Patently Mathematical

This book describes how mathematics contributes to inventions and has been incorporated into many — perhaps thousands — of patents. Most of the book treats the mathematics and associated algorithms that enable the inventions. It is accessible to a general reader with at most a modest mathematical background. Key concepts are carefully explained with explicit examples and a minimum of abstraction.

Although patents are identified for various inventions, the question of patentability is mostly deferred to a short discussion at the end of the book. I’ll say more about this later. Each chapter addresses a general area where mathematics plays a role in an invention. Most of these involve software.

The first chapter introduces some of the ideas behind document searching, indexing and information retrieval and describes some of the algorithmic techniques that can be used. The author builds on this in the second chapter to describe how search engines work and some of the issues that they face. He calls this chapter “The Trillion Dollar Equation”, perhaps referring to Google’s amazing financial success. But of course there is no such single equation, and Google’s Page and Brin did not become wealthy just because of a few patents.

Other chapters consider image comparison, dating services and personality matching, identity protection with passwords, optimization under constraints, protecting communications with encryption, social networks, data compression, breaking codes, and antenna design. Associated with many of these subjects are patents that the author identifies and briefly describes.

Among the more unusual inventions the author describes are a computer-generated method of building instructions for LEGO constructions, a fractal antenna whose design is generated by assembling replicas of various well-know fractal shapes (Koch, Cantor, Mandelbrot, Julia, etc.) and a computer-implemented method of designing craters for the surface of a golf ball.

Patent law in the United States does not allow patent protection for “abstract ideas” and this has been interpreted to include algorithms. Until fairly recently, however, patents have been granted that claim processes using algorithms to manipulate existing information to generate other information. This has always seemed (to me) to be a very thinly disguised way of giving patent protection to algorithms. In 2014 the Supreme Court ruled that this was improper, and in so doing effectively called into question a considerable number of issued patents. More recently, the courts have tried to clarify when software based on mathematical algorithms might be patentable. It will probably take a long time for any kind of resolution.

In an epilogue the author briefly addresses this question and its possible consequences for patent protection of inventions that involve mathematics. He points out that algorithms, whatever their origin or specific application, are often very broadly applicable and thus are capable — in principle — of establishing a kind of patent monopoly over very large technical areas.

This book would appeal to anyone interested in the interaction between mathematics and the world of patents. It also offers a number of examples of the applicability of mathematics in unusual places.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.