteing the first summer. I participated in policy debates as a high school student, and learned to use the Readers' Guide to Periodical Literature to find articles on topic. Construction and usage of the Interstate Highway System gave a continuous sense of national progress. The NSF sponsored science and mathematics programs for high school juniors. Students learned a little calculus and a bit of physics. I wrote a report on electroluminescence based on readings at the Carnegie library downtown. We also spent a few weeks on college campuses with classes presented by professors. Each of my parents had a cousin in academia, so that was my career goal. Advisors told me to get three academic degrees and to learn to read French, German and Russian mathematics.

## University

First year sciences were physics and psychology. Later there were courses in economics and astronomy. Another course introduced programming in Fortran and SNOBOL using punched cards. The University of Michigan had some professors teaching out of their own texts: Raymond Wilder on foundations of mathematics and Irving Copi on logic. Studying through the summmers, soon I graduated with a degree. On the basis of a very good grade point average and faculty recommendations University of California at Berkeley admitted me for graduate study in mathematics. Alfred Tarski was my advisor and courses included measure theory with John L. Kelley. But graduate students were not deferred from the Vietnam War so I applied for recognition as a conscientious objector but was turned down. It seemed that all the learning was going for nought while the Selective Service System provided young men for induction into the Army. In California I saw for the first time that regular consumption of animal parts (carnivorism) was unnecessary for human vitality. On Sproul Plaza many groups promoting civil rights had set up card tables. Some of them explained the process of "refusing induction" available for young men standing their ground against conscription. After the usual physical examination, at the moment for entrance into the army, one simply stayed back while willing recruits stepped forward. Neither the Army nor the SSS enforced conscription; that was left to the Department of Justice. I refused induction in Oakland, California. When Eugene McCarthy started his campaign to stop the war I participated in getting out the vote in that California primary; the tragic outcome on election day was a real blow and has discouraged me from politics since.

Frustrated in my course of study, a broader perspective was sought in reading Understanding Media, the novel Siddhartha, and Herb Caen's column in the San Francisco Chronicle. Deeper considerations arose in hexagrams of the I Ching. Travelling to Alaska, I took up study at University of Alaska Fairbanks which provided a course of study including differential geometry and integral geometry (after Luis Santaló) leading to M.Sc. degree. Coursework at university included ring theory courses that provided enough abstraction for wide application. When the book Diet for a Small Planet appeared, it offered a glimpse of sustainable living for Earth. In Anchorage I refused a induction Army second time, and when threatened with prosecution left the country. Some years later President Jimmy Carter granted amnesty for refusal of induction.

Pierre Trudeau was Prime minister when I landed in Canada, which was then a major contributor to United Nations operations in peacekeeping. For science policy, reference is made to Francis Bacon in Canada rather than to Benjamin Franklin.

It was my good fortune to hear F.A. Kaempffer, author of Concepts in Quantum Mechanics (1965) and Elements of Physics (1967), speak in Halifax to a conference on differential geometry and relativity. He reviewed the symmetry of the electromagnetic field that is expressed by spherical wave transformations that form a conformal group on spacetime. Wolfgang Rindler's textbook Essential Relativity was acquired and investigation of conformal maps proceeded at one of Nova Scotia's leading universities. In a university program, I was a teaching assistant and studied functional analysis including topological groups and Haar measure, and attended the functional analysis seminar. Going for the third degree, research was aided by Science Citation Index. My approach to the conformal symmetry was expressed on the projective line over biquaternions. I wrote a report on the projective line over a ring and made some deductions for associative composition algebras, using the title Geometry of Electromagnetism. As J. Arthur Seebach wrote in his telegraphic review for American Mathematical Monthly (volume 85, foot of page 778), "The geometry is that of a homography transformation on the category of projective spaces with two homogeneous coordinates taken from an arbitrary associative ring with 1. The development of these geometries is shown to be relevant to the study of Maxwell's equations." As a "Recent book", it was noted by IEEE Spectrum (page 74, February 1979).

## BC

There were some classroom teaching opportunities in elementary algebra, introductory physics and computer science, particularly after microcomputers became a retail product. So for a few years binary arithmetic and Boolean algebra were in my curriculum. In 1987 I wrote a letter to American Journal of Physics (56:296) : "An Also Known As List for Perplex Numbers", which listed some references for split-complex numbers.

At last I became a vegetarian, relying on whole grains (particularly brown rice) along with legumes. Of the animal protein sources, eggs, milk, and meat, I choose only the middle one.

One day I noticed that if the i, j, and k in the quaternion group were given +1 for their squares, then one obtains a quasi-group of order 8, and the real algebra over this quasi-group has many features of Minkowski space. When I asked Nathan Divinsky, it appeared new to him. But later I found out that Alexander Macfarlane had the same idea for his hyperbolic quaternions in the 1890s! Deeper background appeared with the writings of James Cockle who found tessarines and coquaternions in the 1853 biquaternions that William Rowan Hamilton had developed.

The waters of the Strait of Georgia, sheltered by Vancouver Island, are ideal for sea kayak touring. Buoyed by a double kayak, islands appeared up and down the Strait. And the kayak folded up into package fit for a bus bay: the frame dismantles and skin rolls up. Lago Nahuel Huapi and other lakes in the Andes were explored. In the Atlantic the caravel visited Isla Mujeres, Florianopolis and canals in Tigre Partido. In the Pacific the shores of Chiloé Island, Waiheke Island, and the Marlborough Sounds were viewed and visited. Such explorations were in the wake of Pytheas, who sought amber, but before the new millennium. Returning to the quest for my own amber, new tools had been laid out by the digital revolution. The article "Split-complex Numbers" was posted including a longer list of synonyms of perplex numbers.

I continue to be a bibliophile, enjoying especially biography and history of science. Today I use MathSciNet and Google Book Search in addition to search engines. Scientists of age please note: In writing a biography for Wikipedia, the existence of autobiographical notes makes a significant difference. For examples, Allen G. Debus and Jerald Ericksen wrote about their work and movements. Taking notes when they traveled, Stephen Timoshenko and Willard Quine give an impression of perpetual movement. Too many scientists leave no trace but their academic output in books, articles, or media releases. Recollecting some of the turbulence encountered in academic life may be a meaningful experience.

Vancouver Public Library and the University of British Columbia Library have provided some materials for references. And Biodiversity Heritage Library, Internet Archive, HathiTrust,Project Gutenberg,Project Euclid, Cornell Historical Math Monographs, University of Michigan Historical Mathematics Collection and Jstor (early content) have been found to be vast repositories of reliable sources that may be linked for encyclopedia users to read online.

For students interested in tools, beyond real numbers ℝ and complex numbers ℂ, that can structure mathematical models, the text Associative Composition Algebra is now available at Wikibooks.