[FOM] 714: Foundations of Geometry/2
Harvey Friedman
hmflogic at gmail.com
Sun Sep 25 22:26:23 EDT 2016
Dana Scott pointed out that my condition that a permutation of R^k
that is line preserving and fixes exactly the points on a given line
is distance preserving is false in k = 3. Actually, it is false even
in the crucial dimension k = 2. I neglected to say "order 2". I.e., if
you apply the permutation twice, you get the identity. And this holds
for all dimensions k >= 2.
This is very much in the spirit of what I am trying to do. Namely, you
can see in the mind's eye the usual reflection permutation about any
given line. It is "obviously" a permutation, is "obviously" line
preserving,, and is "obviously" distance preserving.
So we restate the core of
http://www.cs.nyu.edu/pipermail/fom/2016-September/020084.html now,
and follow it by a sketch of the proof that the only such permutation
is reflection about the given line.
There is no change called for in the initial part that characterizes the lines.
***************
In this approach, we first treat R^2 with lines only. Here are the
three conditions on the lines in R^2, which are certain subsets of
R^2.
1. Every line contains at least two points, and has at most one point
in common with every other line.
2. Every line is disjoint from a line.
3. Every line containing (a,b), (c,d) also contains ((a+c)/2,(b+d)/2).
4. If the points in a line come arbitrarily numerically close to point
x in both coordinates simultaneously, then x is on the line.
THEOREM 1. There is exactly one set of subsets of R^2 for "lines"
satisfying conditions 1-4. These are the usual lines.
We now define Betweenness explicitly.
5. (c,d) lies between (a,b) and (e,f) if and only if (a,b), (c,d),
(e,f) are on a line, and (a < c < e or e < c < a or b < d < f or f < d
< b).
The simplest characterization of distance that we know of is as
follows. A function f is of order 2 if and only if for all x in its
domain, f(f(x)) = x.
6. The distance between (0,a),(0,b) is |a-b|.
7. Every line preserving order 2 permutation of R^2 whose set of fixed points
form a line is distance preserving.
The idea here is that now that we know exactly what the lines are, we
can mathematically determine all of the line preserving order 2 permutations
whose set of fixed points form a line, and hence the scope of 7.
However we can avoid having to "do the math' and use the following
alternative form of 7:
7'. Every line is the set of fixed points of some line and distance
preserving order 2 permutation of R^2.
Thus 7' can be readily visualized by picturing the reflection about
the given line, and "see" that it is line and distance preserving and
of order 2.
THEOREM 2. There is exactly one "distance function" satisfying 6 and
7, There is exactly one "distance function" satisfying 6,7'. This is
the usual Euclidean distance function in R^2.
If we have lines and distance, we can define Betweenness explicitly by
5'. y is between x and z if and only if x,y,z are distinct and lie on
a line, and the distance from x to y is less than the distance from x
to z.
INTUITIVE R^k GEOMETRY
k >= 3
We work with R^k and again first determined the lines only, as certain
subsets of R^k.
k1. Every line contains at least two points, and has at most one
point in common with every other line.
k2. Every union of k-1 lines is disjoint from a line.
k3. Every line containing x,y also contains (x+y)/2.
k4. If the points in a line come arbitrarily numerically close to
point x in all k coordinates simultaneously, then x is on the line.
THEOREM 3. There is exactly one set of subsets of R^k for "lines"
satisfying conditions k1-k4. These are the usual lines in R^k.
We now define Betweenness explicitly.
k5. y lies between x and z if and only if x,y,z are on a line, and for
some 1 <= i <= k, x_i < y_i < z_i or z_i < y_i < x_i.
The simplest characterization of distance that we know of is as follows.
k6. The distance between (0,...,0,a) and (0,...,0,b) is |a-b|.
k7. Every line preserving order two permutation of R^k whose set of fixed points
form a line is distance preserving.
We can also use the more directly visual
k7'. Every line is the set of fixed points of some line and distance
preserving order 2 permutation of R^k.
Thus k7' can be readily visualized by picturing the reflection about
the given line, and "see" that it is line and distance preserving and order 2.
Of course, visual clarity does in a sense require k <= 3.
THEOREM 4. There is exactly one "distance function" satisfying k6 and
k7, There is exactly one "distance function" satisfying k6,k7'. This is
the usual Euclidean distance function in R^k.
f we have lines and distance, we can define Betweenness explicitly by
k5'. y is between x and z if and only if x,y,z are distinct and lie on
a line, and the distance from x to y is less than the distance from x
to z.
LEMMA ON REFLECTIONS
THEOREM. Let T:R^k into R^k be an affine permutation whose set of
fixed points is exactly the line L, and of order 2. Then T is
reflection about the line L.
If by order 2 we exclude order 1, which is the identity, then we can
say this in a slightly stronger way:
THEOREM'. Let T:R^k into R^k be an affine permutation which fixes each
point on the line L, and is of order 2. Then T is reflection about the
line L.
We assume k >= 2. The following statement is in pure linear algebra in
a sense to be clarified:
PROPOSITION. Let L be a line in R^k. There is a unique T:R^k into R^k
which is line preserving, of order 2, and fixes all points on L.
It is clear that this Proposition implies Theorem' since reflection
about L is an example.
Noe that this Proposition provides a second order property of lines L
in the following familiar structure: There are two domains, R^k for
the set of points, and V for the set of lines in R^k. There is the
obvious incidence relation, that a given point lies on a given line.
We know that there are plenty of automorphisms of this structure, and
in fact we know what they are. But for now we only need that there is
an automorphism carrying any given line onto any other given line.
Therefore to prove the Proposition, we need only prove it for the line
x_k = 0, a very convenient line to be using.
Now line preservation is the same thing as being an affine
transformation. So let T:R^k into R^k be an affine permutation of
order 2, fixing all (x,0) in R^k. Thus we have the equation T(x,0) =
(x,0).
This immediately tells us something about the general expression
T(x,r), r in R. Obviously T(x,r) = (alpha(x,r),beta(x,r)), where alpha
is k-1 affine combinations of x_1,...,x_k-1 and r, and beta(x,r) is a
single affine combination of x_1,...,x_k and r. We can write this as
T(x,r) = (gamma(x) + delta(r), sigma(x) + tau(r))
where gamma(x) is k-1 linear combinations of x_1,...,x_k-1, delta(r)
is k-1 affine combinations of just r, sigma(x) is a linear combination
of x_1,...,x_k-1, and tau(r) is single affine combination of just r.
If we set r = 0 then we must be getting T(x,0) = (x,0) for the right
hand side. This means that there are no constant coefficients present,
and also gamma(x) is simply x_1,...,x_k-1.Also, sigma(x) must vanish.
Hence we have
T(x,r) = (x + c*r, dr) = T(x,dr) + (c*r,0).
where c* is a constant vector of length k-1, and d is a constant.
We now apply T to T(x,r) and set it equaled to (x,r).
To apply T to (x + c*r, dr), for the first k-1 coordinates, we add the
first k-1 coordinates of (x+ c*r, dr) to c* times the last coordinate
of (x + c*r, dr). This is
x + c*r + c*dr.
For the last coordinate of this evaluation, we multiple d by the last
coordinate of (x + d*r, dr). This is
(d^2)r.
Thus we have the order 2 equations
x + c*r + c*dr = x
(d^2)r = r.
Hence d = +-1. Since the first equation above holds for all r, clearly c* = 0.
So now we have
T(x,r) = (x,r) or
T(x,r) = (x,-r).
The first equation correspond to order 1, and the second equation
corresponds to order 2, the reflection about the line x_k = 0. QED
***********************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 714th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/2316 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
Harvey Friedman
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