### geometry.college - brief descriptions

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#### 1996

1. Penrose Tiling Applet (Geert-Jan van Opdorp) 09/16/96
Announcing a java applet to play with 'Kites' and 'Darts.' [Penrose Tiles]

2. Area of Sphere = 4 * (Area of Circle) (Daniel A. Asimov) 06/25/96
On the Usenet newsgroup "sci.math," someone recently inquired why the area of a sphere is equal to 4 times the area of a disk of the same radius. Using calculus, it is of course straightforward to prove the area formulas area(disk) = pi r^2 and area(sphere) = 4 pi r^2, whence the result follows. But does anyone know a more direct way to see that area(sphere) / area(disk) = 4 ??? [Chinese mathematics, Mikami, Joseph Needham, Frank Swetz]

3. Klein Bottle (GUANSHEN REN) 06/14/96
Could anyone give the reference of Franklin's paper on the proof that only six colors are needed to color every map on the Klein bottle in 1934? [Rouse Ball's Mathematical Recreations and Essays; Geometry and the Imagination]

4. Generalized Napolean's Theorem (John Conway) 04/03/96
I have to make a presentation of Feuerbach's theorem soon, which has a simple and elegant proof using inversion. I'd like to be able to show a Euclidean proof of the theorem for contrast.

5. Professional Development Symposium (Celia L. Adair ) 03/26/96
Announcing a Symposium on Professional Development Strategies for Middle and Secondary School Mathematics Teachers.

6. Test for Congruence (Joe Seeley) 03/13/96
Do you know of a way to tell if two arbitrary polygons are congruent? Is there a better way than brute force?

7. Geometry and Architecture (Tatiana Aburto) 03/09/96
Could anybody tell me where to find information regarding this geometry and architecture? [William Blackwell, Geometry in Architecture]

8. Two-Sided Regular Polygons (William Blackwell) 02/29/96
A finite straight line has two sides and can be considered a regular n-gon where n = 2. (This line is also a rectangle with length, but 0 width, an ellipse with one axis = 0, an isosceles triangle with base = 0, and so on. In every case it has two sides.) Is there a correct "gon" term (besides "line segment") for this shape? [dihedron, digon, bigon]

9. "m" for Slope (TBRUMA) 02/16/96
Why do mathematicians let "m" stand for slope? [modulus]

10. 3D isometries (Andre Deschenes) 01/24/96
Given two isometric solids, what isometry maps the first onto the second? [Coxeter, Regular Polytopes; George Martin, Transformation Geometry; Three-Dimensional Symmetry; James R. Smart, Modern Geometries]

11. Given: Rectangle ABCD (Bill O.) 01/11/96
GIVEN: Rectangle ABCD with P= any point on AB PE is perpendicular to BD, PM is perpendicular to AC, AN is perpendicular to BD. PROVE: PE+PM=AN

#### 1995

1. Math movies and videos (Heidi Burgiel ) 12/12/95
What's a good source of mathematical movies and videos? What companies or organizations would supply such things? ["An Annotated Bibliography of Films & Videotapes for College Mathematics," edited by David Schneider / Great Media Company catalog / The International Film Bureau]

2. Karl Brandon Mollweide (1774-1825) (Guanshen Ren) 12/07/95
What's the first name of a German mathematican, Mollweide, who discovered the Mollweide's formulas: (b-c)/a = (sin((B-C)/2))/(cos A/2), (two more similar ones) for a triangle ABC with sides a, b and c? [Karl Brandon Mollweide / Clark University math history WWW archive (chronology) ]

3. Golden ratio/golden rectangle (Tierney McCarty) 10/13/95
Does anybody have any information on the golden rectangle or the golden ratio? [Jay Kappraff's _Connections_ / The Walt Disney cartoon "Donald Duck in Mathmagic Land"]

4. Napier's Analogies (GUANSHEN REN) 08/28/95
Does any one know any references to a original proof of the Newton's Formulars: For any plane triangle ABC with a, b and c the opposite sides, (b+c)/a = (cos((B-C)/2))/sin(A/2); (c+a)/b = (cos((C-A)/2))/sin(B/2); (a+b)/c = (cos((A-B)/2))/sin(C/2) ?[I believe that these formulae are the ones that were traditionally known as "Napier's Analogies" (not Newton's!).]

5. Set of Circular Discs (John Conway) 08/09/95
A set of circular discs (let's call them "coins") has total area A. Is it always true that they can be fitted into another disc ("the arena") of area 2A?

6. How to Fill N-Dimensional Space with Hoops (Evelyn Sander) 05/26/95
This article describes research results of Daniel Asimov, of NASA Ames Research Center in Mountain View, California. It gives an example of an open region in three-space continuously filled with hoops. It then explains the higher dimensional generalizations of continuous families of hoops. Figures are in ftp:mathforum.org/pictures/articles/khoops or read the WWW version at http://www.geom.umn.edu/docs/forum/hoops_links/khoops.html.

7. Banchoff at the Geometry Center (Bob Hesse) 05/23/95
A description of the work of Prof. Thomas Banchoff (Brown Univ.) at the Geometry Center, and his WWW home page.

8. Lotka-Volterra Equations (Evelyn Sander) 05/02/95
Lotka-Voterra Equations Through Computer Visualization. Population growth - Mary Lou Zeeman, mathematics professor and Drew LaMar, undergraduate, both at the University of Texas, San Antonio, are looking at the question of long term behavior using dynamical systems theory.

9. Angle Trisection (Bob Hesse) 04/14/95
Why tell people it is impossible to trisect an angle via straightedge and compass? Instead we could say it is possible to trisect an angle, just not with a straightedge and a compass. [Archimedes' use of a marked straightedge / Quadratrix of Hippias / Tomahawk]

#### 1994

1. Solving for point of intersection (JK) 12/20/94
Given a circle (0,0) with a radius of 1 and a line segment, given to you in the form of two points (x1,y1 and x2,y2), knowing that the distance between the points is at least the diameter of the circle, how can you tell how many times the line segment crosses the circle (0,1, or 2) -- or, put another way, does the line cross the circle at any point? [simple test]

2. Geometry of the industrial landscape (Brian Hayes) 12/07/94
Request for suggestions: "I'm in the middle of writing a "Field Guide to the Industrial Landscape," a book that's meant to identify and explain all the manmade elements of the modern landscape--everything from smokestacks to telephone poles--in the same way that nature guides introduce birds and trees and sea shells. I want to include a "sidebar" in the book discussing some of the ways that human artifacts differ geometrically from natural features, and I would like to list some of the geometrically interesting forms you might hope to see if you keep your eyes open in an urban or industrial setting. For example, there is the catenary curve of the overhead power line and of the suspension-bridge support cable. Parabolic surfaces are conspicuous in backyard satellite-dish antennas and other focusing reflectors. The natural-draft cooling towers at many nuclear power stations have a hyperboloidal form." [roofs, geodesic domes / most polyhedra not common in nature]

3. Some Applications of Virtual Reality (Bob Hesse) 12/02/94
Eighteen professors from five departments decide to work together and submit a request for a virtual reality system. The administration approves the proposal, provided that the virtual reality system is put to use in the classroom. The faculty acquire the funds to purchase an SGI Onyx 2 Reality Engine and 10 SGI Indigos. The above scenario is not some introduction to a John Grisham suspense novel, but a real story at Clemson University. . . .

4. Symmetrics groups (Mary K. Hannigan) 11/08/94
Request for information about transformation of geometry, way of teaching isometries in secondary school in different countries in the world [book: Symmetries of Culture / 17 wallpaper groups/ fabrics and pottery]

5. College course on conics and inversion (Dave Wilson) 11/01/94
Design of a course - suggestions solicited [access to Sketchpad / imagery and visualisation / hyperbolic and elliptic or spherical geometry tie in well with inversive geometry / Peaucellier cell / stereographic projection]

6. Secondary Statistics List (J. Chappel) 10/19/94
Anyone interested in a K-12 Statistics list? [EdStat-L list / secondary level]

7. HyperGami -- A system for designing paper sculptures (Mike Eisenberg) 10/09/94
Experimental "programmable application" based on the Scheme programming language [middle school and older, mathematicians, artists, designers / design and decorate two-dimensional folding nets on the screen for folding into a variety of three-dimensional shapes / polyhedra: Platonic and Archimedean solids / origami nets]

8. Viewing Four-dimensional Objects in Three Dimensions (Bob Hesse) 09/06/94
Given that humans only visualize three dimensions, how is it possible to visualize four dimensional, or higher, objects? [Flatland / polytopes (polyhedra and polygons / projection - stereographic / 'cut-throughs' and 'fold-downs' / seeing only two 2-dimensional pictures in which corresponding points differ by a horizontal "parallax" / perceptual psychology / vertical disparities]

9. New Group for GSP, Cabri, etc.? (Stephen Weimar) 09/02/94
Forum plan to add a new newsgroup to focus on "dynamic" geometry software--to facilitate the discussion of teaching with such software, notification of related resources, user support, and the exchange of sketches and drawings. As with all Forum newsgroups, it would also be offered as a mailing list. [any way of posting Sketchpad modules? / tell how to send and capture sketches]

10. Learning & Mathematics (Polya) (Sarah Seastone) 08/16/94
An approach to teaching math based on common-sense questions that would naturally occur to an experienced problem-solver. [10 commandments for teaching / baseball, Barbie dolls, and reality problems / classroom visits by real mathematicians / the complexity of interest]

11. World's Largest Icosahedron (Bob Hesse) 08/08/94
Recently the Geometry Center was the construction site of the world's largest icosahedron, a twenty-sided polygon whose sides are equilateral triangles. This one was constructed with triangles having sides approximately 37" long. Sixth through eighth graders in a summer program run by the Special Projects Office of the University of Minnesota decorated the triangles. [the length from vertex to opposing vertex of an icosahedron is equal to 5^(1/4)*tau^(1/2)*s where tau is the golden ratio and s is the side length... proof]

12. Learning & Math: Metacognition (Schoenfeld) (Dan Hirschhorn) 07/12/94
"What metacognition is, why it's important, and what to do about it -- all in clear language that we can understand." Reflecting on how we thinkthrough a discussion of how a problem was solved and what it was about, orwhere and why a difficulty occurred in the process of problem solving. [reconsidering assumptions / discussing ideas in the classroom / army buses and remainders]

13. Learning & Math: Everyday Situations (Lesh) (K. Ann Renninger) 07/11/94
"If students are provided with everyday situations for practicing and learning the important uses of mathematics, they will develop such skills as "making inferences, evaluating reasonableness of results... [and] using references to 'look up' what they need to know." [real life mathematical situations / is more math learned if the subject is of interest? / electronic networks and software / the abstract beauty and power of math / practical problems]

14. Learning & Math: Introduction (Gene Klotz) 07/06/94
Summaries and reactions - some seminal articles dealing with how students learn mathematics.

15. Geomview (Bob Hesse) 07/01/94
What is the most requested software program created at the Geometry Center? [interactive program for viewing and manipulating geometric objects / allows several objects to interact with each other / how to construct three-dimensional objects using Mathematica and view them with Geomview / ftp archive is "geom.umn.edu" / why created? / future plans for the program / modules: "Hinge" & "Interactive Hyperbolic Flythrough"]

16. Descriptive Geometry Info? (John Conway) 05/04/94
Descriptive geometry is the geometry of "plans and elevations", which was formalized by Gaspard Monge at the beginning of the 19th century. [book: Machine Interpretation of Line Drawings / Computer Aided Design (CAD) programs / unsolved geometry problems related to plane drawings, possible constraints, and 'realizability']

17. Reference for 2 Theorems? (Michelle Manes) 02/04/94
(1) Any two rectilinear figures in the plane that are equal in area can be dissected into each other using a finite number of cuts. (2) The theorem that the above is *not* true in R3.[equidecomposability problems / two dimensions: bolyai-gerwien-wallace theorem / three dimensions: Dehn's theorem / Banach-Tarski Paradox / Hilbert / cut polygon into triangles]

#### 1993

1. Squaring the Circle One (Evelyn Sander) 12/21/93
A proof that given an arbitrary circle, it is impossible to construct a square of the same area using only straight edge and compass. [classical Greek problem of squaring a circle / figures are available by anonymous ftp from mathforum.org in file: /pictures/articles/squaring.the.circle/circle.eps]

2. 4D Course Outline (Allen Shepard) 11/09/93
Outline of a course on the fourth dimension given summers at the Hampshire College Summer Studies in Mathematics program for gifted high school students. The underlying principle of this course is to involve students with as much geometric visualization as possible, using topics that are intended to add to their appreciation of more algebraically-oriented material they are likely to be exposed to in the future.

3. Sharkovskii's Theorem (Evelyn Sander) 11/01/93
Knowing one periodic point can indicate the existence of many other periodic points [Li & Yorke, "Period three implies chaos" / strange ordering of integers / continuity]

4. Image Homotopy (Davide P. Cervone) 10/29/93
Request for the relations for the image homotopy semi-group. [defining relations / relevant papers / What IS a tight immersion? / clarification of definition of regular homotopy]

5. Course for Prospective Teachers (Dan Hirschhorn) 10/08/93
If you had only a 1-semester course in geometry for prospective secondary school teachers, what would be the 2 or 3 most important topics to include? [Investigative geometry / connectivity and symmetry / graph theory / valence (for vertices or faces), circuits, Euler's relation, trees, algorithms for spanning trees, travelling salespeople / transformations, including dilations; non-euclidean geometry; solids / more important than "what" is "how" / looking at ideas at multiple levels of rigor and abstraction]

6. Cartesian Coordinates <==> Hyperspherical Coordinates (Djamal Bouzida) 09/04/93
Guess for 4D & n dimensions - is it right? [ sin and cos positions in 2D and 3D different from those in 4D / consistent ordering]

7. Trig Puzzle (Joseph O'Rourke) 06/18/93
Request for proof of an identity. [trig identities / reduce the statement to a polynomial / another proof]

8. 4D Visualization (Evelyn Sander) 05/20/93
Interesting responses to the question how well people's work makes them understand 4D. [interviews with mathematicians / best way by analogy with three dimensions / minimal surfaces / wire frame and soap solution / holographic image / 24-Cell and Klein Bottle / Grids in PDEs / Brakke test for visualizing 4D / Ken Brakke - Jeff Weeks correspondence / Brakke skeptical that anyone can visualize 4D / Snake Test / Opaque Hypercube Test / Illusion Test / Banchoff movie / book: Beyond the Third Dimension]

9. 4D Geometry (Gene Klotz) 05/13/93
Request for information on computer graphics for a course called "Images From the Fourth Dimension." [SIGGRAPH / topology / programs for NeXT computers / do 4D walks while projecting into 3D and graphing with a 3D grapher / Asimov's "grand tour" / Torus Chess / Klein Chess]

10. Sphere Eversion Movie (Bret Jolly) 04/22/93
Nelson Max's pioneering computer animation of Bernard Morin's sphere eversion.

11. Geometry of the Platonic Solids (Mark Rauschkolb) 04/13/93
Request for data for advanced solids--tetrahedron, octahedron, icosahedron, etc.--location of vertices; also coordinates for other non-regular solids (buckyballs). [Coxeter's _Regular Polytopes_ / metrical properties / symmetries via reflections / Magnus Wenninger / Holden, _Shapes, Space, and Symmetry_]

12. Interview with Bob Devaney (Evelyn Sander) 04/01/93
Evelyn Sander talks with a Boston Univ. professor about the lack of communication between high school and college math teachers and researchers. [what's turning stsudents away from math? / how can we popularize mathematics? / incorporating computers into the classroom / teach iteration in high school / chaos, fractals, the Mandelbrot set / general outline for teaching the Mandelbrot set / corrections]

13. Theorema Egregium (Joseph O'Rourke) 02/23/93
Can anyone offer a translation and pronounciation guide? And did Gauss give it this name, or did his peers? [Latin derivation - ex grege / what Gauss wrote]

14. Scientific Visualization (Gene Klotz) 02/09/93
Scientific visualization now lies somewhere between a growth industry and a buzz word. [mathematical content / why important / three-dimensional geometry / list of books / review: Thalmann, _Scientific Visualization and Graphics Simulation_]

15. Volume of a Pyramid (Joseph O'Rourke) 01/21/93
How to show that the volume of a pyramid is 1/3 * area of base * altitude without using calculus? [observation: three identical tetrahedra pack half a cube... / V. Boltyianskii, _Hilbert's Third Problem_ / Euclid / think of a light in the corner of a cubical room]