Paul S. Donchian

Paul Samuel Donchian (1895-1967), a remarkable graduate of Hartford Public High School, had a long association with geometer H.S.M. ("Donald") Coxeter (1907-2003).  Donchian provided wire and paper models photographed in Coxeter's Regular Polytopes (1949; New York:  Dover Publications, Inc., 1973).  The paper models showed the faces of geometric solids, while the wire models showed the edges. The models often represent fourth and higher dimensional objects. Coxeter and Donchian also co-authored an article, "An n-Dimensional Extension of Pythagoras' Theorem," in the Mathematical Gazette, July, 1935.

Paul Donchian's father, Samuel B. Donchian, was born in Diarbekir, Turkey, in 1853, and died in 1910. Samuel founded the Samuel Donchian Rug Co. in Hartford. The building housing the business on Pearl St. was completed in 1910, the year of Samuel's death. Paul's mother was the former Armenouhi A. Davoud.

Coxeter provides biographical information about Donchian on page 260 of Regular Polytopes. (This page may be viewable in the Excerpts at Google Books. There are a limited number of views of these excerpts permitted from any one computer, and the excerpts do not contain the photos showing Donchian's models. The book remains in print at Dover.)   Coxeter states that Donchian's "mathematical training ended with high school geometry and algebra."  The owner of a business in Hartford selling oriental carpets, he began his self-taught studies of geometry at age 30 after a series of "startling and challenging dreams of the previsionary type soon to be described by Dunne in 'An Experiment with Time.'"

For the Wikipedia article about J.W. Dunne, who published his "Experiment with Time" in 1927, click here. Dunne's theory, based on dreams, was that all time exists in the present. Donchian was 30 in 1925, so apparently had his dreams before publication of Dunne's work.

Some of Donchian's models (regrettably unattributed when I saw them in 2001) were on display in the mathematics exhibit at the Franklin Institute in Philadelphia.

Donchian is perhaps best known for his wire model of the 120-Cell. This is a model of a four dimensional dodecahedron. The three dimensional dodecahedron, with 12 pentagonal faces, is one of the Platonic solids. Several of the Platonic solids have analogs in four dimensional space. Just as the tesseract is the four dimensional analog of the cube, the 120-Cell is the four dimensional analog of the dodeacahedron. For further information about the 120 Cell, see John Stilwell, The Story of the 120 Cell, Notices of the AMS, January, 2001; Ed Pegg, Jr., Math Games - Dodecahedral Tilings, December 18, 2006.

Donchian also provided models photographed in W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays (New York: Dover Publications, 13th Ed. 1987). The book doesn't state that Donchian made the models, but several of Donchian's contributions to the study of polyhedra are noted on page 141.

Donchian at Hartford Public High School

1910 Owl Yearbook 1911 Owl Yearbook 1912 Owl Yearbook 1913 Owl Yearbook 1914 Owl Yearbook

Donchian is listed in each of the five Owl yearbooks for Hartford Public High School between 1910 and 1914. Thus, it took him five years to complete what would normally be four years of high school. To see these yearbook entries, including photographs of Donchian, and his cartoons (such as the one above from 1912), click on the entries above.

Although Donchian is listed as class of '13 in the 1910, 1911 and 1912 yearbooks, he is listed as class of '14 in the 1913 and 1914 yearbooks, and graduated in 1914.  He was active in school affairs in 1910 (Boys'Mandolin club), 1911 (Boys'Mandolin club), and 1912 (captain of tennis team; manager of Boys' Mandolin Club). He won the first prize in his class for scholarship in 1910 (reported in the 1911 yearbook), and repeated his first prize in 1911 (reported in 1912 Hartford Municipal Register). He contributed a cartoon to the 1911 Owl (which he signed "PSD '13"), and numerous cartoons to the 1912 Owl (which he signed "PSD," dropping his year affiliation). He did not contribute signed cartoons to the 1913 or 1914 yearbook, and is not listed with the mandolin club during those years.  The 1913 Yearbook cryptically states he was elected captain of the tennis team for the 1912 season but then resigned. There may have been some event or difficulty in 1912 or 1913 which prevented his graduation in 1913.

[Note: His delay in graduation may have been caused by a need to assist in running the business, or perhaps was related to difficulties within the family, ultimately stemming from his father's death in 1910. Paul's older brother Arthur, who had graduated from Hartford Public High School in 1910, entered Yale but dropped out in 1911 on account of Samuel's death in order to run the business. The Hartford Courant reported on September 29, 1915 that Arthur's mother sought a conservatorship over Arthur's assets. In court proceedings, she contended that Arthur had "squandered" much of the $30,000 he had inherited from his father. A doctor stated Arthur had a mental disorder which had "caused him to make threats against his mother and brothers." On October 13, 1915 the Courant reported that the Court had appointed a conservator, but had dismissed for lack of evidence the attempt to have Arthur committed to an institution. Eventually Arthur resumed helping to run the business, as vice president and treasurer. Arthur died in 1938.]

The 1914 Yearbook lists Paul Donchian ("Donchian '14") as manager of the tennis team for 1913, and also lists him as Chairman of the Pin Committee, and a member of the Color Committee. The "Rogues Gallery," which had a writeup of ten members of the class, states his ambition to be a veterinarian.

(Note: Some of the above materials from Hartford Public High School may be seen in the Gutman Library, Harvard Graduate School of Education. Their Special Collections department has Owl Yearbooks for 1906-10, 1913-16, and 1920, as well as the Catalog for 1925 which contains an Alumni Directory.)

The Yale Records

I first learned from an alumni directory for the high school (Catalog, 1925) that after Donchian graduated in 1914, he received an A.B. from Yale. Coxeter does not mention Donchian's Yale degree (granted in 1918), nor do other writers discussing the models. I thought it would be interesting to determine from Yale whether Donchian took mathematics classes there.  A search of the Yale records has not yet determined this question, but shows no reason to doubt Coxeter's assertion, if construed narrowly to mean that Donchian took no mathematics courses after high school. These records do show science courses which likely had mathematical content. The documents cast additional light on Donchian's activities at Yale and later work.

He was a member of Phi Beta Kappa, was appointed to the Philosophical Oration his Junior Year, and received Honors of 1st Grade his freshman year. Especially given this outstanding academic record (which was all the more remarkable, given the turmoil the Donchian family evidently was undergoing during at least a portion of that time), Coxeter's failure to mention Donchian's Yale education is surprising.

Donchian was a member of several mandolin and banjo/mandolin clubs and also played freshman tennis.  His freshman courses were chemistry, English, French, German and physics.

A complete course list for years after the freshman year is unavailable.  However, he listed his most inspiring professor as William Lyon Phelps [English], and his hardest course as Financial History of the U.S., taught by F.R. Fairchild [Economics] (Economics B70 is described here in the 1917-18 Yale Catalog.). His easiest subject was Pictorial Art, with Professor Taylor. His most valuable subject was Distributing Systems, with Westerfield. [Note - Ray B. Westerfield was an instructor of political economy from 1913 to 1916, and then Assistant Professor of Political Economy. His course in Distributing Systems, Economics C38, is described here.]

Donchian's mention of two upper level economics courses as his hardest and most valuable suggests he may have majored in Economics. The records provided by Yale do not set forth his major, but Economics would be a plausible major for a student like Donchian planning to enter the business world. At the time, the study of economics was not as dependent upon mathematics as it is today. There was no math requirement for Economics majors. Yale's general requirement was to take one course in either math or science both freshman and sophomore years. The records provided by Yale do not indicate how Donchian satisfied his sophomore math or science requirement, but it was entirely possible for Donchian to have completed his degree without taking a mathematics course there.

Donchian served as president of the Yale Armenian students' association. He had contributions published in the News, Record and Sheffield Scientific Monthly.  Hopefully back issues of these publications remain available.

Perhaps the most important information in the Yale records is his 40th year class report, in 1954.  Donchian wrote for his classmates that he had been puzzled by the ability to foretell the future with great precision through dreams.  He believed the fourth dimension offered a possible explanation, and therefore sought to simplify the concept of hyperspace by constructing mathematical models.  He also mentioned the importance in his life of the search for religious truth.  It is unclear from the report alone whether the prophetic dreams were a personal experience, or whether he was relating the experience of others. However, given Coxeter's reference to Donchian's "previsionary" dreams, it seems clear that the experience related in the class report was a personal one.  I have been unable to find further detail regarding these dreams or what he believed they showed about the future.

The award to Donchian at graduation of the honor of philosophical oration was reported in the N.Y. Times, June 20, 1918. This honor was among the highest available, perhaps the equivalent of summa cum laude. (See explanation of Yale honors here). Elliot B. Cohen, also listed in the Times article as receiving Philosophical Orations, went on to found Commentary magazine.

After graduating, Donchian joined the service and went to Pelham Bay Naval Training Station.  He remained there through the end of 1918, reaching the rank of Seaman 2d Class. (Perhaps a course in navigation would have been an additional source for Donchian's acquaintance with aspects of geometry, prior to the self-study leading to construction of his models.)

Following his military service, he wroked for the Samuel Donchian Rug Co., starting in 1919.  For a few months in 1920, he worked in the rug department of Scholle Furniture Co. in Chicago, and then returned to the Samuel Donchian Rug Co. in October, 1920, where he remained until his death in 1967.

In 1921, he reported to the alumni office that he had made a buying trip to London and France, where he had run into a classmate.

He was listed as a donor to the Yale University Library in 1931-32, p. 27. One could suppose that he spent time in that library doing research leading up to the making of his models.

He was elected to the American Mathematical Society in 1932.

In 1934 he displayed his models at the World's Fair in Chicago, which led to his meeting with Coxeter. Marc Pelletier, Coxeter's Model Maker, Paul Donchian. Photos of some of the models, as well as Donchian, are shown here. Later that year, the models were displayed at an AAAS conference in Pittsburgh.

An article appeared in the N.Y. Times, July 21, 1935, showing photographs of Donchian and several of his mathematical models.

Coxeter introduced Donchian to the Indian astrophysicist and cosmologist Subrahmanyan Chandrasekhar (see Wali's biography of Chandrasekhar here). This led to Donchian hosting Chandrasekhar for the astrophysicist's first Christmas in America in 1935. Wali, Chandrasekhar, La Naissance de l'astrophysique, p. 173. Chandrasekhar (1910-1995) had received his Ph.D. at Cambridge in 1933. He would be awarded the Nobel Prize in Physics in 1983.

In May, 1940, Donchian reported to the alumni office the showing of his figures of "four dimensional and higher dimensional space" in Chicago and Pittsburgh. Significantly, he also reported that his greatest interest was in the field of "mathematics and metaphysics."

Paul Donchian's Patents

Donchian was listed as the inventor in several patents, including a camera (Patent 1,588,666; application filed 1923); rug fastener (1,770,879; application filed 1927); and dish (2,121,654; application filed 1937); and design patents for trays (D110,175; D110176; D 110177; applications filed 1937); lamp base (D106,183; application filed 1937); lamp shade (D110,173; application filed 1937); lamp base (D110,174; application filed 1937); plaque (Patent D112060; application filed 1938); and plate (D110,538; application filed 1938). Especially in the dish patent (above), one can see the influence of Donchian's study of the dodecahedron. Several of the patents rely on a pentagon motif.


An obituary appeared in the Hartford Times, May 26, 1967, which mentioned his mathematical models, as well as his hobby of rug weaving. A report of his funeral service in the Yale records indicates he had been a member of the Congregationalist church in Hartford for over 50 years. He had been active in prison outreach, teaching Bible and Christianity to inmates.

Why Didn't Coxeter's Regular Polytopes Mention Donchian's Yale Education?

As noted above, it is puzzling why Coxeter's biographical sketch of Donchian in Regular Polytopes fails to mention Donchian's Yale education. This omission is all the more surprising given Donchian's high academic achievements there. Instead Coxeter states on p. 260 that Donchian's "mathematical education ended with high school geometry and algebra." Why were Donchian's studies at Yale, including at least freshman chemistry and physics, omitted from the biography? Such courses likely had considerable mathematical content.

One possibility that comes to mind is that Donchian's Yale experience might have been an unhappy one, and Donchian chose not to discuss his education there for that reason. Yet there is no evidence of such dissatisfaction. Donchian kept in regular touch with the alumni office.

To establish a more likely reason for the omission would depend on whether it was Donchian's or Coxeter's decision to omit Donchian's Yale degree from the biographical note. Each might have his own reasons for deciding that Donchian's educational background was irrelevant to the biographical sketch. The first possibility is that the decision was Donchian's (if, for example, he never revealed his Yale background to Coxeter). If the choice was Donchian's, the explanation may lie in his motivation for constructing the models - to make the fourth and higher dimensions intelligible to others who did not have the advantages of education. This is mentioned both in Coxeter, p. 260 - "so that anyone like himself with only elementary mathematical training could follow every step," and in Donchian's 40th year alumni report - "a set of models that make it possible for any high school graduate to comprehend the subject." To state Donchian's achievements at Yale might detract from the thesis that those untrained in mathematics, or lacking any college education, could understand the models and the fourth dimension. Donchian considered the models, as well as their comprehensibility to the public, to be metaphysically significant. They were not mere dry mathematical constructs.

If the decision to omit the information was Coxeter's, it may be noteworthy that the contribution of non-professionals to the field made several appearances in Coxeter's book. Such appearances include the discussion at p. 258 of Alicia Boole Stott (1860-1940), with whom Coxeter also had collaborated. The story of the self-taught amateur contributing to mathematics might be expected to have a fascination for the reader. Perhaps when Coxeter was writing about Stott and Donchian, he was recalling the story of Ramanujan (1887 - 1920), the self-taught Indian mathematician who achieved spectacular results in number theory. Ramanujan communicated some of these to the world's leading number theorist, Professor G.H. Hardy in Cambridge. Hardy recognized Ramanujan's genius and arranged to bring Ramanujan to Cambridge for further study and collaboration with Hardy.

There is an interesting biographical link between Coxeter and Ramanujan. This link is recounted in Siobhan Roberts' biography of Coxeter, King of Infinite Space, New York: Walker & Co., 2006. Through Bertrand Russell, Coxeter while a teenager met mathematician Eric H. Neville. Neville was instrumental in guiding Coxeter in his pre-university studies, eventually leading to Coxeter's admission to Cambridge. Neville had earlier acted as Hardy's emissary, in travelling to India to persuade Ramanujan to attend Cambridge.

There is a parallel between Ramanujan and Donchian in the role played by dreams in giving rise to their mathematical studies. Ramanujan had initially resisted, based on religious reasons, the attempt of Hardy and Neville to convince him to leave India and move to Cambridge. Ramanujan changed his mind after his mother had a dream, in which her son was surrounded by Europeans and the family goddess commanded the mother to withdraw her objections to Ramanujan's travel to England. Robert Kanigel, The Man Who Knew Infinity, New York: Charles Scribner's Sons (1991), p. 188, quoting Neville.

It is of course speculative to consider whether an analogy with Ramanujan might have consciously or unconsciously led Coxeter to omit any mention of Donchian's educational background in writing the biographical sketch. Nonetheless, for Coxeter to emphasize or even mention Donchian's academic achievements, if they were known to Coxeter, might have seemed to weaken the compelling nature of the story. Donchian the rug-merchant somehow seems more interesting than Donchian the graduate of Yale with highest honors.

The failure to mention Donchian's Yale background was not unique to Coxeter. Several press articles from 1934 and 1935, giving publicity to the model displays, fail to set forth Donchian's educational background as well. Donchian's graduation from Yale was mentioned in a newspaper article when he first entered his late father's business (Hartford Courant, Feb. 1, 1919), and in Donchian's obituaries, but may not have been mentioned in the press during the interim.

Given Donchian's metaphysical inclinations, he would have doubtlessly found satisfaction in recent theories that the 120-Cell may be a key to understanding the structure of the universe. L. Marek-Crnjac, "Higher Dimensional dodecahedra as models of the macro and micro universe in E-infinity Cantorian space-time," Chaos, Solutions & Fractals, Vol. 32, Issue 3, May 2007, pp. 944-950.

There is a review of Regular Polytopes mentioning Donchian's "remarkable" models in the Journal: Bull. Amer. Math. Soc. 37 (2000), 107-108

About Professor Coxeter:

Lecture by Marc Pelletier: Paul Donchian, Modeler of Higher Dimensions (site of the Fields Institute), February, 2002.

Announcement of Lecture by Marc Pelletier, Coxeter's Model Maker, Paul Donchian at Coxeter Colloquium, Princeton, November, 2006.

The following pages contain references to Donchian's models:

George Hart website

More about polytopes:

Polyhedra and Polytopes by David Eppstein, UC Irvine
Hyperspace Star Polytope Slicer by Mark Newbold

Geer's Hartford Directory 1918

Paul's brother Richard Donchian (1905-1993) was a leading figure in the field of technical stock and commodity analysis. Graduating from Yale in 1928, he went to work for Samuel Donchian Rug Co. before entering the investment field.

Threads from the Oriental Loom, by V. Gurdji (1901), revised by John B. Donchian (1913), is a book about oriental rugs. I do not know if John B. Donchian was related to Paul. Like Samuel, John was born in Diarbekir, Turkey. He owned Donchian & Co., a rug store on Fifth Avenue in New York City. After his death in 1921, the New York Times reported on August 25, 1922 that his $233,000 estate would eventually used for charitable purposes in Diarbekir. The estate was the subject of extended court proceedings. In re Donchian's Estate, 199 NYS 107 (Surr.1923), aff'd, 204 NYS 2d 1924 (Sup. 1924); In re Donchian's Estate, 217 NYS 2d 318 (Surr. 1926). As a result of the litigation, the charitable bequest was reduced, but not eliminated. John Donchian's widow had contended that political events in Turkey had made the bequest impossible of fulfillment, but this contention was rejected by the Court.



This page was last updated March 30, 2009.