Metric
vector spaces |
A metric vector space is a vector space which has a (usually
indefinite) scalar product. I first became fascinated with these spaces
as a beginning master's student. Geometrically interesting in their own
right (as Euclidean n-space or Minkowski spacetime, for example), they are
also invaluable as coordinate spaces: it's quite extraordinary just how
many classical geometries can be coordinatized by n-tuples subject to some
indefinite scalar product. And looking at these geometries through their
coordinate spaces often makes obvious the isomorphisms between different
models of the same geometry, or even between different geometries: the same
coordinate space implies the same or related geometries. |
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On Null-Cone Preserving Mappings. (with
M. A. McKiernan) Math. Proc. Camb. Phil. Soc. 81 (1977) 455
- 462 |
Cone Preserving Mappings for Quadratic Cones over
Arbitrary Fields. Canad. J. Math. 29 (1977) 1247 - 1253 |
Some Properties of Non-Positive Definite Real Metric
Vector Spaces. Utilitas Math. 12 (1977) 327 - 333 |
A Characterization of Non-Euclidean, Non-Minkowskian
Inner Product Space Isometries. Utilitas Math. 16 (1979) 101
- 109 |
Transformations of n-Space Which Preserve a Fixed
Square-Distance. Canad. J. Math. 31 (1979) 392 - 395 |
Conformal Spaces. J. Geom. 14 (1980) 108
- 117 |
Transformations Preserving Null Line Sections of
a Domain: the Arbitrary Signature Case. Resultate Math. 9 (1986)
107 - 118 |
A Metric Vector Space Proof of Miquel's Theorem.
C. R. Math. Rep. Acad. Sci. Canada 9 (1987) 59 - 62 |
The Octahedron Theorem in Minkowski Three-Space:
A Metric Vector Space Proof of Miquel's Theorem in the Laguerre Plane. J.
Geom. 30 (1987) 196 - 202 |
Geometric
characterization problems |
The basic characterization problem: determine those transformations
of a geometric space which preserve some particular feature or measurement
of the space, without recourse to regularity assumptions (linearity, continuity,
. . .) or even without assuming bijectivity. The classical example is the
Beckman-Quarles theorem: functions from the Euclidean plane to itself preserving
pairs of points a distance 1 apart must be Euclidean motions (the functions
need not be assumed bijective, or even single valued). |
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There are characterization theorems for many other spaces and spacetimes:
for a survey, see a book chapter I wrote:
Distance Preserving Transformations. Chapter 16 of Handbook of
Incidence Geometry, p. 921 - 944, ed. F. Buekenhout, Elsevier Science
B. V., 1995.
|
On Null-Cone Preserving Mappings. (with M. A.
McKiernan) Math. Proc. Camb. Phil. Soc. 81 (1977) 455 - 462 |
Cone Preserving Mappings for Quadratic Cones over
Arbitrary Fields. Canad. J. Math. 29 (1977) 1247 - 1253 |
A Characterization of Non-Euclidean, Non-Minkowskian
Inner Product Space Isometries. Utilitas Math. 16 (1979) 101
- 109 |
Transformations of n-Space Which Preserve a Fixed
Square-Distance. Canad. J. Math. 31 (1979) 392 - 395 |
On Distance Preserving Transformations of Lines in
Euclidean Three-Space. Aequationes Math. 28 (1985) 69 - 72 |
Euclidean Plane Point Transformations Preserving
Unit Area or Unit Perimeter. Archiv Math. (Basel) 45 (1985) 561
- 564 |
Transformations Preserving Null Line Sections of
a Domain: the Arbitrary Signature Case. Resultate Math. 9 (1986)
107 - 118 |
A Characterization of Motions as Bijections Preserving
Circumradius or Inradius 1. Monatsh. Math. 101 (1986) 151 - 158 |
Martin's Theorem for Euclidean n-Space and a Generalization
to the Perimeter Case. J. Geom. 27 (1986) 29 - 35 |
Orthogonal Spheres. C. R. Math. Rep. Acad. Sci.
Canada. 8 (1986) 231 - 235 |
On Line Mappings Which Preserve Unit Triangles. Utilitas
Math. 31 (1987) 81 - 84 |
Angle-preserving Transformations of Spheres. Aequationes
Math. 32 (1987) 52 - 57 |
A Beckman-Quarles-Type Theorem for Coxeter's Inversive
Distance. Canad. Math. Bull. 34 (1991) 492 - 498 |
Many of my theorems are also discussed in two books by Walter Benz:
Geometrische Transformationen unter besonderer Berücksichtigung
der Lorentztransformationen. B.I. Wissenschaftsverlag, Mannheim 1992
Real Geometries. B.I. Wissenschaftsverlag, Mannheim 1994
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Spacetime
geometry |
Both of the previous topics come together here: the simplest spacetime,
Minkowski spacetime is an example of a metric vector space, while Alexandrov's
theorem characterizes its transformations (Lorentz transformations) by
the fact that they preserve pairs of points connected by light signals.
Another example: Zeeman's theorem characterizes the causality-preserving
transformations of Minkowski spacetime, for example. I've done generalizations
of these and other theorems for several other spacetimes.
In another, disjoint spacetime direction, my favourite paper Does Matter Matter? uses some ideas from classical inversive
geometry to construct a spacetime model in which the location of infinity
is relative (i.e. observer-dependent). Under some very natural assumptions
about proper time, the model predicts cosmological redshifts with unexpectedly
realistic properties. It also predicts an age for the universe of about
25 billion years (comfortably more than the stars in it, unlike the situation
in more classical theories).
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The Beckman-Quarles Theorem in Minkowski
Space for a Spacelike Square-Distance. Arch. Math. (Basel) 37 (1981)
561 - 568 [summary in C. R. Math. Rep. Acad. Sci. Canada 3 (1981)
59 - 61] |
Alexandrov-type Transformations on Einstein's Cylinder
Universe. C. R. Math.Rep. Acad. Sci. Canada 4 (1982) 175 - 178 |
Transformations of Robertson-Walker Spacetimes Preserving
Separation Zero. Aequationes Math. 25 (1982) 216 - 232 |
A Physical Characterization of Conformal Transformations
of Minkowski Spacetime. Ann. Discrete Math. 18 (1983) 567 - 574 |
Conformal Minkowski Spacetime I: Relative Infinity
and Proper Time. Il Nuovo Cimento 72B (1982) 261 - 272 |
Conformal Minkowski Spacetime II: A Cosmological
Model. Il Nuovo Cimento 73B (1983) 139 - 149 |
Separation Preserving Transformations of de Sitter
Spacetime. Abh. Math. Sem. Univ. Hamburg 53 (1983) 217 - 224 |
The Causal Automorphisms of de Sitter and Einstein
Cylinder Spacetimes. J. Math. Phys. 25 (1984) 113 - 116 |
Relative Infinity in Projective de Sitter Spacetime
and its Relation to Proper Time. Ann. Discrete Math. 37 (1988)
257 - 264 |
Zeeman's Lemma on Robertson-Walker Spacetimes. J.
Math. Phys. 30 (1989) 1296 - 1300 |
The Effect of a Relative Infinity on Cosmological
Redshifts. Astrophysics and Space Science 207 (1993) 231 - 248 |
Does Matter Matter? Physics
Essays 11 (1998) 481 - 491 |
Complex
triangle and polygon geometry |
This topic began as a minor recreational problem and expanded
into a major research project. Basically what I've done is to develop a
rather productive complex cross ratio formalism for triangle geometry. First,
I use a single complex number, called shape, to describe Euclidean
triangles and prove theorems about similar triangles. Second, I use another
complex number to coordinatize the plane relative to a given triangle and
to prove theorems about it. Third, I relate the two: the coordinate of any
special point of a triangle is a corresponding function of its shape. This
function can be used to discover and prove theorems about special points
by reducing the proofs to complex algebra. |
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Triangles I: Shape. Aequationes Math.
52 (1996) 30 - 54 |
Triangles II: Complex Triangle Coordinates. Aequationes
Math.52 (1996) 215 - 245 |
Triangles III: Complex Triangle Functions. Aequationes
Math. 53 (1997) 4 - 35 |
A generalization of Napoleon's theorem to n-gons.
C. R. Math. Soc. Canada 16 (1994) 253 - 257 |
This work has spawned several other projects. For example,
computer experimentation with Geometer's Sketchpad provides very convincing
evidence for a rather nice generalization of Morley's theorem. I can use
my formalism to prove a special case, but the general case remains thus
far elusive. Another example: iterations of the abovementioned functions
can be used to study iterated pedal triangles, a dynamical systems problem.
The iterations lead to fractal-like structures which can be plotted. The
work is yet in a very preliminary stage: the pictures are there but the
reasons for them are not. My functions can also be used to examine other
geometric topics such as animated triangle centres. |
My triangles work has been extended to other planes: see for
example Shapes of Polygons, R. Artzy, J. Geom. 50 (1994) 11
- 15 and Shape-Regular Polygons in Finite Planes, R. Artzy and G.
Kiss, J. Geom. 57 (1996) 20 - 26. |