1. Poster Of Schlafli Ludwig Schläfli. lived from 1814 to 1895. Schläfli's polyhedra. Find outmore at http//math.ichb.ro/history/ Mathematicians/Schlafli.html. http://math.ichb.ro/History/Posters2/Schlafli.html

2. Poster Of Schlafli Ludwig Schläfli. died 105 years ago. 20th March 1895. Find out more at http//wwwhistory.mcs.st-andrews.ac.uk/history/Mathematicians/Schlafli.html. http://www.math.hcmuns.edu.vn/~algebra/history/history/Posters/320.html

3. Poster Of Schlafli Ludwig Schläfli. was born 186 years ago. 15th January 1814. Find out more athttp//wwwhistory.mcs.st-andrews.ac.uk/history/Mathematicians/Schlafli.html. http://www.math.hcmuns.edu.vn/~algebra/history/history/Posters/115.html

4. References For Schlafli Biographisches und Kulturhistorisches aus Briefen und Akten von ludwig Schläfli,Gesnerus 36 wwwhistory.mcs.st-andrews.ac.uk/history/References/schlafli.html. http://www-gap.dcs.st-and.ac.uk/~history/References/Schlafli.html

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/history/References/Schlafli.html

5. 4D Platonic Solids - Polytopes ludwig schlafli, a Swiss, is credited with discovering the regular polytopes in ndimensional space. http://westview.tdsb.on.ca/Mathematics/4DPlatonicSolids.html

Revised August 14, 1998 4D Platonic Solids - Polytopes From: rustybel@foothill.net (Russell Towle) Newsgroups: geometry.research Subject: 4D Platonic solids (Schlafli symbols) Date: 6 Aug 1998 DGoncz writes, ...What are these numbers? I think of a tetrahedron as having 4 faces each of which is a triangle. So the platonic solids would be I see the brackets above. They must have some meaning. Is it the set of something?... Ludwig Schlafli, a Swiss, is credited with discovering the regular polytopes in n-dimensional space. He did this ca. 1850. His notation is followed to this day. The five "Platonic solids" in 3-space, which are the only regular *convex* polyhedra, have Schlafli symbols as follows: "Polytope" being the generalization of polygon (a 2-polytope) and polyhedron (a 3-polytope), when we begin to try to characterize what makes a polytope "regular," several criteria become very important: The regular n-polytope must be bounded by regular (n-1)-polytopes; these are called "cells." The cells must be all equal.

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

7. Schlafli Biography of ludwig Schläfli (18141895) ludwig Schläfli. Born 15 Jan 1814 in Grasswil, Bern, Switzerland st- andrews. ac. uk/ history/ 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 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.

8. Schlafli Double Six This is about getting a glimpse of transcendence. I'm talking aboutthe schlafli Double 6 . ludwig schlafli was a mathematician. http://www.jackstrawsstudios.com/Archives/superam/Archives/schlafli/schlafli_art

Schlafli is Alive and Well and Living in a Hollywood Studio. I have been wondering for some time about when I was going to transcend this life as I know it. Wondering what impact reading books like Bart Kosko's Fuzzy Thinking and The Secret Life of Salvador Dali would have on me. I longed to transcend the clarity of linear black and white thinking and rise above into the fog of not only circular but into spherical thinking. This is about getting a glimpse of transcendence. I'm talking about the "Schlafli Double 6". Ludwig Schlafli was a mathematician. He lived and worked throughout the eighteen hundreds when information was transferred by the written word and delivered by horses and boats. He started out as a translator in the field of mathematics. When the French, Italians, Germans or Americans wanted to share information about mathematics, they took Ludwig along to translate for them. As a result he had the opportunity to learn from the leading mathematicians of his time. Imagine transferring such information from one genius to another. Imagine the skill he must have obtained in understanding new concepts. As a result of this, Schlafli developed some of math's most far reaching theories. Theories with names like The Theory of Manifold Continuity and Euclidean space Rn of n dimensions "Transcending Math", as it were. Schlafli was said by the experts in his field, "to have had the sad misfortune of those who are ahead of their time." Sadly even in the field of mathematics his main theories were "rejected (until after his death) by the academies of Vienna and Berlin because of their great length" . Although he laid the groundwork for the scientific explosions that followed his death.

9. Schlafli Portrait ludwig Schläfli. JOC/EFR August 2001 st andrews. ac. uk/ history/ PictDisplay/ schlafli. html http://www-history.mcs.st-and.ac.uk/history/PictDisplay/Schlafli.html

JOC/EFR August 2001 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Schlafli.html

10. Polytopes An introduction to the subject of regular polytopes (generalizations of polygons and polyhedra). With Category Science Math Geometry Polyhedra and Polytopes...... Some examples are given below, labeled with their schlafli symbols (ludwig schlafliwas a pioneer in the study of regular polytopes, and devised these symbols http://personal.neworld.net/~rtowle/Polytopes/polytope.html

11. 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 n http://mathforum.com/epigone/geometry-research/claxspotar/v03102801b1ef701add79@

4D Platonic solids (Schlafli symbols) by Russell Towle reply to this message post a message on a new topic Back to messages on this topic Back to geometry-research Subject: 4D Platonic solids (Schlafli symbols) Author: rustybel@foothill.net Date: The Math Forum

12. Polytopes examples are given below, labeled with their schlafli symbols (ludwig schlafli was a pioneer in the study of regular 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

13. Java Examples regular solids in three dimensions. ludwig schlafli proved in 1901that there are exactly six regular solids in four dimensions. http://www.uoregon.edu/~koch/java/FourD.html

Java Examples Below is a program which can display all possible three and four dimensional regular solids. Euclid proved around 200 B.C. that there are exactly five regular solids in three dimensions. Ludwig Schlafli proved in 1901 that there are exactly six regular solids in four dimensions. Schlafli also proved that the only regular solids in dimensions greater than or equal to five are the generalized tetrahedron, cube, and octahedron. Richard Koch Department of Mathematics University of Oregon

14. Java Examples are exactly five regular solids in three dimensions. ludwig schlafli proved in 1901 that there are exactly six regular http://darkwing.uoregon.edu/~koch/java/FourD.html

Java Examples Below is a program which can display all possible three and four dimensional regular solids. Euclid proved around 200 B.C. that there are exactly five regular solids in three dimensions. Ludwig Schlafli proved in 1901 that there are exactly six regular solids in four dimensions. Schlafli also proved that the only regular solids in dimensions greater than or equal to five are the generalized tetrahedron, cube, and octahedron. Richard Koch Department of Mathematics University of Oregon

15. HyperSolids In 1901, ludwig schlafli proved that there are exactly six regular solids in fourdimensions, and only three regular solids in each dimension five or higher. http://www.uoregon.edu/~koch/hypersolids/hypersolids.html

HyperSolids About HyperSolids: The Greeks proved that there are exactly five regular solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In 1901, Ludwig Schlafli proved that there are exactly six regular solids in four dimensions, and only three regular solids in each dimension five or higher. The program HyperSolids can show and rotate all regular solids in dimensions 3 and 4, and show the ``faces'' of these solids. The program is distributed under the GPL public license, and thus free. To obtain it using either Internet Explorer or OmniWeb, click on the link "Program" below and hold the mouse button down until a dialog appears. Choose the option of saving the file "hypersolids.tar.gz" to disk. Move the file to your home directory if it is not already there. Open a terminal window and type gzip -d hypersolids.tar.gz and then tar -xf hypersolids.tar and then rm hypersolids.tar Move the resulting program to the Applications directory. Program Source Code Richard Koch Department of Mathematics University of Oregon Eugene, Oregon 97403

16. Java Examples are exactly five regular solids in three dimensions. ludwig schlafli proved in 1901 that there are exactly six regular http://darkwing.uoregon.edu/~koch/java/FourDSmall.html

Java Examples Below is a program which can display all possible three dimensional regular solids, and all but two of the regular four dimensional solids. Euclid proved around 200 B.C. that there are exactly five regular solids in three dimensions. Ludwig Schlafli proved in 1901 that there are exactly six regular solids in four dimensions. Schlafli also proved that the only regular solids in dimensions greater than or equal to five are the generalized tetrahedron, cube, and octahedron. Richard Koch Department of Mathematics University of Oregon

17. Schlafli ludwig Schläfli first studied theology, then turned to science. Heworked for ten years as a school teacher in Thun. schlafli.html. http://math.ichb.ro/History/Mathematicians/Schlafli.html

18. Topology dodecahedron. In 1901, ludwig schlafli showed that there are onlysix regular polychora 1 , or polytopes in hyperspace. One may http://temporal_science.tripod.com/introduction/special1.htm

A polytope is any convex, geometric figure; they are found in all the dimensions. There are an infinite number of regular polytopes in two dimensions, in which they are known as polygons. Around 200 B.C., Euclid's extraordinaire proved that there are only five regular polyhedra, polytopes in three spatial dimensions: the tetrahedron, cube, octahedron, icosahedron, and the dodecahedron. In 1901, Ludwig Schlafli showed that there are only six regular polychora , or polytopes in hyperspace. One may believe with more dimensions, there are more regular polytopes. However, the fourth dimension is as complex as it gets. This is because each dimension's increasing "freedom" nullifies its complexity. In fact, all higher dimensions each only have three regular polytopes. A Geometric Approach Differentiating between figures in the first few dimensions can be quite a task. One of the simplest ways to view higher dimensions is by slicing. A simple three-dimensional figure, a cube, can be sliced parallel to its sides to give a square, a two-dimensional figure. A hypercube, the cubic equivalent in four dimensions, therefore can be sliced to give cubes. When you take a sheet of paper and look at it from the top, you see a rectangle. If you turn in on its side, you see a line. Between the rectangle and line, you see rhombuses, parallelograms, and other quadrilaterals.

19. NRICH | October 2000 | Article | Classifying Solids Using Angle Deficiency edges of each polygon meeting at a vertex of a regular or semiregular tessellationor solid, was devised by the Swiss mathematician ludwig schlafli (1814-1895 http://www.nrich.maths.org.uk/mathsf/journalf/oct00/art2/

Classifying Solids using Angle Deficiency Warwick Evans (28 September 1939 to 7 July 2000) NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News This article is based on a mathematics workshop for 12 year olds given by Warwick Evans and Alison Clark Jeavons in Cambridge in February 2000. The photos were taken at the ATE / NRICH Mathematics Superweek holiday for 10 to 13 year olds at Southam in August 2000. Use your mouse to have a close look at each of these, the platonic solids

20. Www.math.niu.edu/~rusin/known-math/98/sliced_cake Heather M. Shannon Coxeter Introduction to Geometry (second edition, p.183)has the following quote attributed to ludwig schlafli (18141895) If i http://www.math.niu.edu/~rusin/known-math/98/sliced_cake

From: Per Erik Manne Newsgroups: sci.math Subject: Re: Find a formula for the series : 1, 2, 4, 8, 16, 31 Date: Tue, 12 May 1998 14:46:30 +0200 H. M. Shannon wrote: > > In article , > Per Erik Manne

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