Axiomatic expressions of Euclidean and Non-Euclidean geometries
Contrary to traditional works on axiomatic foundations of
geometry, the object of this section is not just to show that some
axiomatic formalization of Euclidean geometry exists, but to
provide an effectively useful way to formalize geometry; and not
only Euclidean geometry but other geometries as well.
Precisely, this section aims to combine the following goals:
- Give a nice and useful axiomatic expression of Euclidean
geometry
- Also provide nice axiomatic foundations of other
(non-Euclidean) geometries
- Show the deep similarities between these diverse geometries,
right from their axiomatic foundations
- Let this be a good start towards higher theories of
physics
However, while this section tries to be relatively self-sufficient,
it is not completely so : it rather comes after a (still incomplete)
series of previous studies on the foundations
of mathematics (see the main page for the whole table of
contents). This already included some other studies of geometry,
especially a general
introduction to geometry, a study of affine geometry,
and of vector
spaces in duality.
We may say that affine and vector spaces were already cases of
geometrical spaces, however they still did not look so much like our
usual Euclidean space. Now finally we are coming to this familiar
geometry, as a particular case of a diversity of similar geometries
(somehow more similar to the Euclidean one than vector or affine
spaces were).
Inner product spaces
While the value of the dimension n sufficed to classify
affine geometries, it does not suffice with inner products. Inner
products are classified by an oriented pair of natural numbers (p,q)
with p+q=n, called the signature,
which roughly means that "scalar squares are positive in p
dimensions and negative in q dimensions". The geometry is
Euclidean if scalar squares of all vectors have the same sign
(usually taken as positive, that is q=0, but the opposite
sign convention of negative scalar squares, that is p=0,
essentially gives the same geometry).
Notice that the set of "rotations" (transformations which preserve
the inner product) has dimension n(n-1)/2 for any signature.
Common aspects to all these geometries
Remember that like any mathematical
theory, a geometry is made of the following hierarchy of
components:
Now all geometric theories we are going to consider, will have the
same following list of types
- ℝ = the field of real numbers, with its usual structures (0,1,
+,-,⋅), and axioms.
- S = a 1-dimensional ℝ-vector space (meant as the type
of surface densities, i.e. lengths to the power -2), thus with
its structures 0,+,-, and the multiplication by real numbers.
- E = a finite-dimensional ℝ-vector space, whose elements
may be called "coordinates", "affine forms"
- M = the set of points
and the following common list of structures and axioms:
- An operation from E×M to ℝ, that makes the pair
(P,E) form an ℝ-duality system and
identifies M as a subset of the dual E* of E
; equivalently, E is identified as an ℝ-subspace of ℝM.
Formally, this means the axioms
- : ∀x,y∈E, (∀u∈M, x(u)
= y(u)) ⇒ x=y (Separation of E
by M)
- ∀u,v∈M, (∀x∈E, u(x)
= v(x)) ⇒ u=v (Separation of M
by E)
- ∀x,y∈E,∀u∈M, u(x+y)
= u(x) + u(y)
- ∀x∈E,∀a∈ℝ,∀u∈M, (ax)(u)
= a x(u)
- A bilinear, symmetric operation ⋅ from E×E to S,
called the metric :
- ∀x,y∈E, x⋅y = y⋅x
∈ S
- ∀x,y,z∈E, (x+y)⋅z = x⋅z
+y⋅z
- ∀x,y∈E,∀a∈ℝ, (ax)⋅y,=a(x⋅y)
The separation of E by M, i.e. (∀x,y∈E,
(∀u∈M, x(u) = y(u)) ⇒ x=y),
may be seen here as implicit by the functional notation x(u).
It may also be seen as a consequence of later axioms, and dispenses
us of the need to declare the vector space properties of E
as axioms (though I'm not sure both virtues may be added up).
The next structures and axioms will vary between geometries.
The case of flat geometry
For flat geometries, the last needed structure will be a symbol of
constant called the mass, that is m∈E, with
the axioms
- m ≠ 0E
- ∀x∈E, m⋅x = 0
- M={u∈ E*| m(u)=1}, i.e.∀u∈
E*, m(u)=1 ⇔ u∈M
Note that through the last axiom, the axiom m ≠ 0E
is just equivalent to M≠ ∅ and thus may be seen as redundant
with previous axioms. We may also notice that this axiom may serve
as a definition of m, which is thus not really a new
structure.
We can define the set of vectors, as V={u∈
E*| m(u)=0}.
Except for the geometry of the Galilean space-time, useful
geometries (in particular the Euclidean and the Minkowski geometry)
also obey the following non-degeneracy axiom:
- ∀x∈E, (∀a∈ℝ, x ≠ am)⇒(∃y∈E,
x⋅y ≠ 0)
that may also be written in the contrapositive form ∀x∈E,
(∀y∈E, x⋅y = 0)⇒(∃a∈ℝ, x
= am)
and may be condensed with the above axiom m⋅x = 0,
to form a single axiom
- ∀x∈E, (∀y∈E, x⋅y =
0)⇔(∃a∈ℝ, x = am)
The Euclidean geometry is specified by the more precise axiom
∀x∈E, (∃a∈ℝ, x = am)⇔( x⋅x
= 0)
which more precisely implies that the sign of x⋅x
is fixed (for topological reasons : it cannot switch sign without
cancelling somewhere), and thus may be held as positive (defining
the order in S).
Study of the metric and related structures in geometries of
Euclid and Minkowski - how geometric structures are definable from
each other
The below is a draft being
reworked from old versions. Not sure when it will be
ready...
The work of correct reasoning on incorrect figures
(mentioned in the relativity
page) to rebuild the different geometrical concepts, is
processed as follows:
Orthogonality is such that for any point of a circle
(sphere), the tangent line (plane) is orthogonal to the
radius (line through the center).
Distances are related to the notion of sphere, in these
ways: a sphere is a set of points at equal distance to the
center; the measure of any volume given by a geometrical
definition depending on 2 points, is proportional to the n-th
power of the distance between them; in a straight line,
distances just add up.
The notion of rotation can be defined as "The affine
transformations that map circles (or spheres) into
circles";
The measure of angles is also related to the notions of
distance and circles, in different ways: an angle can be
defined as the length of an arc of circle (where the
length is the limit of the sum of distances for small
divisions of the arc of circle). Or, since rotations (as
defined by the preservation of circularity) also preserve
angles, we can define angles as the number of "small unit
angles" they are made of, for an arbitrary given small
unit angle that is moved to different directions using
rotations. See other ways to define the measure of
angles by its relation to the notion of circle that
we already expressed. the fact they preserve angles, gives
a general way to compare angles.
Now re-apply the same rules : we cannot anymore use the area
inside a circle (the volume inside a sphere) for getting the
length of its radius, because circles (spheres), being
hyperbolas (hyperboloids), no more contain a finite area
(volume) but an infinite one; however we can still use
almost the same method through the choice of a fixed angle
from the center of this "circle" that cuts there a portion
of "disk" (ball) with finite area.
Using split-complex numbers for Minkowski plane
geometry
The system of split-complex numbers are a nice tool to
study the 2-dimensional Minkowski geometry (signature
(1,1)) in the same way as complex numbers can be used to
study Euclidean plane geometry.
|
The case of curved geometry
In this case, the mass is replaced by a relative mass, that is a
function from M to E, with a constant c∈S
called the curvature of this geometry, with the axioms
- ∀u∈M, mu(u)=1
- ∀x∈E, ∀u∈M, mu⋅x
= x(u) c
- (one more axiom, to be completed)
Note that flat geometry comes as the particular case where c=0.
(This page will be completed later)
The metric may also be expressed in its curried form as the slope
function s∈Mor (E,E*)...
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