### geometry.research - brief descriptions

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#### 1996

1. Euclid's 47th Proposition (Ron Hodgin) 11/17/96
Can anyone assist me in finding Euclid's 47th Proposition? [Pythagorean Theorem]

2. 2 Million Point Problem (Kathryn Redfern) 11/11/96
Two million points are randomly scattered in a circle. Must there always be a straight line that passes through the circle and has a million points on each side? Explain your answer/reasoning.

3. Geometry Paper (Charles Grady) 11/07/96
For my Euclidean/Non-Euclidean Geometry class, I am required to write a paper over some subject dealing with geometry. Since I am going to teach at the high-school level, I'd like something dealing with geometry in that realm. Any suggestions?

4. Arc Problem (Isaac Thaler) 10/27/96
Trying to find the formula of a small arc in a circle. The length of the chord is known and the radius of the circle is known.

5. Packing and Covering Problems (Marty Adickes) 07/30/96
Does anyone know of any current research that is being done with respect to non-convex regions or volumes and covering them with circles or packing them with spheres? [GOSSET]

6. Theorem of Pohlke (Caspar Curjel) 07/18/96
The theorem of Pohlke (1853) reads as follows: Given any four points in the plane, not all of them on a straight line, there exist an orthonormal coordinate system OXYZ and a direction vec u in space so that the four points in the plane are the parallel projection of OXYZ in the direction vec u. Where can I find a reference to this theorem in English? [Gauss, Dorrie]

7. Rectangular Lattice Hiding in Regular Triangular Lattice (Daniel A. Asimov) 06/14/96
A question about rectangular and triangular lattices.

8. Knots in Strings (John Furey) 03/01/96
I was wondering if anyone was interested in the problem of a string tying itself into knots. I have thought it best to work this problem backwards, by seeing which motions of the string are required to form a particular knot, and then determining the probability that the string had that motion. However, I need to know the following things: a) is the indexing set for types/numbers of knots discrete or continuous; b) how to determine if a particular type of knot is/is not able to be formed from a particular string; c) how a knot restricts the motions of a string. [Colin Adams; Tamar Schlick; Wilma Olson; Crowell and Fox; Lee Neuwirth]

#### 1995

1. Rhombic Dodecahedron (Douglas Zare) 12/15/95
What does one call a rhombic dodecahedron with the six vertices of degree 4 truncated?

2. Hundred-Sided Polygon (Laurel A. White) 12/13/95
What is a hundred-sided polygon called? [ hectogon / hectagon ]

3. Polyhedra Which Tile Hyperbolic 3-Space (Douglas Zare) 12/13/95
Which polyhedra are Vornoi cells of cosets of co-compact subgroups of the isometries of H^3? What other polyhedra tile H^3? Is it still open whether the Dehn invariant of any polyhedron tiling H^3 is 0?

4. Map coloring on orbifolds (Heidi Burgiel) 11/07/95
Does anybody out there know if any research has been done on orbifolds as they relate to colored tilings? For example, the torus is the orbifold of one of the seventeen crystallographic groups. There is a tiling of the torus that cannot be "colored" with fewer than seven colors. What can be said about colorings of the tiling of the plane (or of tilings of other covering spaces of the torus) that arise from that tiling of the torus?

5. Geometry of Sound (Wavz) 10/22/95
Does anyone know of any work done with sound and geometry? Actually music and geometry is what I'm looking for. Since music is all about ratios and math, seems to me like there should be some really fun ways to visually describe the realtime mathmatical relationships between the frequencies of the tones. [fractal music site]

6. The Game of Sprouts (Bill Haloupek) 09/22/95
My Finite Math class is studying tournament graphs. Yesterday we had a tournament using "The Game of Sprouts", invented by John H. Conway and Michael Paterson. I found out about this game in an article in the Amer. Math. Monthly, May 1993. It really provided a fun way to teach some graph concepts. I was wondering if any work has been done on strategies for this game. Is there a winning strategy for the first (or second) player? [If you can divide the board into k regions bounded by curves already drawn, in such a way that each region has a live spot in its interior, then the number of spots that are still live at the end of the game will be at least k.]

7. Dynamic Geometry - Is anyone interested? (ZYXZYXZYX) 08/24/95
I have invented a new geometry outside the category of Euclidean, Lobechevski and Riemann geometries, in which the change in area of a finite circle is not directly dependent upon the the change in radius of such a circle. Is anybody interested?

8. Double bubbles most efficient (Eric Sasson) 08/08/95
Of all the possible shapes in the world, the "double bubble" is the most efficient at enclosing two equal volumes, say two mathematicians from the University of California, Davis, and Real Software in Santa Cruz, who report having solved this problem that began 2,000 years ago. "There are infinitely many possible shapes for enclosing volumes - cubes, inner tubes, cell walls, gas tanks," says UC Davis mathematics professor Joel Hass. "As it turns out, nature's soap bubbles are the best." Using a computer, Hass and colleague Roger Schlafly proved that two spherical bubbles optimally attached to each other require the least surface area necessary to enclose two equal volumes. Hass is presenting the results on Sunday, Aug. 6, at a special session on soap bubble geometry at the 1995 Burlington Mathfest in Burlington, VT.

9. 3D-shape Ontology (Classification, Taxonomy) (Joaquin.A.Delgado) 07/28/95
I'm trying to build a classification of basic 3-D shapes: sphere, cube, pyramid, etc... I need a more or less complete vocabulary of primitives and their mathematical representation. Apparently, in the literature there is no formal taxonomy for 3d shapes. Is there any book or paper I can refer to for a list of basic solids or 3d shapes?

10. Area of a circle - history (Steven Kirshner) 07/05/95
Is there indication as to where the area of a circle was first discovered?

11. Circles on Spheres (Ben Backus) 04/27/95
Can anyone recommend a general treatment of coordinate systems on spheres, and/or where the coordinate systems end up under various perspective projections onto a plane? I'd like to identify the coordinate systems on a sphere that consist of two sets of mutually perpendicular circles. [latitude/longitude system / direction circle system / Berger, Geometry II, 1987]

12. Fourth Dimension (Sarah Barclay) 03/03/95
I completed a mostly visual-based project and presentation on the fourth dimension in first term and would like to elaborate on this same topic for a presentation this term. We have recently been studying spherical geometry and I want to extend this onto the hypersphere. Any information people have about the fourth dimension and especially the hypersphere would be much appreciated. [all points in 4-space whose distance from the origin is 1 / ordinary 3-space R^3 with one more point added "at infinity" / the group of unit quaternions / Smale's proof of the Generalized Poincare Conjecture]

13. Why do the angles of a triangle total 180 degrees? (James T. Cargile) 02/24/95
A question about the merit of an informal intuitive "argument" to explain why the angles of a triangle total 180 degrees. [a good track to finding one characterization of curvature / What this argument depends on is what physicists call the possibility of "parallel transport" - the existence of a canonical way of moving one's frame of reference from one point to another, so that the orientation it arrives in in the new place is independent of that path / why should it follow that the three angles add up to pi? / Instead of whole great circle sides, think of tangent vectors]

14. A geometric interpretation of the 'kth trace' (Lee Rudolph) 02/21/95
The "kth trace" of a linear transformation A of a vectorspace V of dimension n is the kth elementary symmetric function of the eigenvalues of A; e.g., the "first trace" is the trace, and the "nth trace" is the determinant. Recently, there have been two enlightening threads concerning the geometric interpretation of the trace and of the determinant. But can anyone provide a geometric interpretation for some other invariants, e.g. some quadratic invariants...

Penrose tiles are currently available (in various forms-- the chicken and bird, kite and dart, 2 rhombi) from a British company, Pentaplex, address as follows: Pentaplex Ltd. Royd House, Birds Royd Lane, Brighouse, West Yorkshire, HD6 1LQ, England

16. Jig saw Puzzle (pivoting metal rod) (Evelyn Sander) 01/03/95
Bolt a metal rod to the end of the blade of a jig saw so that the rod can pivot in one plane. The rod and blade are facing to the right. However, with the motor off, since the rod can pivot, if it is not held in this position, it will fall and point downward. What would you expect to happen if with the motor running you point the object so that the rod and the blade face directly upward?

#### 1994

1. Geometry of seashells (David Fowler) 12/15/94
Does anyone know of an upcoming [or recently held] conference on the mathematical modelling of sea shells? The conference may be [or have been] held in Australia, and called "Conchoid Geometry." [Conchology Conference and Art Exhibition, Jan 2-7, Murwillumbah, New South Wales, Hosted by the Science-Art Research Centre; An International Conference on Computer Modelling of Seashells and Bioforms / to be held in conjunction with an international exhibition of computer graphics, Order and Chaos in Nature / modelling of sea shells and bioforms using the most advanced possible mathematics and computer graphics]

2. Bernouilli effect (Peter Clark) 11/08/94
What theorem did the Bernouilli brothers make? [the Bernouilli family accomplishments / Bernouilli effect: the pressure in a fluid reduces as the speed increases / Bernouilli distribution in statistics / Bernouilli numbers / "Calculus Gems" by George Simmons]

3. Rotations in 3D (Mortenson) 08/07/94
In Simon L. Altmann's book 'Rotations, Quaternions, and Double Groups,' Altman states (p.28), "Rotations, however, are an accident of three-dimensional space. In spaces of any other dimensions, the fundamental operations are reflections (mirrors)." I know that a rotation can be represented by reflection transformations, that reflections are more general, and that rotations can be performed in 2, 3, or N dimentions. So, what do you suppose Altman means when he says rotations are an 'accident of three-dimensional space?' [Can all the rotations that can be performed in three dimensional space be represented by reflections? / Consider a rotation of angle theta about an axis A / Any member of the n-dimensional orthogonal group (that is to say, and congruence that fixes the origin) can be written as the product of at most n reflections / Euler's theorem]

4. Rigidity of Convex Polytopes (John M. Sullivan) 06/16/94
There's a famous theorem of Cauchy (proving a claim of Euclid it seems) that two convex polyhedra with the same shape faces, assembled the same way, are in fact congruent. In other words, if you're building a polyhedron from its faces, it may be floppy when you're part way through, but when you finish, it is rigid. Is the same true in higher dimensions? [Alexandrov's theorem / mixed volumes, for convex polytopes / three accessible references / two lemmas can also be used for the higher dimensional case / generic rigidity / non-convex realizations]

5. Medial Axis Algorithm (Alex Nicolaou) 06/01/94
Looking for an implementation of the "Medial Axis" algorithm for convex simple (non-self intersecting) polygons. Does anyone know of code or -detailed- papers for this case? [what is the Medial Axis of a polygon? / term derives from computer vision / "grassfire transformation / boundary of the Voronoi cell decomposition / Question: for a polygonal boundary, is there always a finite set whose Voronoi diagram matches the medial `axis' construct? / For a convex polygon, the medial axis consists of straight edges. But reflex vertices (those whose internal angle exceeds pi) generate parabolic arcs.]

6. Automatic Groups (Heidi Burgiel) 05/20/94
Are quotient groups of Euclidean groups Automatic? [classification of regular maps / on compact surfaces / all such maps are obtained as quotients of regular maps in one of (sphere, Euclidean plane, Hyperbolic plane)]

7. Tiling a Polygon (Andreas Bernig) 05/04/94
Is it possible to tile any polygon into congruent parts? [no unless one congruent part / related problems / divide the sphere into N lunes...]

8. Tonsoidal Torus? (Suresh Krishnajois) 04/05/94
Is there such a thing as a tonsoidal torus? [seems VERY unlikely to mean anything / OED does suggest some sort of meaning / a torus that has been trimmed down from a doughnut shape, and if the word was chosen really carefully, it was trimmed down to a twin-lobed shape / probably a misprint / self-fertilizing polyhedron / from a dream]

9. Archimedean Solids (Barbara Hausmann) 03/15/94
Are there finitely many Archimedean solids in each dimension, and if so how can this be prooved? [definition of "Archimedean" / cells must be Archimedean of the next lower dimension, and the group be transitive on the vertices / symmetric embeddings of regular maps / n-gon x m-gon is still "Archimedean" / vertex-figures / icosahedron has 4 irreducible "regular representations"]

10. Reference for 2 Theorems? (Joe Malkevitch) 02/15/94
Two-dimensions: Bolyai-Gerwien-Wallace theorem; 3-dimensions: Dehn and Syndler theorems [references / Can every rectangle be broken up into a finite number of squares? / repeatedly take off the largest square you can / problem? Mrs. Perkins' Quilt]

11. Convex Hull (Jay Mookherje) 02/11/94
What is the most efficient way to find out the convex hull of a group of polygons? Those polygons could be concave or convex. Once the convex hull is found, what is the best way to find out the shortest bounding rectangle which encloses this convex hull?

12. Polyhedra Questions (Walter Whiteley) 01/25/94
1) When does Euler's formula for polyhedra (V + F = E + 2) NOT work? 2) Why is a rhombicuboctahedron, or a rhombi-anything for that matter, called such when it has no rhombi for faces? 3) How many axes of symmetry does a stellated octahedron have? 4) How many edges does a small stellated dodecahedron have? [works for 'connected spherical map', connected planar graph / intersection of a rhombic dodecahedron and a cuboctahedron in dual position]

13. Why 'snub cube'? (Annie Fetter) 01/07/94
Does anyone know why the snub cube is called the snub cube? [free translation of Kepler's latin word "simus", meaning "squashed" / principles used in naming the Platonic and Archimedean polyhedra / Greek stems / truncated versions / a die with snubbed corners]

#### 1993

1. 1st-order Theory of the Reals (Joseph O'Rourke) 11/08/93
D.S. Arnon says in a paper on geometric reasoning, that, "It is known for any sentence phi of L [the first-order theory of the reals], either phi is in Th(R) or !phi is in Th(R)," where "Th(R) is the collection of all sentences of L which are true for the real numbers." (Here I am using ! for not.) Can someone cite the theorem from which this claim derives? [A. Tarski, A decision method for elementary algebra and Geometry / for any collection of polynomial inequalities in n real variables, you can tell whether or not there are real numbers satisfying them / Sturm's theorem / Tarski's completeness theorem / Another question: One sometimes see Go:del's theorem stated informally as this: "In any formal system adequate for number theory there exists an undecidable formula" (this is from the Encylopedia of Philosophy). Is not Th(R) "adequate for number theory"? If so, shouldn't Th(R) contain undecidable formulae? / not possible to define "r is a natural number" (or "r is a rational number") in the language of fields / first order theory of R is decidable]

2. Sketchpad Statistics (Bill Finzer) 10/19/93
I've been interested in visualizations of statistical concepts. This and the following two files contain Geometer's Sketchpad sketches that show geometric interpretations of the mean, the standard deviation, least squares regression, and the correlation coefficient.

3. Cabri n'est-il qu'un jouet? (Beth Bruch) 06/23/93
Bilingual (French/English) posting: Editorial -- Is Cabri nothing but a toy for professors regressing back to childhood? . . . And worse! Cabriole, isn't it the expression of a regression to childhood with respect to the very history of mathematics?[direct manipulation / reasons for using Sketchpad and Cabri]

4. General 3D 'jigsaw puzzle' (Joseph O'Rourke) 06/09/93
There has been some work done on the "Jackstraws" problem, which asks, given a collection of compact manifolds in R^n, can they be separated to infinity one at a time? [Snoeyink and Stofi: "Objects that cannot be taken apart by two hands" / Bob Dawson constructed a collection of 12 convex objects that can only be taken apart by explosion / Fejes-Toth and Heppes established a somewhat stronger result]

5. Simple Geometry Question (Joseph O'Rourke) 05/29/93
Given four points p1, p2, q1, q2 in a plane and suppose that there is a line segment from p1 to p2 and from q1 to q2. How will I know if the two line segments intersect? Is there a simple formula for this given only the coordinates of the four points? [formula written as boolean expression / VAX FORTRAN code]

6. The Form of the Formless (Steven L Combs) 05/11/93
Introduction to "Unrepentent Synergetics: Synergetic Atomic Model" [expanding on the work of R. Buckminster Fuller / duality is similar to that of Schrodinger's Wave Mechanics and Heisenberg's Matrix Mechanics / space waves, fractal fields, and vector stars can all be deduced from one another / a very large number of people... can sense total nonsense from miles away]

7. Geometry of 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_]

8. Tait Flyping Conjecture (Joseph O'Rourke) 02/09/93
Has the century-old Tait flyping conjecture that two reduced alternating projections of the same knot are equivalent on the sphere iff they can be converted into one another by flypes been proven? Any references? [W. Menasco and M. Thistlethwaite]

#### 1992

1. Dodecahedron in S3 (Jim Buddenhagen) 11/20/92
Given a (regular pentagonal faced) dodecahedron of edge e living in a 3-sphere of radius r, how does one compute the dihedral angle in terms of s = e/r ? [table of symbolic dihedrals sin(theta) and tan(theta) for the Platonic and quasi-regular Archemidean solids / space packing, Kepler's 'snub' solids / Sullivan article in the Mathematica Journal 1:3]