Robert Berger (born 1938) is an applied mathematician, known for discovering the first

aperiodic tiling An aperiodic tiling is a non-periodic Tessellation, tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic set of prototiles, aperiodic if copie ...

using a set of 20,426 distinct tile shapes.

Contributions to tiling theory

The unexpected existence of aperiodic tilings, although not Berger's explicit construction of them, follows from another result proved by Berger: that the so-called

domino problem Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang (academic), Hao Wang in 1961, is a class of formal systems. They are modeled visually by square tiles with a color on each side. A set of such til ...

is undecidable, disproving a conjecture of Hao Wang, Berger's advisor. The result is analogous to a 1962 construction used by Kahr,

Moore Moore may refer to: Language * Mooré language, spoken in West Africa People * Moore (surname) ** List of people with surname Moore * Moore Crosthwaite (1907–1989), a British diplomat and ambassador * Moore Disney (1765–1846), a senior ...

, and Wang, to show that a more constrained version of the domino problem was undecidable.

Education and career

Berger did his undergraduate studies at

Rensselaer Polytechnic Institute Rensselaer Polytechnic Institute (; RPI) is a private university, private research university in Troy, New York, United States. It is the oldest technological university in the English-speaking world and the Western Hemisphere. It was establishe ...

, and studied

applied physics Applied physics is the application of physics to solve scientific or engineering problems. It is usually considered a bridge or a connection between physics and engineering. "Applied" is distinguished from "pure" by a subtle combination of fac ...

Harvard Harvard University is a private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher lear ...

, earning a master's degree, before shifting to applied mathematics for his doctorate. Along with Hao Wang, Berger's other two doctoral committee members were Patrick Carl Fischer and

Marvin Minsky Marvin Lee Minsky (August 9, 1927 – January 24, 2016) was an American cognitive scientist, cognitive and computer scientist concerned largely with research in artificial intelligence (AI). He co-founded the Massachusetts Institute of Technology ...

. Later, he has worked in the Digital Integrated Circuits Group of the

Lincoln Laboratory The MIT Lincoln Laboratory, located in Lexington, Massachusetts, is a United States Department of Defense federally funded research and development center chartered to apply advanced technology to problems of national security. Research and dev ...

Publications

Berger's work on tiling was published as "The Undecidability of the Domino Problem" in the '' Memoirs of the AMS'' in 1966.. This paper is essentially a reprint of Berger's 1964 dissertation at

Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

. In 2009, a paper by Berger and other Lincoln Laboratories researchers, "Wafer-scale 3D integration of InGaAs image sensors with Si readout circuits", won the best paper award at the IEEE International 3D System Integration Conference (3DIC). In 2010, a

CMOS Complementary metal–oxide–semiconductor (CMOS, pronounced "sea-moss ", , ) is a type of MOSFET, metal–oxide–semiconductor field-effect transistor (MOSFET) semiconductor device fabrication, fabrication process that uses complementary an ...

infrared Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...

imaging device with an

analog-to-digital converter In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a Digital signal (signal processing), digi ...

in each pixel, coinvented by Berger, was one of ''R&D Magazines

R&D 100 Award Research and development (R&D or R+D), known in some countries as experiment and design, is the set of innovative activities undertaken by corporations or governments in developing new services or products. R&D constitutes the first stage of d ...

recipients.MIT Lincoln Laboratory receives five R&D 100 Awards
Lincoln Laboratory, retrieved 2011-09-30.

References

{{DEFAULTSORT:Berger, Robert Living people 20th-century American mathematicians 21st-century American mathematicians Rensselaer Polytechnic Institute alumni Harvard John A. Paulson School of Engineering and Applied Sciences alumni 1938 births MIT Lincoln Laboratory people