Lumiste - Betweenness plane geometry and its relationship with convex linear and projective plane geometries.pdf
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Proc. Estonian Acad. Sci. Phys. Math., 2007, 56, 3, 233–251
Betweenness plane geometry and its relationship
with convex, linear, and projective plane geometries
Ülo Lumiste
Institute of Pure Mathematics, Faculty of Mathematics and Computer Science, University of
Tartu, J. Liivi 2, 50409 Tartu, Estonia; lumiste@math.ut.ee
Received 5 April 2007, in revised form 8 June 2007
Abstract. In a previous paper the author recapitulated betweenness geometry, developed in
1904–64 by O. Veblen, J. Sarv, J. Hashimoto, and the author. The relationship of this geometry
with join geometry (by W. Prenowitz) was investigated. Now this relationship will be extended
to convex and linear geometry. The achievements of the well-developed projective plane
geometry are used to enrich betweenness plane geometry with coordinates, ternary operation,
algebraic extension, Lenz–Barlotti classification, translation, and Moufang type. The final
statement is that every Moufang-type betweenness plane is Desarguesian.
Key words: betweenness plane, ordered projective plane, ternary operation, Lenz–Barlotti
classification, Moufang-type plane.
1. INTRODUCTION
In the foundation of geometry the betweenness relation has fascinated the
investigators for a long time. Already C. F. Gauss, in his letter to F. Bolyai (6 March
1832; see [
1
], p. 222), pointed to the absence of betweenness postulates in Euclid’s
treatment. Elimination of this defect was started 50 years later by Pasch [
2
]. Further
development in the 19th century (through the works of G. Peano, F. Amodeo,
G. Veronese, G. Fano, F. Enriques, and M. Pieri) led to Hilbert’s fundamental
Grundlagen der Geometrie
[
3
], where the betweenness relation is subject to the
axioms of connection and of order
(I 1–7, II 1–5 of Hilbert’s list), called by
Schur [
4
] the
projective axioms
of geometry. Here the axioms of order II 1–5 are
presented as dependent on the axioms I 1–7.
In the first decade of the 20th century Hilbert’s projective axioms were
investigated by Moore [
5
] and Veblen [
6
]. Moore indicated some redundancy in
233
Hilbert’s axioms of order, which Hilbert took into consideration in the following
editions (e.g. in the seventh edition of [
3
]). But in these editions there was
not considered the question asked by Henri Poincaré in 1902: “Ne setait-il pas
préférable de donner aux axiomes du deuxième groupe une forme qui les affranchit
de cette dépendance et les séparât complètement du premier groupe?” (see [
7
],
Appendice. Ch. 1:
Les foundaments de la géométrie
, page 112; first published in
Journal des savants
, Mai 1902).
1
Veblen [
6
] was the first to respond to this question in 1904. He gave independent
axioms of betweenness relation and showed that the lines and planes can then be
defined as special sets of points.
Huntington [
8
¡
11
] gave an elaborated system of axioms for the betweenness
relation, but only in dimension one, i.e. for the case of a line.
This standpoint was developed further in Estonia, first by Nuut [
12
] in
1929 (for dimension one, as a geometrical foundation of real numbers). Then
Sarv [
13
] proposed in 1931 an axiomatics for the betweenness relation for
an arbitrary dimension
n
, extending the Moore–Veblen approach so that all
axioms of connection, including also those concerning lines, planes, etc., became
consequences. This self-dependent axiomatics was simplified and then perfected
by Nuut [
14
] and Tudeberg (from 1936 Humal) [
15
]. As a result, an extremely
simple axiomatics was worked out for
n
-dimensional geometry using only two
basic concepts: “point” and “between”.
The author of the present paper developed a comprehensive theory of the
models of betweenness
, based on this axiomatics [
16
]. At the same time he
established [
17
] that in dimension
>
2this model reduces to a convex domain in
an
n
-dimensional linear space over an ordered skew field.
2
As a whole, the theory
of these models, including also the Huntington–Nuut theory for dimension 1, can
be called
betweenness geometry
. The same term was introduced independently in
a similar situation by Hashimoto [
19
] in 1958.
In 1970–80 Rubinshtein [
20
¡
22
] developed (together with Rutkovskij) a theory
of
axial structures
. This theory is tightly connected with betweenness geometry
and uses some of its results (with exact references to [
16
;
17
]).
Independently there evolved also another approach, independent of the axioms
of connection. In 1909, Schur [
23
] tried to work out a part of geometry based on
the basic concepts “point” and “line segment” (Ger.
Strecke
). This approach was
elaborated in 1961 by Prenowitz [
24
] (see also [
25
;
26
]). The segment was considered
as the “join” of its endpoints, and so the
join operation
was introduced in the set
of points. This approach was then developed as the theory of
convexity spaces
in
the 1970s by Bryant and Webster [
27
], Doignon [
28
] and others (summarized in
papers [
29
¡
31
], and in monographs [
32
;
33
]).
1
“Wouldn’t it be better to give the axioms in the second group a form which makes them
free of this dependence and separates them from the first group?”
2
Later Pimenov [
18
] (in Appendix: Local betweenness relation) called this perfected
axiomatics the
Humal–Lumiste axiomatics
and its model in dimension 2, when the above
result cannot be used, the
Lumiste plane
.
234
In a recent paper [
34
] the author studied the relationship of betweenness
geometries with join geometries, treated in [
32
]. He proved there Theorems 14
and 15, according to which betweenness geometry coincides with the Pasch–Peano
geometry of [
28
]. Due to Theorems 16 and 18 of [
34
], betweenness geometry
coincides with convex geometry, according to the Theorem in Sec. 1.3 of [
28
].
Betweenness and convex geometries were developed by several authors to so-
called
linear geometry
(in the sense of [
32
;
33
;
35
;
36
]).
Betweenness geometry is tightly connected also with projective geometry (and
also with absolute geometry; see [
37
]). Already in [
16
], §20 and [
17
] it was
established that in the 3-dimensional betweenness space every bundle of lines
through a fixed point has the structure of a Desarguesian projective plane (see
also [
34
], Sec. 8). In the further study of the betweenness planes (which can also
be non-Desarguesian), which follows below, several constructions of the theory of
projective planes will be useful. First, however, betweenness geometry must be
recapitulated.
2. TOWARDS BETWEENNESS GEOMETRY
Recall that the author worked out betweenness geometry, considered here, more
than 40 years ago in [
16
;
17
], being guided by [
13
;
15
]. (Independently it was initiated
by Hashimoto [
19
].) Recently this geometry was recapitulated in [
34
] as follows.
Let
S
be a set, and let
B
be a subset in
S£S£S
(i.e. a
ternary relation
for
S
).
Further(
abc
)will mean that(
a;b;c
)
2
B
, then
b
is said to be
between
(or
inter
)
a
and
c
. Moreover, let us denote
habci
=(
abc
)
_
(
bca
)
_
(
cab
); [
abc
]=
habci_
(
a
=
b
)
_
(
b
=
c
)
_
(
c
=
a
)
:
(1)
The triplet(
a;b;c
)is said to be
correct
if
habci
, and
collinear
if[
abc
].
Definition 1.
The pair
(
S;
B
)
is called an
interimity model
and
B
the
interimity
relation
if
B1:(
a6
=
b
)
)9c;
(
abc
);B2:(
abc
)=(
cba
);
B3:(
abc
)
):
(
acb
);B4:
habci^
[
abd
]
)
[
cda
];B5:(
a6
=
b
)
)9c;:
[
abc
]
:
x2bc
L
ax
, is called a
plane
through
a
,
b
,
c
. Here
Q
b
and
Q
c
are obtained
by reordering
a;b;c
in the definition of
Q
a
; hence
P
abc
does not depend on this
reordering.
A one-to-one map
f
:
S!S
of an interimity model onto itself is said to
be a
cointerrelation
if(
abc
)
)
(
f
(
a
)
f
(
b
)
f
(
c
)), i.e. if the interimity relation
235
L
ab
[L
ac
S
For any two different
a;b;a6
=
b
the subset
ab
=
fxj
(
axb
)
g
is called an
interval
with ends
a
and
b
, and the subset
L
ab
=
fxj
[
xab
]
g
is said to be a
line
through
a
and
b
.
For any non-collinear
a;b;c
the subset
P
abc
=
Q
a
[Q
b
[Q
c
, where
Q
a
=
remains valid by
f
. Then this
f
is also a
collineation
, because due to (1),
[
abc
]
)
[
f
(
a
)
f
(
b
)
f
(
c
)], i.e. every collinear point-triplet maps into a collinear
point-triplet, hence every line maps into a line.
The basic concept will be introduced by the following
Definition 2.
If in an interimity model
,
in addition
,
B6:
:
[
abc
]
^
(
abd
)
^
(
bec
)
)9f;
((
afc
)
^
(
def
))
;
then this model is called a
betweenness model
and its relation is said to be the
betweenness relation (see [
16
;
17
]).
A subsidiary concept gives now the following
Definition 3.
If in an interimity model
B6
is replaced by
B6
0
:
:
[
abc
]
^
(
abd
)
^
(
aec
)
)9f;
((
bfc
)
^
(
dfe
))
;
then this model is called a
betwixtness
3
model
and its relation is said to be the
betwixtness relation.
The connecting instrument for the betweenness and betwixtness models is the
so-called
Pasch postulate
P:
:
[
abc
]
^
(
bec
)
^
(
d2P
abc
)
^
(
d62L
bc
)
^
(
a62L
de
)
)9f;
(
f2L
de
)
^
[(
afb
)
_
(
afc
)]
:
The above postulatesB6andB6
0
can be considered as forms of this Pasch
postulate in terms of the initial betweenness and betwixtness relation, respectively.
4
It is natural to start with the interimity model.
Lemma 4.
In an interimity model
(
abc
)
implies thata;b;care three distinct points.
Proof
. Indeed,B3excludes
b
=
c
, and together withB2excludes also
b
=
a
.
Finally,
a
=
c
is impossible as well, because if
c
=
a
, then
b6
=
a
and due to
B5
9d;:
[
abd
], but on the other hand(
abc
)
)hacbi
and(
a
=
c
)
)
[
acd
], and
these together imply due toB4that[
dba
]=[
abd
], but this contradicts
:
[
abd
]a
nd
finishes the proof.
3
Here the word “betwixt” has been in mind (which, according to dictionaries, is now archaic
except in the expression
betwixt and between
), as well as the word “interim”.
4
They are called the
outer
and
inner Pasch axioms
, respectively, and denoted by OP and IP
(see [
36
], where it is proved that IP does not imply OP, while OP does imply IP, accord-
ing to [
6
]; see also [
16
], Theorem 13, and [
34
], Theorem 11; this means that every between-
ness model is also a betwixtness model, but not vice versa).
236
If a triplet
a, b, c
is correct, i.e.
habci
, then due to Lemma 4 here
a;b;c
are three
different points, and due toB2,B3only one of them is between the two others.
Recall that if[
abc
], then
a, b, c
are said to be collinear. It is obvious that correctness
and collinearity of any three
a;b;c
does not depend on their order, i.e.
habci
=
hbcai
=
hcabi;
[
abc
]=[
bca
]=[
cab
]
:
(2)
Lemma 5.
In an interimity model leta;b;cbe collinear
,
i.e.
[
abc
]
and so
(1)
holds.
Here only the following four possibilities occur
:
1)(
a
=
b
)
_
(
b
=
c
)
_
(
c
=
a
), 2)(
abc
), 3)(
bca
), 4)(
cab
)
:
Each of them excludes the three others.
Proof
. The first possibility follows from Lemma 4. Due toB2,B3,(
abc
)=
(
cba
)
):
(
cab
),(
abc
)
):
(
acb
)=
:
(
bca
). Due to the same Lemma 4,
(
abc
)
):
[(
a
=
b
)
_
(
b
=
c
)
_
(
c
=
a
)].
Lemma 6.
In an interimity model there hold
:
[
abc
]
^habdi):
[
acd
]
;
(3)
:
[
abc
]
^
[
abd
]
^
[
adc
]
)
(
a
=
d
)
;
(4)
(
abc
)
^
(
bcd
)
)habdi;
(5)
:
[
abc
]
^
(
adb
)
^
(
aec
)
)d6
=
e:
(6)
Proof
. Let us suppose for (3), by
reductio ad absurdum
, that[
acd
]. Then due to (1)
andB4,
habdi^
[
acd
]=
hadbi^
[
adc
]
)
[
bca
]=[
abc
], but this is impossible.
For (4),
:
[
abc
]
)
(
a6
=
b
),
:
[
abc
]
^
[
adc
]
)
(
b6
=
d
); now by
reductio ad
absurdum
,
[
abd
]
^
(
a6
=
b
)
^
(
b6
=
d
)
^
(
a6
=
d
)
)habdi
=
hadbi
and then due toB4
hadbi^
[
adc
]
)
[
bca
]=[
abc
]
;
but this is impossible.
For (5), due to Lemma 4,(
abc
)
^
(
bcd
)
)
(
a6
=
b
)
^
(
b6
=
d
). Also
d6
=
a
,
because otherwise, due toB2,(
bcd
)=(
bca
)=(
acb
)and, now due toB3,
:
(
abc
),
which is impossible. Further, due to (1),(
abc
)
^
(
bcd
)=
hbcai^
[
bcd
], and now
due toB4,[
adb
], which is, due to (1), equivalent with[
abd
], but this together with
(
a6
=
b
)
^
(
b6
=
d
)
^
(
d6
=
d
)implies
habdi
, as needed.
For (6),(
adb
)
)
[
abd
], and now, by
reductio ad absurdum
, if one supposes
d
=
e
, then(
aec
)=(
adc
)
)
[
adc
], and (4) would yield
a
=
d
. On the other hand,
due to (1),(
adb
)
)a6
=
d
, which gives a contradiction.
This finishes the proof.
For a line the following assertions can be proved, which show that in an
interimity model the points
a;b
are not some specific points of a line
L
ab
, but can
be exchanged by every two of its different points
c;d
. Indeed, there holds
237
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