Duncan M. Y. Sommerville | |
---|---|

Born | 24 November 1879 |

Died | 31 January 1934 Wellington, New Zealand |

Nationality | Scottish |

Citizenship | United Kingdom |

Alma mater | St Andrews University |

Known for | Textbooks on geometry Dehn-Sommerville relations |

Scientific career | |

Fields | Mathematics |

Institutions | St Andrews University (1904–13) Wellington University (1913–34) |

**Duncan MacLaren Young Sommerville** FRSE FRAS (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote *Introduction to the Geometry of N Dimensions*, advancing the study of polytopes. He was a co-founder and the first secretary of the New Zealand Astronomical Society.

Sommerville was also an accomplished watercolourist, producing a series New Zealand landscapes.

The middle name 'MacLaren' is spelt using the old orthography **M'Laren** in some sources, for example the records of the Royal Society of Edinburgh.^{ [1] }

Sommerville was born on 24 November 1879 in Beawar in India, where his father the Rev Dr James Sommerville, was employed as a missionary by the United Presbyterian Church of Scotland. His father had been responsible for establishing the hospital at Jodhpur, Rajputana.

The family returned home to Perth, Scotland, where Duncan spent 4 years at a private school, before completing his education at Perth Academy. His father died in his youth. He lived with his mother at 12 Rose Terrace.^{ [2] } Despite his father's death, he won a scholarship, allowing him to continue his studies to university level.^{ [3] }

He then studied mathematics at the University of St Andrews in Fife, graduating MA in 1902. He then began as an assistant lecturer at the university. In 1905 he gained his doctorate (DSc) for his thesis, *Networks of the Plane in Absolute Geometry* and was promoted to lecturer. He continued teaching mathematics at St Andrews until 1915.^{ [4] }

In projective geometry the method of Cayley–Klein metrics had been used in the 19th century to model non-euclidean geometry. In 1910 Duncan wrote "Classification of geometries with projective metrics".^{ [5] } The classification is described by Daniel Corey^{ [6] } as follows:

- He classifies them into 9 types of plane geometries, 27 in dimension 3, and more generally 3
^{n}in dimension n. A number of these geometries have found applications, for instance in physics.

In 1910 Sommerville reported^{ [7] } to the British Association on the need for a bibliography on non-euclidean geometry, noting that the field had no International Association like the Quaternion Society to sponsor it.

In 1911 Sommerville published his compiled bibliography of works on non-euclidean geometry, and it received favorable reviews.^{ [8] }^{ [9] } In 1970 Chelsea Publishing issued a second edition which referred to collected works then available of some of the cited authors.^{ [10] }

Sommerville was elected a Fellow of the Royal Society of Edinburgh in 1911. His proposers were Peter Redford Scott Lang, Robert Alexander Robertson, William Peddie and George Chrystal.^{ [11] }

In 1912 he married Louisa Agnes Beveridge.

In 1915 Sommerville went to New Zealand to take up the Chair of Pure and Applied Mathematics at the Victoria College of Wellington.

Duncan became interested in honeycombs and wrote "Division of space by congruent triangles and tetrahedra" in 1923.^{ [12] } The following year he extended results to *n*-dimensional space.^{ [13] }

He also discovered the Dehn–Sommerville equations for the number of faces of convex polytopes.

Sommerville used geometry to describe the voting theory of a preferential ballot.^{ [14] } He addressed Nanson's method where *n* candidates are ordered by voters into a sequence of preferences. Sommerville shows that the outcomes lie in *n* ! simplexes that cover the surface of an *n*− 2 dimensional spherical space.

When his *Introduction to Geometry of N Dimensions* appeared in 1929, it received a positive review from B. C. Wong in the American Mathematical Monthly.^{ [15] }

Sommerville was co-founder and first secretary of the New Zealand Astronomical Society (1920). He was President of Section A of the Australasian Association for the Advancement of Science meeting, Adelaide (1924). In 1926 he became a fellow of the Royal Astronomical Society.

He died in New Zealand on 31 January 1934.

- 1914:
*The Elements of Non-Euclidean Geometry*, William P. Milne editor, Bell's Mathematical Series for Schools and Colleges, G. Bell & Sons. - The Elements of Non-Euclidean Geometry, link from University of Michigan Historical Math Collection.
- 1930:
*An Introduction to the Geometry of N Dimensions*, New York, E. P. Dutton, (Dover Publications edition, 1958) - 1933: Analytical Conics from Google Books
- 1934:
*Analytical Geometry of Three Dimensions.*Cambridge University Press.

**Harold Scott MacDonald** "**Donald**" **Coxeter**, was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.

**Discrete geometry** and **combinatorial geometry** are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

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In group theory and geometry, a **reflection group** is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections, it is a continuous group, not a discrete group, and is generally considered separately.

The **quarter cubic honeycomb**, **quarter cubic cellulation** or **bitruncated alternated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.

**George Bruce Halsted**, usually cited as **G. B. Halsted**, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and his many important translations. Especially noteworthy were his translations and commentaries relating to non-Euclidean geometry, including works by Bolyai, Lobachevski, Saccheri, and Poincaré. He wrote an elementary geometry text, *Rational Geometry*, based on Hilbert's axioms, which was translated into French, German, and Japanese.

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In mathematics, **convex geometry** is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.

In geometry, a **Schlegel diagram** is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.

**Henry Frederick Baker** FRS FRSE was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations, and Lie groups.

In geometry, a **flexible polyhedron** is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex.

In geometry, by Thorold Gosset's definition a **semiregular polytope** is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as *The Semiregular Polytopes of the Hyperspaces* which included a wider definition.

In geometry and combinatorics, a **simplicial**** d-sphere** is a simplicial complex homeomorphic to the

In elliptic geometry, two lines are **Clifford parallel** or **paratactic lines** if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions. Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, although the "lines" of elliptic geometry are geodesic curves and, unlike the lines of Euclidean geometry, are of finite length.

In mathematics, a sequence of *n* real numbers can be understood as a location in *n*-dimensional space. When *n* = 7, the set of all such locations is called **7-dimensional space**. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product.

**William Marshall Smart** was a 20th-century Scottish astronomer.

In geometry, the **Gram–Euler theorem**, **Gram-Sommerville, Brianchon-Gram** or **Gram relation** is a generalization of the internal angle sum formula of polygons to higher-dimensional polytopes. The equation constrains the sums of the interior angles of a polytope in a manner analogous to the Euler relation on the number of d-dimensional faces.

* Convex Polytopes* is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.

* Regular Figures* is a book on polyhedra and symmetric patterns, by Hungarian geometer László Fejes Tóth. It was published in 1964 by Pergamon in London and Macmillan in New York.

- ↑ Waterston, Charles D; Macmillan Shearer, A (July 2006).
*Former Fellows of the Royal Society of Edinburgh 1783–2002: Biographical Index*(PDF). Vol. II. Edinburgh: The Royal Society of Edinburgh. ISBN 978-0-902198-84-5. Archived from the original (PDF) on 4 October 2006. Retrieved 5 February 2011. - ↑ Perth Post Office Directory 1895
- ↑ "Duncan Sommerville - Biography".
- ↑
*Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*(PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 4 March 2016. Retrieved 6 August 2018. - ↑
*Proceedings of the Edinburgh Mathematical Society*28:25–41 - ↑ MR 550670
- ↑ D. Sommerville (1910) On the Need of a Non-Euclidean Bibliography,
*Report*of the British Association - ↑ G. B. Halsted (1912) "Duncan M.Y. Sommerville", American Mathematical Monthly 19:1–4, includes portrait, MR 1517626
- ↑ G. B. Mathews (1912) Bibliography of Non-Euclidean Geometry from Nature 89:266 (#2220)
- ↑ MR 270890
- ↑
*Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*(PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 4 March 2016. Retrieved 6 August 2018. - ↑
*Proceedings of the Royal Society of Edinburgh*43:85–116 - ↑ D. Sommerville (1924) "The regular divisions of space of
*n*dimensions and their metrical constants", Rendiconti del Circolo Matematico di Palermo 48:9–22 - ↑ D. Sommerville (1928) "Certain hyperspatial partitionings connected with preferential voting", Proceedings of the London Mathematical Society 28(1):368 to 82
- ↑ B.C. Wong (1931) "Recent publications", American Mathematical Monthly 38(5):286–7

- H.W. Turnbull (1934) "Duncan M. Y. Sommerville", Journal of the London Mathematical Society 9(4):316–18.
- B. A. Rosenfeld (1979) "The Works on Geometry of Duncan Sommerville",
*Istoriko-Matematicheskie Issledovania*, MR 550670.

- O'Connor, John J.; Robertson, Edmund F., "Duncan Sommerville",
*MacTutor History of Mathematics archive*, University of St Andrews - Obituary, Monthly Notices of the Royal Astronomical Society Vol. 95, pp. 330–331.
- C.J. Seelye (1974) Mathematics At Victoria In Retrospect, Notes for a talk to the Mathematics and Physics Society, from NZETC.

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