Bibliography: Balanced Ternary

Any evidence of signed digit arithmetic in ancient Indian mathematics (?)

A quoted book Vedic Mathematics by Swami Bharati Krsna Tirtha is not of Vedic origin and I did not find balanced ternary in that book anywhere.

The most important book by Fibonacci is the Liber Abaci published in 1202 and 1228. The book has 15 sections a large part of it on his trying to introduce the new Hindu ciphers to Middle Age Europe, its mathematical operations etc. The Section 12 is the most significant and also the most voluminous. In that section, he discussed three problems, the first of which is the much celebrated "rabbit reproduction" which today is known as the Fibonacci sequence and thats how all of us know him. But the next problem was indeed the Balanced Ternary problem. The following is from this reference.

The next problem considered by Fibonacci is called the "problem about choice of the best system of standard weights for weighing on the beam balance" or simply the "weighing problem". In the Russian historical-mathematical literature the "weighing problem" is known under the title of "Bashet-Mendeleev's problem", called so in honor of the French 17th century mathematician Bachet de Mesiriaque, who placed this problem in the "Collection of pleasant and entertaining problems" (1612), and of the outstanding Russian chemist Dmitry Mendeleyev, who interested in this problem when he worked as a director of the Main Standard and Weight Bureau of Russia.

Since Fibonacci got his mathematical education from the Arabic Education system which was heavily influenced by the Hindu system, it is possible that this problem was present in the mathematical folklore of that tradition in those days.

Johannes Kepler, German mathematician and astronomer (1571-1630) uses balanced ternary scheme modeled on Roman numbers (?)

John Colson (1680-1760) England, the fifth Lucasian Professor of Mathematics at Cambridge University, is a relative unknown in the history of the field. He is remembered for producing several of Sir Isaac Newton's works including De Methodis Serierum et Fluxionum in English in 1736

"The method of negativo-affirmative arithmetic is still interesting. Colson found a way to mix negative and positive digits to make up a number. He devised a set of rules to create a negativo-affirmative number and another set of rules to return the number to a common one. There are sets of rules for performing the arithmetic as well. Once learned, the method does, in fact, speed up the multiplication of large numbers."

John Leslie 1820

The Philosophy of Arithmetic, Exhibiting a progressive view of the theory and practice of calculation, with Tables for the multiplication of Numbers as far as one thousand. Edinburgh: Willliam and Charles Tait.

A large, wooden calculating machine was built in 1840 by Thomas Fowler in his workshop in Great Torrington, Devon, England. In what may have been one of the first uses of lower bases for computing machinery, Fowler chose balanced ternary to represent the numbers in his machine. The machine was shown to a number of notable scientists and mathematicians of the day, including Charles Babbage, Augustus DeMorgan and the then Astronomer Royal, Professor George Airy.

Since 1997, two current Devon residents, Pamela Vass and David Hogan, have been researching Thomas Fowler and his inventions. They discovered a two-page description of Fowler’s calculating machine, written in 1840 by a prominent mathematician of the day, Augustus DeMorgan. Working together with Vass and Hogan, Mark Glusker designed and built a model, based primarily on the information in DeMorgan's description.

What is also notable is that Augustus DeMorgan had realised its importance enough to document it. Fowler was a printer by profession, only a self taught mathematician.

In 1840, Augustin Cauchy discussed signed-digit numbers in various bases and Leon Lalanne immediately followed up with a discourse on the special virtues of balanced ternary.

Leon Lalanne, 1840: Note sur quelques propositions d'arithmithmologie elementaire. Comptes rendus hebdomadaires des seances de l'Academie des sciences 11:903-905

It appears that there was considerable interest in this subject during that epoch of the 1840s.

Axel Thue, Norway, 1912
Uber die gegenseitige lage gleicher teile gewisser Zeichenreihen.
In Selected Mathematical Papers of Axel Thue, pp. 413-477. Oslo: Universitetsforlaget

Axel Thue proved that unbounded square free ternary sequences exist and gave a method to construct one.

The design of small digital machine "Setun" (Setun is the little river which flows into the river "Moscow" near the University) was initiated by member of the academy of Sciences S. L. Sobolev at 1956

All references here

Brousentsov N. P. Origins of informatics. – Moscow, The New Millennium Foundation, 1994. (In Russian).

Brousentsov N. P. Computing machine "Setun" of Moscow State University // "New developments on computer technology". – Kiev , 1960, pp. 226-234. (In Russian).

Small digital computing machine "Setun" / N. P. Brousentsov, S.P.Maslov, V. P. Rosin, A. M. Tishulina – Moscow State Univ., 1965. (In Russian).

Brousentsov N. P., Zhogolev E.A. The structure and algorithmic description of small computing machine // "Computers and problems of cybernetics", Issue 8. – Leningrad State Univ., 1971, pp. 34-51. (In Russian).

Brousentsov N. P., Zhogolev E. A., Maslov S. P. General characteristic of small digital machine "Setun 70" // "Computers and problems of cybernetics", Issue 16. – Leningrad State Univ., 1974, pp. 3-20. (In Russian).

Brousentsov N. P. Threshold realization of threevalued logic on electromagnetic elements // "Computers and problems of cybernetics", Issue 9. – Moscow State Univ., 1972, pp.3-35. (In Russian).

Brousentsov N. P., Ramil Alvarez J. The structured programming on small digital machine // "Computers and problems of cybernetics", Issue 15. – Moscow State Univ., 1978, pp. 3-8. (In Russian).

Yourdon E. The second structured revolution // "Software World", 1981, v.12, n.3.

Brousentsov N. P., Maslov S. P., Ramil Alvarez J. Didactic microcomputer system "Nastavnik". – Moscow, "Nauka", 1990. (In Russian).

Dialogue system of structured programming DSSP-80 / N. P. Brousentsov, V. B. Zakharov, I. A. Rudnev, S. A. Sidorov // "Dialogue microcomputer systems" – Moscow State Univ., 1986, pp. 3-21. (In Russian).

Conceptual characteristic of the RIIIS-processor / N. P. Brousentsov, S. P. Maslov, J. Ramil Alvarez, S. A. Sidorov // "Integrated system for teaching, constructing of programs and developing of didactic materials" – Moscow State Univ., 1996, pp. 16-43. (In Russian).

Brusentsov N.P., Vladimirova Yu.S. Solution of Boolean equations // "Computational Mathematics and Modeling", 1998, v.9, n.4, pp. 287-295.

http://www.computer-museum.ru/histussr/12.htm

http://www.icfcst.kiev.ua/museum/Brusentsov.html

Engineering Research Associates, Inc. 1950. High-speed Computing Devices. New York: McGraw-Hill.

"A symmetrical notation for numbers", American Mathematical Monthly 57:90-93, in 1950

Gardner, Martin. 1964.
The "tyranny of 10" overthrown with the ternary number system. Scientific American 210(5):118–124.

Dijkstra E.W.

Notes on structured programming. EWD 249 – Technical University, Eindhoven, Netherlands, 1969

Gideon Frieder, A Fong, CY Chow, 1973
A balanced ternary computer, Conference Record of the 1973 International Seminar on Multiple Valued Logic, pp 68-88

Frieder and his colleagues at State Univ of New York, Buffalo designed a base 3 machine called Ternac. And also a software emulator for it.

In 1981, in his book "The Art of Computer Programming", Vol 2: Seminumerical Algorithms. Second Edition. Reading Mass: Addison-Wesley, pp 190-193 calls it the "prettiest number system of all" and writes that the replacement of "flip-flop" for "flip-flap-flop" will nevertheless happen on a good day.

He writes about the merits of trinary in containing the sign within the number, rounding is mere truncation.

P Erdos and R. L. Graham. 1980.
Old and New Problems and Results in Combinatorial Number Theory. Geneva: L'Enseignement Mathmatique Universit de Geneve.

David C Rine, 1984
Computer Science and Multiple Valued Logic: Theory and Applications, Second Edition. Amsterdam: North Holland Publishing Company.

Ilan Vardi, 1991
The digits of 2ⁿ in base three. Computational recreations in Mathematica, Reading Mass: Addison-Wesley. pp 20-25

Herbert R. J. Grosch, 1991.
Computer: Bit Slices from a Life. Novato, Calif.: Third Millennium Books.

Had proposed Whirlwind computer project at MIT in 1950s. The system that was deployed was however not a ternary computer.

Came across this idea in 1987 as a student.
First printed publication in a campus weekly of the College of Military Engineering, India in 1995.
Incorporated tristate in an algorithm for software encryption in a widely circulated software. PCQuest magazine CDROMs, Jan 1998, Mar 1998. A Microprocessor Simulator for 8085
Published on web since 1998.
This publication in July 2006 offers detailed solution for fractional numbers.

Steve Grubb, 1995(?)

Some hardware suggestions for ternary computers have been provided by Steve Grubb in his website http://trinary.cc
Steve talks about trinary and not balanced ternary, however the hardware suggestions could be valuable.

His company Deteministic Data Systems paid for a broad patent search on this and has obtained references of the following IBM hardware patents.

The premise for the above patents may be ternary and not balanced ternary but could be anyway valuable. The patents may have expired.

Shalosh B Ekhad and Zeilberger Doron, 1998.
Journal of Integer Sequences, Vol 1, Article 98.1.9

Discovered that there are more than 2^n/17 n letter ternary square free words.

General Balanced Ternary (GBT) spatial addressing scheme (Gibson. and Lucas, 1982)
Gibson, L., Lucas, D. (1982) Spatial Data Processing Using Generalised Balanced Ternary, IEEE Pattern Recognition and Image Processing, pp. 566 - 571.

James Alright maintained a website promoting balanced ternary, 1996. The website is no more accessible.

Dinesh Sarvate of College of Charleston has been encouraging his students to research on balanced ternary systems.

"If we marvel at the patience and the courage of the pioneers, we must also marvel at their persistent blindness in missing the easier ways through the wilderness and over the mountains. What human perversity made them turn east to perish in the desert, when by going west they could have marched straight through to ease and plenty?" - E T Bell

US Patent No.	Date of issue	Title
3,129,340 *	04/14/64	Logical and Memory Circuits Utilizing Tri-Level Signals
3,176,154 *	03/30/65	Three State Memory Device
3,207,922 *	09/21/65	Three Level Inverter and Latch Circuits
3,660,678	05/02/72	Basic Ternary Logic Circuits
3,671,763	06/20/72	Ternary Latches
3,671,764	06/20/72	Auto-Reset Latches
3,909,634	09/30/75	Three State Latch