1. Schlafli Double Six This is about getting a glimpse of transcendence. I'm talking about the "Schlafli Double 6". Ludwig Schlafli was a mathematician. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

2. Legacy LUDWIG SCHLAFLI Bern, Switzerland 18141895 1863 - Full professor University of Bern http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. 4D Platonic Solids (Schlafli Symbols) By Russell Towle Is it the set of something? Ludwig Schlafli, a Swiss, is credited with discovering the regular polytopes in ndimensional space. He did this ca. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. Poster Of Schlafli Ludwig Schläfli. lived from 1814 to 1895. Schläfli s work was in geometry,arithmetic and function theory. Mathematicians/Schlafli.html. http://www-groups.dcs.st-and.ac.uk/~history/Posters2/Schlafli.html

lived from 1814 to 1895 Find out more at http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Schlafli.html

5. Schlafli, Ludwig (1814-1895) Schl fli, Ludwig (18141895) A German mathematician whose worked centered on geometry, arithmetic and the theory of functions. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Famous Mathematicians With An S Schafer Robert Schatten Juliusz Schauder Henry Scheffe Georg Scheffers Wilhelm Schickard Ludwig Schlafli Oscar Schlomilch Erhard http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Polytopes Some examples are given below, labeled with their Schlafli symbols (Ludwig Schlafli was a pioneer in the study of regular polytopes, and http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. Collected Works S-T-U-V Schlafli, Ludwig, 18141895 Gesammelte mathematische Abhandlungen / Hrsg. vom Steiner-Schlafli-Komitee der Schweizerischen Naturforschenden http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Food For Thought Biographies Schlafli, Ludwig (Swiss mathematician) 18141895. Schlafly, Phillis n e Steward (American activist) b.1924 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Ciberoteca - Ndice De Autores Schlafli, Ludwig Schlagintweit, Emil Schlegel, August Wilhelm von. Schlegel, Nikolai Schleich, Joachim Schleussner, Ellie http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Java Examples Ludwig Schlafli proved in 1901 that there are exactly six regular solids in four dimensions. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Schlafli Ludwig Schläfli first studied theology, then turned to science. http//wwwhistory.mcs.st-andrews.ac.uk/history/Mathematicians/Schlafli.html. http://202.38.126.65/mirror/www-history.mcs.st-and.ac.uk/history/Mathematicians/

13. Schlafli, Ludwig (1814-1895) Schläfli, ludwig (18141895). A German mathematician whose worked centered ongeometry, arithmetic and the theory of functions. http://www.daviddarling.info/encyclopedia/S/Schlafli.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Schläfli, Ludwig (1814-1895) A German mathematician whose worked centered on geometry, arithmetic and the theory of functions. He made an important contribution to non-Euclidean geometry when he proposed that spherical three-dimensional space could be thought of as the surface of a hypersphere in Euclidean four-dimensional space. Schläfli started out as a schoolteacher and amateur mathematician. He was also was an expert linguist and spoke many languages, including Sanskrit. In 1843 he served as a translator for the great mathematicians Jakob Steiner , Karl Jacobi , and Peter Dirichlet during their visit to Rome and learned a great deal from them. Ten years later he became professor of mathematics at Bern. However, his true importance was only appreciated following the publication of his magnum opus Theory of Continuous Manifolds in 1901, several years after his death.

14. Schlafli Symbol A notation, devised by ludwig Schläfli, which describes the number of edges ofeach polygon meeting at a vertex of a regular or semiregular tiling or solid http://www.daviddarling.info/encyclopedia/S/Schlafli_symbol.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Schläfli symbol A notation, devised by Ludwig Schläfli , which describes the number of edges of each polygon meeting at a vertex of a regular or semi-regular tiling or solid. For a Platonic solid p q p is the number of sides each face has, and q is the number of faces that touch at each vertex. BACK TO TOP var site="s13space1234"

15. Schlafli Biography of ludwig Schläfli (18141895) The URL of this page ishttp//www-history.mcs.st-andrews.ac.uk/Mathematicians/schlafli.html. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schlafli.html

Born: 15 Jan 1814 in Grasswil, Bern, Switzerland Died: 20 March 1895 in Bern, Switzerland Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing first studied theology, then turned to science. He worked for ten years as a school teacher in Thun. During this period he studied advanced mathematics in his spare time. Steiner Jacobi and Dirichlet Bessel function and of the gamma function . He also worked on elliptic modular functions. Theory of continuous manifolds was published in 1901 after his death and only then did his importance become fully appreciated. He received the Steiner Prize from the Berlin Academy for his discovery of the 27 lines and the 36 double six on the general cubic surface. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites SuperAm Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index History Topics Societies, honours, etc.

16. Schlafli ludwig Schläfli first studied theology, then turned to science. The URL of this pageis http//wwwhistory.mcs.st-andrews.ac.uk/Mathematicians/schlafli.html. http://www-history.mcs.st-and.ac.uk/Mathematicians/Schlafli.html

Born: 15 Jan 1814 in Grasswil, Bern, Switzerland Died: 20 March 1895 in Bern, Switzerland Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing first studied theology, then turned to science. He worked for ten years as a school teacher in Thun. During this period he studied advanced mathematics in his spare time. Steiner Jacobi and Dirichlet Bessel function and of the gamma function . He also worked on elliptic modular functions. Theory of continuous manifolds was published in 1901 after his death and only then did his importance become fully appreciated. He received the Steiner Prize from the Berlin Academy for his discovery of the 27 lines and the 36 double six on the general cubic surface. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites SuperAm Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index History Topics Societies, honours, etc.

17. Polytopes ludwig schlafli, a Swiss, made his great advances in the study of higher spacein the middle of the 19th century. His work went largely unnoticed, http://home.inreach.com/rtowle/Polytopes/polytope.html

Polygons, Polyhedra, Polytopes Polytope is the general term of the sequence, point, segment, polygon, polyhedron, ... So we learn in H.S.M. Coxeter 's wonderful Regular Polytopes (Dover, 1973). When time permits, I may try to provide a systematic approach to higher space. Dimensional analogy is an important tool, when grappling the mysteries of hypercubes and their ilk. But let's start at the beginning, and to simplify matters, and also bring the focus to bear upon the most interesting ramifications of the subject, let us concern ourselves mostly with regular polytopes. You may wish to explore my links to some rather interesting and wonderful polyhedra and polytopes sites, at the bottom of this page. Check out an animated GIF (108K) of an unusual rhombic spirallohedron. Yes, we shall be speaking of the fourth dimension, and, well, the 17th dimension, or for that matter, the millionth dimension. We refer to Euclidean spaces, which are flat, not curved, although such a space may contain curved objects (like circles, spheres, or hyperspheres, which are not polytopes). We are free to adopt various schemes to coordinatize such a space, so that we can specify any point within the space; but let us rely upon Cartesian coordinates, in which a point in an n -space is defined by an n -tuplet of real numbers. These real numbers specify distances from the origins along

18. Polytopes: Shadows Of Hypercubes The discovery of the regular polytopes is rightly credited to one ludwig schlafli,a Swiss, about 1850. He considered the vectors to the vertices of regular http://home.inreach.com/rtowle/Polytopes/Chapter2/Polytopes2.html

Polytopes, Continued an exploration of themes unifying the theory of higher space Although this section is intended primarily for those already familiar with the broad outlines of the theory of regular polytopes, let us recall a few salient points: In n-dimensional space, there are always the three primitive polytopes: the simplex, the cross polytope, and the hypercube. The simplex (generalization of the triangle and tetrahedron) is the simplest polytope (with n-dimensional content) which can be constructed in an n-space. An n-simplex is bounded by (n-1)-simplexes, and has an (n-1)-simplex for its vertex figure. Regular simplexes are self-dual. The cross polytope (generalization of the square and Platonic octahedron) is named after the "cross" or frame of mutually perpendicular Cartesian coordinate axes which can be constructed in an n-space. There are just n such axes, and points equidistant from the origin along each axis, in both directions, are the vertices of a cross polytope. Cross polytopes are bounded by regular (n-1)-simplexes and have for vertex figures (n-1)-cross polytopes. The dual of an n-dimensional cross polytope is an n-cube. Hypercubes (generalizations of the square and cube) are specially simple zonotopes. They are bounded by (n-1)-cubes and have for vertex figure an (n-1)-simplex. Taken as zonotopes, they are "determined" by a star of n mutually perpendicular equal vectors. Thst is, they are "hyper-solids of translation" along those n vectors, "traced out" by a series of orthogonal translations. The dual of an n-cube is an n-dimensional cross polytope. They close-pack to fill the n-space, and moreover, when of unit edge length, the hypercube is called a measure polytope, and provides the measure for the unit of n-dimensional content.

19. References For Schlafli ludwig Schläfli, Gesnerus 36 (34) (1979), 277-299. JOC/EFR December 1996, The URLof this page is http//www-history.mcs.st-andrews.ac.uk/References/schlafli. http://www-groups.dcs.st-andrews.ac.uk/history/References/Schlafli.html

Version for printing Biography in Dictionary of Scientific Biography (New York 1970-1990). Books: Elemente der Mathematik Articles: Mitt. Naturforsch. Ges. Bern 1942 Gesnerus Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/References/Schlafli.html

20. Bibliography schlafli, ludwig, 18141895, Gesammelte mathematische Abhandlungen / hrsg.vom Steiner-schlafli-Komitee der Schweizerischen Naturforschenden Gesellschaft http://www.library.cornell.edu/math/bibliography/display.cgi?start=S&

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