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XXIX.—Semi-regular Networks of the Plane in Absolute Geometry

Published online by Cambridge University Press:  06 July 2012

Duncan M. Y. Sommerville
Duncan M. Y. Sommerville
Affiliation:
University of St Andrews
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Extract

§ 1. The networks considered in the following paper are those networks of the plane whose meshes are regular polygons with the same length of side.

When the polygons are all of the same kind the network is called regular, otherwise it is semi-regular.

The regular networks have been investigated for the three geometries from various standpoints, the chief of which may be noted.

1. The three geometries can be treated separately. For Euclidean geometry we have then to find what regular polygons will exactly fill up the space round a point. For elliptic geometry we have to find the regular divisions of the sphere, or, what is the same thing, the regular polyhedra in ordinary space. The regular networks which do not belong to either of these classes are then those of the hyperbolic plane.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 41 , Issue 3 , 1906 , pp. 725 - 747
Copyright
Copyright © Royal Society of Edinburgh 1906

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References

page 725 note * See V. Schlegel, “Theorie der homogenen zusammengesetzten Raumgebilde,” Nova Acta, Bd. xliv., Nr. 4, 1883.

page 725 note † W. Dyck, “Gruppentheoretische Studien,” Math. Annalen, xx. 1–44 (1882), and W. Burnside, “Theory of Groups,” ch. xii., xiii. Also Klein and Fricke, “Theorie der elliptischen Modulfunctionen.” (For these references I am indebted to the referee.)

page 742 note * A correspondence between the regular polyhedra and certain general classes of polyhedra was considered by C. Jordan, “Recherches sur les polyedres,” Comples Rendus, lx. 400–403, lxi. 205–208, lxii. 1339–1341, 1865–66.

page 744 note * If two opposite groups of five triangles are replaced by pentagons we get the simple type 3310512 (like fig. 25).

page 744 note † If both groups are replaced we get the simple type 423812 (like fig. 24).

page 745 note * Asymmetrical. Two enantiomorphic forms, and .

page 746 note * See also T. L. Heath, The Works of Archimedes (Camb. 1897), p. xxxvi.

page 746 note † Collectio, lib. v. pars 2.

page 746 note ‡ Harmonices Mundi (1619), lib. ii. pp. 61–65.

page 746 note § Sammlung geometrischer Aufgaben (Berlin, 1805–7), vol. ii. pp. 139–185.

page 746 note ∥ Elemente der Mathematik (1862), Bd. ii., Buch v. § 7.

page 746 note ¶ Published 1867, under the title “Des Polyèdres semi-réguliers, dits solides d'Archimède,” Mém. de la Soc. des Sciences phys. et nat. de Bordeaux, v. 319–369.

page 747 note * Loc. cit., pp. 51–55.

page 747 note † Keppler and Baltzer, loc. cit.; Meier Hirsch, loc. cit., pp. 186–196; J. H. L. Müller, Trigonometrie (1852), p. 345.