Volume 4, Number 4A  -  January 1999 Discussions

27 January 1999 Vol. 4, No. 4A THE MATH FORUM INTERNET NEWS - JANUARY 1999 DISCUSSIONS This special issue of the Math Forum's weekly newsletter highlights interesting conversations taking place during January of 1999 on Internet math discussion groups. For a full list of these groups with links to topics covered and information on how to subscribe, see: http://mathforum.org/discussions/ Replies to individual discussions should be addressed to the appropriate group rather than to the newsletter editor. ______________________________ + ______________________________ JANUARY SUGGESTIONS: CALC-REFORM - a mailing list hosted by e-MATH of the American Mathematical Society (AMS) and archived at http://mathforum.org/epigone/calc-reform/ - Revisionist Simpson's Rule (10 Jan. 1999) http://mathforum.org/epigone/calc-reform/flegrolpex/ "Some recent calculus books approach Simpson's rule by looking at the ratio: (E-T)/(E-M) where E is the exact area under some curve and T, M are the trapezoidal and midpoint estimates respectively. Experimentally one sees that this ratio is (usually) very close to -2. Setting it equal to -2 and solving for E gives: E = (2M+T)/3. Thus, the exact answer is very nearly a certain weighted average of the trapezoidal and midpoint rules. Writing this out explicitly results in Simpson's formula. When I first saw this I thought it was kind of neat; I even had my students 'discover' it using a spreadsheet lab to experiment with the ratio. However, after thinking about it for a while, I decided that, yes, it was cute, but perhaps a bit too cute.... I have decided to go back to the straightforward motivation that Simpson used, and give the error ratio guess *afterward*, as an interesting computer lab: I didn't feel right about having the students 'discover' it. What do you think?" - Mark Bridger "Doug Kuhlmann asked about Richardson Extrapolation, so I'll sketch the main idea...." - David A. Olson The conversation continued, spreading to: - (E-T)/(E-M) (11 Jan. 1999) http://mathforum.org/epigone/calc-reform/frahprimpdwang/ and - Elegance is not the issue (11 Jan. 1999) http://mathforum.org/epigone/calc-reform/yerdsnoxplul/ ______________________________ + ______________________________ GEOMETRY-PUZZLES http://mathforum.org/epigone/geometry-puzzles/ - Tori...Torae composed of straight lines (9 Jan. 1999) http://mathforum.org/epigone/geometry-puzzles/sermfloozou/ "Anyone know where I can find a picture of a one-holed torus defined by the minimum number of straight lines? I guess the lines would have to form a minimum of seven planar polygons, but the polygons would not have to be convex." - Bob Underwood "The example I know of with seven faces has seven (non-convex) hexagons (each meeting all the other faces) and 21 edges. The dual - with seven vertices - also has 21 edges. The standard torus formed from a ring of three triangular prisms has few edges - 18 edges and 9 quadrilateral (actual trapezoidal and rectangular) faces. Anyone have a smaller number of EDGES?" - Walter Whiteley "...I see a clear diagrammatic illustration of the Szilassi polyhedron, a 7-edged toroid of seven hexagons (6 non-convex, 1 convex) on p. 28 of Scientific American, Nov, 1978. It has 21 edges (as I count them from the diagram)." - Mary Krimmel "John Conway has proposed the problem of finding "holyhedra," meaning every face contains a hole. Here are some constructions of mine..." - Antreas Hatzipolakis ______________________________ + ______________________________ GEOMETRY-PRE-COLLEGE http://mathforum.org/epigone/geometry-pre-college/ - Hinge Theorem (21 Jan. 1999) http://mathforum.org/epigone/geometry-pre-college/swontanplil/ "If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second." - Antreas P. Hatzipolakis A conversation that began on January 21 with posts by John Conway and Antreas Hatzipolakis, stating the theorem and its converse, discussing how the name came about, and providing a proof. ______________________________ + ______________________________ SCI.MATH, a discussion group focused on general and advanced mathematics that can be read as a Usenet newsgroup or on the Web: http://mathforum.org/epigone/sci.math/ - Cutting equilateral triangle (20 Jan. 1999) http://mathforum.org/epigone/sci.math/vumjimpkoi/ "How does one prove that an equilateral triangle cannot be subdivided into finitely many smaller equilateral triangles, such that no two are congruent? Is it known if there exists a tiling of the plane by pairwise noncongruent equilateral triangles?" - David Radcliffe "It's not possible. This was originally discussed in an article that I discuss at my site this week." - Ed Pegg Jr, http://www.mathpuzzle.com "See http://www.seanet.com/~ksbrown/kmath153.htm" - David Eppstein ______________________________ + ______________________________ We hope you will find these selections useful, and that you will browse and participate in the discussion group(s) of your choice. CHECK OUT OUR WEB SITE: The Math Forum http://mathforum.org/ Ask Dr. Math http://mathforum.org/dr.math/ Problems of the Week http://mathforum.org/pow/ Internet Resources http://mathforum.org/~steve/ Teacher2Teacher http://mathforum.org/t2t/ Discussion Groups http://mathforum.org/discussions/ Join the Math Forum http://mathforum.org/join.forum.html Send comments to the Math Forum Internet Newsletter editors

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