- Archimedean solids:
John Conway describes some
interesting maps among the Archimedean
Eric Weisstein lists
and pictures of the Archimedean solids.
- 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
- The charged particle
model: polytopes and optimal packing of p points in n dimensional spheres.
- 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.
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.
- Cube triangulation.
Can one divide a cube into congruent and disjoint tetrahedra?
And without the congruence assumption,
how many higher dimensional simplices are needed to triangulate a hypercube?
For more on this last problem, see
an n-dimensional cube, S. Finch, MathSoft,
triangulations of the d-cube, Orden and Santos.
- DeVicci's Tesseract.
Higher-dimensional generalizations of Prince Rupert's cube,
from MathSoft's favorite constants pages.
- Explore the 120-cell!
Free Windows+OpenGL+.Net software.
hyperspace with the geometric product. Thomas S. Briggs explains
some four-dimensional shapes.
A Romance of Many Dimensions.
- Four dice hypercube visualization.
- 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.
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.
- Hermite's constants.
Are certain values associated with dense lattice packings of spheres
Part of Mathsoft's
- Hyper hyper! Extra dimensions
- The HyperSphere, from an Artistic point of View,
- Hypercube's Home Page. Speculations on the fourth dimension collected by Eric Saltsman.
- Hypercube fun.
John Atkeson finds a nice recursive drawing pattern for high dimensional
hypercubes in two dimensional planes.
- Hypercube game.
Experience the fourth dimension with an interactive, stereoscopic
java animation of the hypercube.
- Hypercube visualization,
in hyper perspective. Red-blue 3d visualizations produced with the
virtual flower system.
- Hyperdimensional Java.
Several web applets illustrating high-dimensional concepts, by
- Hyperspace star
polytope slicer, Java animation by Mark Newbold.
- Hyperspace structures. Exploring the fourth dimension.
- Hyperspheres. Eric Weisstein calculates volumes and surface areas of hyperspheres, which curiously reach a maximum for dimensions around 5.257 and 7.257 respectively.
hyperspace, and the fourth spatial dimension. M. R. Feltz
views the universe as a closed cosmic hypersphere.
open-source software for visualizing Cayley graphs of Coxeter groups
as symmetric 4-dimensional polytopes.
- Keller's cube-tiling conjecture is false in high dimensions,
J. Lagarias and P. Shor, Bull. AMS 27 (1992).
Constructs a tiling of ten-dimensional space by unit hypercubes
no two of which meet face-to-face, contradicting a
that any tiling included two face-to-face cubes.
non-adjacent vertices on 120-cell. Sci.math discussion on the size
of the maximum independent set on this regular 4-polytope.
it is known to be between 220 and 224 inclusive.
area cross-section of a hypercube.
- N-dimensional cubes, J. Bowen, Oxford.
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
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.
- Pictures of 3d and 4d regular solids, R. Koch, U. Oregon.
Koch also provides some
4D regular solid visualization applets.
solids and quaternion groups, J. Baez.
- 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.
polyhedra, polytopes, R. Towle.
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
- Realization Spaces of 4-polytopes are Universal,
G. Ziegler and J. Richter-Gebert, Bull. AMS 32 (1995).
4d polytope foldouts. Java animations by Andrew Weimholt.
Also includes some irregular polytops.
polytopes in higher dimensions. Russell Towle uses Mathematica to
slice and dice simplices, hypercubes, and the other high-dimensional
Russell's 4D star
polytope quicktime animations.
- Regular polytopes in Hilbert space.
Dan Asimov asks what the right definition of such a thing should be.
hypercube. 3x3x3x3 times as much
puzzlement. Windows software from Melinda Green and Don Hatch,
now also available as Linux executable and C++ source.
- Ruler and Compass.
Mathematical web site including special sections on the
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
Sausage Conjecture Page).
pentatope video by Chris Edward Dupilka. A four-dimensional analogue
of the Sierpinski triangle.
Simplex: Minimal Higher Dimensional Structures.
- 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.
library of polytopes encoding the solutions to optimization problems
such as the TSP.
- Soap bubble 120-cell
from the Geometry Center archives.
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
(on coloring just the rational points on a sphere), as well as this
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.
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.
- Speculations on the
fourth dimension, Garrett Jones.
- 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.
Story of the 120-cell, John Stillwell, Notices of the AMS. History,
algebra, geometry, topology, and computer graphics of this
regular 4-dimensional polytope.
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...
- Student of
Hyperspace. Pictures of 6 regular polytopes, E. Swab.
Polyhedra Page and
Information and images on universal polyhedra and higher dimensional polytopes.
- 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).
talk abstract by Petr Lisonek (and paper in JCTA 77 (1997) 318-338)
describe some related results.
A somewhat generalized definition of 4d polytopes,
investigated and classified by J. Bowers, the polyhedron dude.
See also the dude's pages on
uniform polyhedron nomenclature.
- Visualizations of 4d hypercube (Java applet).
- Visualizing the hypersphere via 3d slices, and
four-dimensional thoughts by Jeff Fuquay.
rendering of a 4-cube. Rick Mabry animates a 3d projection that has
a nice symmetrical 2d projection.
From the Geometry Junkyard,
and recreational geometry pointers.
Send email if you
know of an appropriate page not listed here.
from a common source file.