Brahmagupta's formula.
A "Heron-type" formula for the maximum area of a quadrilateral,
Col. Sicherman's fave. He asks if it has higher-dimensional
generalizations.
Circumcenters of triangles.
Joe O'Rourke, Dave Watson, and William Flis
compare formulas for computing
the coordinates of a circle's center from three boundary points,
and higher dimensional generalizations.
Convex
Archimedean polychoremata, 4-dimensional analogues of the
semiregular solids, described by Coxeter-Dynkin diagrams
representing their symmetry groups.
A Counterexample to Borsuk's Conjecture, J. Kahn and G. Kalai,
Bull. AMS 29 (1993). Partitioning certain high-dimensional polytopes
into pieces with smaller diameter requires a number of pieces
exponential in the dimension.
The Fourth
Dimension. John Savard provides a nice graphical explanation of the
four-dimensional regular polytopes.
Four-dimensional visualization.
Doug Zare gives some pointers on high-dimensional visualization
including a description of an interesting chain of successively higher
dimensional polytopes beginning with a triangular prism.
Grid subgraphs.
Jan Kristian Haugland looks for sets of lattice points that induce
graphs with high degree but no short cycles.
Ham
Sandwich Theorem: you can always cut your ham and two slices of
bread each in half with one slice, even before putting them together
into a sandwich.
From Eric Weisstein's treasure trove of mathematics.
Hyperspheres. Eric Weisstein calculates volumes and surface areas of hyperspheres, which curiously reach a maximum for dimensions around 5.257 and 7.257 respectively.
Octacube.
Stainless steel 3d model of the 24-cell (one of the six regular
polytopes in four dimensions), by Adrian Ocneanu, installed as a
sculpture in the Penn State Math Department. Includes also a shockwave
flythrough of the model.
Odd squared distances. Warren Smith considers point sets
for which the square of each interpoint distance is an odd integer.
Clearly one can always do this with an appropriately scaled regular simplex;
Warren shows that one can squeeze just one more point in,
iff the dimension is 2 (mod 4).
Moshe Rosenfeld has published a related paper in Geombinatorics
(vol. 5, 1996, pp. 156-159).
Packings in Grassmannian spaces, N. Sloane, AT&T.
How to arrange lines, planes, and other low-dimensional spaces into
higher-dimensional spaces.
Pairwise
touching hypercubes. Erich Friedman asks how to partition the unit cubes
of an a*b*c-unit rectangular box into as many connected polycubes as
possible with a shared face between every pair of polycubes.
He lists both general upper and lower bounds as functions of a, b, and
c, and specific constructions for specific sizes of box.
I've seen the same question asked for d-dimensional hypercubes
formed out of 2^d unit hypercubes; there is a lower bound of roughly
2d/2 (from embedding a 2*2d/2*2d/2 box
into the hypercube)
and an upper bound of O(2d/2 sqrt d)
(from computing how many cubes must be in a polycube
to give it enough faces to touch all the others).
Peek, software for visualizing high-dimensional polytopes.
Penumbral shadows of polygons
form projections of four-dimensional polytopes.
From the Graphics Center's graphics archives.
Plücker coordinates.
A description by Bob Knighten of this useful and standard way
of giving coordinates to lines, planes, and higher dimensional
subspaces of projective space.
PolyGloss.
Wendy Krieger is unsatisfied with terminology for higher dimensional geometry
and attempts a better replacement.
Her geometry works
include some other material on higher dimensional polytopes.
A quasi-polynomial bound for the diameter of graphs of polyhedra,
G. Kalai and D. Kleitman, Bull. AMS 26 (1992). A famous open conjecture in polyhedral
combinatorics (with applications to e.g. the simplex method in linear
programming) states that any two vertices of an n-face polytope are
linked by a chain of O(n) edges. This paper gives the weaker bound
O(nlog d).
Rubik's
hypercube. 3x3x3x3 times as much
puzzlement. Windows software from Melinda Green and Don Hatch,
now also available as Linux executable and C++ source.
Sausage
Conjecture. L. Fejes Tóth conjectured that, to minimize the volume of the convex hull of
hyperspheres in five or more dimensions, one should line them up in a row.
This has recently been solved for very high dimensions
(d > 42) by Betke and Henk
(see also Betke et al., J. Reine Angew. Math. 453 (1994) 165-191
and the
MathWorld
Sausage Conjecture Page).
Sierpinski
pentatope video by Chris Edward Dupilka. A four-dimensional analogue
of the Sierpinski triangle.
Simplex/hyperplane intersection.
Doug Zare nicely summarizes the shapes that can arise on intersecting
a simplex with a hyperplane: if there are p points on the hyperplane,
m on one side, and n on the other side, the shape is
(a projective transformation of)
a p-iterated cone over the product of m-1 and n-1 dimensional simplices.
Skewered lines.
Jim Buddenhagen notes that four lines in general position in R3
have exactly two lines crossing them all, and asks how this generalizes
to higher dimensions.
SMAPO
library of polytopes encoding the solutions to optimization problems
such as the TSP.
Solution
to problem 10769. Apparently problems of coloring the points of a
sphere so that orthogonal points have different colors (or so that each
set of coordinate basis vectors has multiple colors) has some relevance
to quantum mechanics; see also papers
quant-ph/9905080 and
quant-ph/9911040
(on coloring just the rational points on a sphere), as well as this
four-dimensional construction
of an odd number of basis sets in which each vector appears an even
number of times, showing that one can't color the points on a
four-sphere so that each basis set has exactly one black point.
Squares
are not diamonds. Izzycat gives a nice explanation of why
these shapes should be thought of differently, even though they're
congruent: they generalize to different things in higher dimensions.
Stella and Stella4d,
Windows software for visualizing regular and semi-regular polyhedra and
their stellations in three and four dimensions, morphing them into each other, drawing unfolded nets for
making paper models, and exporting polyhedra to various 3d design packages.
The
Story of the 120-cell, John Stillwell, Notices of the AMS. History,
algebra, geometry, topology, and computer graphics of this
regular 4-dimensional polytope.
Structors.
Panagiotis Karagiorgis thinks he can get people to pay large sums of
money for exclusive rights to use four-dimensional regular polytopes
as building floor plans. But he does have some pretty pictures...
Two-distance sets.
Timothy Murphy and others discuss how many points one can have
in an n-dimensional set, so that there are only two distinct
interpoint distances. The correct answer turns out to
be n2/2 + O(n).
This
talk abstract by Petr Lisonek (and paper in JCTA 77 (1997) 318-338)
describe some related results.
