Notes on Face-listings for
“Classification of Superpotentials"
by A. Dancer and M. Wang
These
notes provide a listing of possible 2-dimensional faces of conv(W), where
W
is the set of weight vectors for the scalar curvature function of a compact
homogeneous
space G/K where the isotropy representation is multiplicity free.
While
they may be of independent interest, these notes are mainly intended as a
supplement
to section 6 of our paper "Classification of Superpotentials".
Accordingly,
we shall use the notation and definitions given there. In particular,
these
notes also list the possible triangles that can occur in Theorem 6.12 of
the
above paper.
GENERAL
REMARKS
We
proceed by studying all possible triples of vectors in W whose affine span
is
a 2-plane. In most cases there are no further vectors in this plane, so we
obtain
a triangle. In some cases there are further vectors in the 2-plane; it
turns
out that the full set of possible vectors in the 2-plane may give a
hexagon,
square, trapezium or parallelogram.
For
each possible triangle (including subtriangles of the hexagons,
parallelograms
etc) we test whether it may give a triangle as in Thm 6.12(i).
We
list the possible vectors
x'' x
x' 3c 3a
3a'
and
test whether it satisfies the conditions of Thm 6.12(i), i.e., that
c,a,a'
are all null and that x'' is orthogonal to a,a'. The cases where this
condition
can hold give examples (Tr1)-(Tr22).
We
recall that if x''xx' gives a triangle satisfying the conditions of
Thm
6.12(i) then x'' cannot be type I. Moreover if x'' is type III, say
(-2^i,
1^j), then x_i = x'_i iff x_i = x'_j.
In
some cases the shape in the 2-plane cannot be a face. In some of these
cases
it is possible that a subtriangle may still be a face, so for these we
still
have to check if the conditions of Thm 6.12(i) can hold.
We
note that configurations involving the following (column) vectors will
never
give a face besides case 0) (see below):
-2 1
or 1 -1
1
-1 -1
-1 -1
This
is because the face condition forces a spanning set for the 2-plane in
0)
to lie in the given 2-plane. So if the latter has a further vector not in
0)
we arrive at a contradiction.
Also
if the configuration
-1
1 -1
-1 1
-1
occurs,
then the face condition implies that
-1
-1 1
1 -1
-1
also
lies in the 2-plane. Many cases can be eliminated by this observation.
Recall
from Remarks 6.13, 6.14 that no triangle containing points of W in the
interior
of an edge can satisfy the conditions of Thm 6.12. Hence we do not
need
to treat such triangles.
We
frequently make use without comment of symmetries in the configuration to
reduce
the number of cases that need be checked.
Finally,
we also check which triangles can satisfy the conditions of Theorem
6.12(ii).
These conditions are symmetric with respect to x'', x, x'. Recall
that
now one vector, say x'', must be type I.
Moreover c,a,a' must be null.
Writing
x'' = (-1^i), we find that nullity of a,a' implies x_i = x'_i.
CONTENTS
The
different cases can be grouped according to the types of the vectors in a
spanning
set.
0) to 15) Three type III
16) to 22) Two type III and a type I
23) to 79) Two type III and a type II
80) to 90a) A type III, a type II and a type I
91) to 154) A type III and two type II
155)
Two type I and a type II
156) to 177) Two type II and one type I
178) to end Three type II vectors
Note
that the case of three type I is included in 0). Also, the case
of
a type III and two type I is dealt with in the comment after 90a).
0) If all three vectors are zero outside a
common set of three indices, we
have
the hexagon (H1) lying in the 2-plane
X_1 + X_2 + X_3 =-1 X_i =0 for i > 3.
The only way to get subtriangles of the
hexagon with no interior points
of
edges is by taking the three type I vectors.
x'' x
x' 3c 3a
3a'
-1 0
0 1 -1
-1 a,c not both null
0 -1
0 -2 -4
2
0
0 -1 -2
2 -4
***************
In
future, therefore, we need only consider triples which between them
involve
nonzero entries in more than three places.
We
first consider 2-planes including three type III vectors.
***************
x'' x
x' 3c 3a
3a'
1)
triangle
-2 -2
-2 -6 -6
-6
1 0
0 -1 1
1 a,c not both null
0 1
0 2 4
-2
0 0
1 2 -2
4
***************
2) -2 -2
0
1 0
1
0 1
0
0 0
-2
The 2-plane is given by
X_1 + X_4 =-2, X_2 + X_3 =1,
X_i =0 : i > 4
and contains in addition the vectors
0 -1
-1
0 1
0
1 0
1
-2 -1
-1
This is the rectangle (P17). We must
consider subtriangles
x'' x
x' 3c 3a
3a'
-2 -2
-1 -4 -8
-2
1 0
1 1 -1
5 a,c not both null
0 1
0 2
4 -2
0 0
-1 -2 2
-4
-2 -2
0 -2 -10
2
1 0
1 1 -1
5 ditto
0 1
0 2 4
-2
0 0
-2 -4 4
-8
-2 -1
0 0 -6
0
1 0
1 1 -1
5
0 1
0 2 4
-2 a',c not both null
0 -1
-2 -6 0
-6
-2 -1
0 0 -6
0
1 1
0 1 5
-1 ditto
0 0
1 2 -2
4
0 -1
-2 -6 0
-6
-1 -2
-1 -5 -7
-1 orthogonality implies
1 0
0 -1 1
1 (d_1,d_2)=(3,1) so a
0 1
1 4 2
2 not null
-1 0
-1 -1 1
-5
-1 -2 0
-3 -9 3
as above
1 0
0 -1 1
1
0 1
1 4 2
2
-1 0
-2 -3 3
-9
-1 -2
-2 -7 -5
-5 orthogonality
1 1
0 1 5
-1 conditions cannot
0 0
1 2 -2
4 both hold
-1 0
0 1 -1
-1
-1 -2
-1 -5 -7
-1
1
1 0 1
5 -1 a,c not both null
0 0
1 2 -2
4
-1 0
-1 -1 1
-5
-1 -2
0 -3 -9
3
1 1
0 1 5
-1 ditto
0 0
1 2 -2
4
-1 0
-2 -3 3
-9
**************
3) -2
-2 0 now 0 0 also
lie in the 2-plane
1 0
-2 0 -1
0 1
0 -2 -1
0 0
1 1 1
X_2+ X_3 + 3 X_4 =1, X_1 + X_2 + X_3 +
X_4 =-1, X_i = 0 : I > 4.
This is the trapezium (T1).
we must consider subtriangles
0 -2
0 -4 -8
4
-2 0
-1 0 0
-6
0 1
-1 0 6
-6 a,c not both null
1 0
1 1 -1
5
0 -2
0 -4 -8
4
-2 1
0 4 2
-4 a',c not both null
0 0
-2 -4 4
-8
1 0
1 1 -1
5
0 -2
0 -4 -8
4
-2 1
-1 2
4 -8
0 0
-1 -2 2
-4 ditto
1 0
1 1 -1
5
-2 -2
0 -2 -10
2
0 1
-1 0 6
-6
1 0
-1 -3 3
-3 a,c not both null
0 0
1 2 -2
4
-2 -2
0 -2 -10
2
0 1
0 2 4
-2 ditto
1 0
-2 -5
5 -7
0 0
1 2 -2
4
0 -2
-2 -8 -4
-4
-1 0
1 3 -3
3
-1 1
0 3 3
-3 ditto
1 0
0 -1 1
1
0 0
-2 -4 4
-8
-1 -2
0 -3 -9
3 ditto
-1 0
1 3 -3
3
1 1
0 1 5
-1
0
0 -2 -4
4 -8
-1 -2
1 -1 -11
-7 ditto
-1 0
0 1 -1
-1
1 1
0 1 -5
-1
***************
4)triangle -2
0 0 2
-2 -2
1 -2
-2 -9 -3
-3 orthogonality fails as
0 1
0 2 4
-2 d_1 is not 1
0 0
1 2 -2
4
***************
5)triangle -2
0 0
2 -2 -2
1 0
0 -1 1
1
0 -2
-2 -8 -4
-4 This is (Tr1)
0 1
0 2 4
-2
0 0
1 2 -2
4
***************
6)triangle
-2 0
0 2 -2
-2
1 0
0 -1 1
1
0 -2
0 -4 -8
4 a,c not both null
0 1
0 2 4
-2
0 0
-2 -4 4
-8
0 0
1 2 -2
4
***************
7) -2 1
0 -1 0
1 -2
0 now 0
-1 also occur.
0 0
-2 0 0
0 0
1 0 0
This
is a triangle with two interior points on one side. By Remark 6.13, 6.14
the
subtriangles to consider are
-2 0
-1 0 0
-6
1 0
0 -1 1
1 a', c not both null
0 -2
0 -4 -8
4
0 1
0 2 4
-2
0 -2
-1 -6 -6
0
0 1
0 2 4
-2
-2 0
0 2 -2
-2 ditto
1 0
0 -1 1
1
0 -1
0 -2 -4
2
0 0
-1 -2 2
-4 ditto
-2 0
0 2 -2
-2
1 0
0 -1 1
1
***************
8)
triangle
0 -2
0 -4 -8
4
-2 1
0 4 2
-4 a', c not both null
1
0 -2 -5
5 -7
0 0
1 2 -2
4
***************
9) -2
0 0
1 1
0 equivalent to 3)
0 -2 1
0 0 -2
***************
10) -2
0 1
1 -2 0
equivalent to 8)
0 1 0
0 0 -2
***************
11) -2
0 0 -1
1 -2 1
now 1 is present.
0 0 -2 0
0 1
0 -1
This
is a triangle with a midpoint of one side. Subtriangles are
-2 0 -1
0 0 -6
1 -2
1 -3 -9
9 a, c not both null
0 0
-1 -2 2
-4
0 1
0 2 4
-2
-1 0
-2 -3 3
-9
1 -2
1 -3 -9
9 ditto
-1 0
0 1 -1
-1
0 1
0 2 4
-2
0 -2
0 -4 -8
4
-2 1
1 6 0
0
0 0
-2 -4 4
-8 orthogonality fails
1 0
0 -1 1
1
0 -2
-1 -6 -6
0
-2 1
1 6 0
0
0 0
-1 -2 2
-4 a,c not both null
1 0
0 -1 1
1
***************
12) -2
0 0 -1
0 -1
1 1
1 Now 1
1 1
0 -2
0 0 -1
-1
0 0
-2 -1 -1 0
are
also in the 2-plane X_2 =1 : X_i =0 for i > 4. This is a triangle with
midpoints
of all sides.
Subtriangles are
-2 0
0 2 -2
-2
1 1
1 3 3
3
0 -2
0 -4 -8
4 a,c not both null
0 0
-2 -4 4
-8
-2 0
0 2 -2
-2
1 1
1 3 3
3
0 -2
-1 -6 -6
0 this is (Tr2)
0 0
-1 -2 2
-4
-2
-1 -1 -2
-4 -4
1 1
1 3 3
3 a,c not both null
0 -1
0 -2 -4
2
0 0
-1 -2 2
-4
0 -2
0 -4 -8
4
1 1
1 3 3
3 ditto
-1 0
-2 -3 3
-9
-1 0
0 1 -1
-1
-1 -2
-1 -5 -7
-1
1 1
1 3 3
3 ditto
-1 0
0 1 -1
-1
0 0
-1 -2 2
-4
-1 -2
0 -3 -9
3
1 1
1 3 3
3 ditto
-1
0 -1 -1
1 -5
0 0
-1 -2 2
-4
0 -2
-1 -6 -6
0
1 1
1 3 3
3
-1 0
-1 -1 1
-5 orthogonality fails
-1 0
0 1 -1
-1
-1 0
-1 -1 1
-5 nullity and
1 1
1 3 3
3 othogonality are
-1 -1
0 -1 -5
1 contradictory
0 -1
-1 -4 -2
-2
***************
13)
triangle -2 0
0 2 -2
-2
1 0
0 -1 1
1
0 -2
0 -4 -8
4
0
1 -2 -2
8 -10 a,c not both null
0 0
1 2 -2
4
***************
14) -2
0 0 now
-1 is present.
1
1 0 1
0
-2 0 -1
0
0 -2 0
0
0 1 0
This
is a triangle with midpoint of one edge.
-2 0
-1 0 0
-6
1 0
1 1 -1
5
0 0
-1 -2 2
-4 a,c not both null
0 -2
0 -4 -8
4
0 1
0 2 4
-2
-2 0
0 2 -2
-2
1 0
1 1 -1
5
0 0
-2 -4
4 -8 ditto
0 -2
0 -4 -8
4
0 1
0 2 4
-2
-1 0
-2 -3 3
-9
1 0
1 1 -1
5
-1 0
0 1 -1
-1 ditto
0 -2
0 -4 -8
4
0 1
0 2 4
-2
0 -2
-1 -6 -6
0
0 1
1 4
2 2
0 0
-1 -2 2
-4 ditto
-2 0
0 2 -2
-2
1 0
0 -1 1
1
0 -2
0 -4 -8
4
0
1 1 4
2 2
0 0
-2 -4 4
-8 this is (Tr3)
-2 0
0 2 -2
-2
1 0
0 -1 1
1
***************
15) -2
0 0
1
0 0
0 -2 1
equivalent to 13)
0
1 0
0
0 -2
***************
Now
we consider two type III and one type I.
16) -2
-2 0 now
0 0 are present in the 2-plane
1
0 0 1 -1
0
1 0 -1
1
0
0 -1 -1
-1
given
by X_1 + X_2 + X_3 + X_4 = -1, X_2 + X_3 - X_4 = 1, X_i = 0: i > 4.
This
is trapezium (T2). The type I must be present; if one type II is present
the
other is. So the only way to obtain a triangle without midpoints is to
consider
the type I and the two type III. This is ruled out by Remark 6.15
***************
17)
triangle
0 -2
0 -4 -8
4
-2 1
0 4 2
-4 a',c not both null
1 0
0 -1 1
1
0 0
-1 -2 2
-4
***************
18) -2
0 0
1
0 0
0 -2 -1 equivalent to 7)
0
1 0
0
0 0
***************
19) -2
0 0
1
0 0
0 -2
0 equivalent to 7)
0
1 -1
0
0 0
***************
20) -2
0 0 Now
-1 1 -1 are also in the 2-plane
1
1 0 0
0 1
0 -2 0
1 -1 -1
0
0 -1 -1 -1 0
X_1
+ 2X_2 + X_3 = 0, X_2 - X_4 =1, X_i =0 for i>4.
This
is the parallelogram (P16). Subtriangles to consider are
-2 0
-1 0 0
-6
1 0
1 1
-1 5 a,c not both null
0 0
-1 -2 2
-4
0 -1
0 -2 -4
2
-1 0
-2 -3 3
-9
1 0
1 1 -1
5 ditto
-1 0
0 1 -1
-1
0 -1
0 -2 -4
2
***************
21)
triangle -2 0 0
1 0 0
0 -2 0
0 1 0
0 0 -1
-2 0
0 2 -2
-2
1 0
0 -1 1
1
0 -2
0 -4 -8
4 a,c not both null
0 1
0 2 4
-2
0
0 -1 -2
2 -4
***************
If
there are two type I in the 2-plane then there are two type III. So we do
not
need to consider type III and two type I further.
For
example
22) -2
0 0
1
0 0 is equivalent to 7)
0 -1
0
0
0 -1
***************
Next
we consider 2-planes including two type III and a type II.
Observe
that any 2-plane including -2 and -1 will also include 0
1 1 1
0 -1 -2
so
we do not consider further examples with
two type III and a type II of
the
above form.
For
example:
23) -2 -2
-1 or -1
1
0 0 1
is equivalent to 2).
0
1 1 0
0
0 -1 -1
***************
24) -2
-2 -1 now
-1 is in the plane
1
0 0 -1
0
1 -1 0
0
0 1 1
X_2 + X_3 + 2X_4 =1, X_1 + X_2 + X_3 +
X_4 = -1, X_i =0 : i > 4.
This is parallelogram (P1). Subtriangles
to consider are:
-2 -2
-1 -4 -8
-2
1 0
-1 -3 3
-3 a,c not both null
0 1
0 2 4
-2
0 0
1 2 -2
4
-1 -2
-2 -7 -5
-5
0 1
0 2 4
-2 a', c not both null
-1
0 1 3
-3 3
1 0
0 -1 1
1
-1 -2
-1 -5 -7
-1
0 1
-1 0 6
-6 a, c not both null
-1 0
0 1 -1
-1
1 0
1 1 -1
-5
-1 -2
-1 -5 -7
-1
0 0
-1 -2 2
-4 ditto
-1 1
0 3 3
-3
1 0
1 1 -1
5
***************
25)
triangle
-1 -2
-2 -7 -5
-5
0 1
0 2 -2
4 orthogonality implies
0 0
1 2 4
-2 d_1 \leq 5, which
1
0 0 -1
1 1 contradicts nullity of c
-1 0
0 1 -1
-1
***************
26) -2
-2 1 now 1 is
also present.
1
0 0 -1
0
1 -1 0
0 0
-1 -1
This
is a parallelogram, but not a face by remark ..
-2 -2
1 0 -12
6
1 0
-1 -3 3
-3
0 1
0 2 4
-2 a,c not both null
0
0 -1 -2
2 -4
1 -2
1 -3 -9
9
0 0
-1 -2 2
-4
-1 1
0 3 3
-3 ditto
-1 0
-1 -1 1
-5
1 -2
-2 -9 -3
-3
0 0
1 2 -2
4 ditto
-1 1
0 3 3
-3
-1 0
0 1 -1
-1
1 1
-2 -3 9
-9
0 -1
1 0 -6
6 ditto
-1 0
0 1 -1
-1
-1 -1
0 -1 -5
1
***************
27)
triangle
1 -2
-2 -9 -3
-3
0 1
0 2 4
-2
0 0
1 2 -2
4 orthogonality fails
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
28) -2
-2 0 Now
0 is in 2-plane; equivalent to
3).
1
0 -1 0
0
1 -1 -2
0
0 1 1
***************
29) -2
-2 0
1
0 1 This is equivalent to 16).
0
1 -1
0
0 -1
***************
30) -2
-2 0 Now
0 is also present.
1
0 0 1
0
1 1 0
0
0 -1 -1
0
0 -1 -1
This
is parallelogram (P2). Subtriangles to consider are
-2 -2
0 -2 -10
2
1 0
1 1 -1
5
0 1
0 2 4
-2 a,c not both null
0 0
-1 -2 2
-4
0
0 -1 -2
2 -4
0 -2
0 -4 -8
4
0 0
1 2 -2
4
1 1
0 1 5
-1 ditto
-1 0
-1 -1 1 -5
-1 0
-1 -1 1
-5
0 -2
0 -4 -8
4
0 1
1 4 2
2
1 0
0 -1 1
1 orthogonality fails
-1 0
-1 -1 1
-5
-1 0
-1 -1 1
-5
0 -2
-2 -8 -4
-4
0 1
0 2 4
-2
1 0
1 1 -1
5 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
31)
-2 -2 0
Now 0 is in the 2-plane
1 0 -1
0
0 1
0 -1
0 0 -1
-1
0 0
1 1
This
is parallelogram (P3). Subtriangles to consider are
-2 -2
0 -2 -10
2
1 0
-1 -3 3
-3
0 1
0 2 4
-2
0 0
-1 -2 2
-4 a,c not both null
0 0
1 2 -2
4
0 -2
0 -4 -8
4
0 0
-1 -2 2
-4
-1 1
0 3 3
-3 ditto
-1 0
-1 -1 1
-5
1 0
1 1 -1
5
0 -2
-2 -8 -4
-4
0 0
1 2 -2
4 ditto
-1 1
0 3 3
-3
-1 0
0 1 -1
-1
1 0
0 -1 1
1
0 0
-2 -4 4
-8
0 -1
1 0 -6
6
-1 0
0 1 -1
-1
-1 -1
0 -1 -5
1 ditto
1 1
0 1 5
-1
***************
32)
triangle
0 -2
-2 -8 -4
-4
0 1
0 2 4
-2
0 0
1 2 -2
4
1 0
0 -1 1
1 this is (Tr11)
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
33)
-2 1
-1
1
-2 0
0
0 -1
0
0 1
This
lies in the 2-plane X_1 + X_2 =-1; X_3 + X_4 =0; X_i =0: i>4.
Now -1 0
-1 are also in the 2-plane.
0
-1 0
1
0 0
-1
0 0
This
is hexagon H2. There is no way to get a subtriangle without midpoints.
***************
34) -2
1 1 Now
0 -1 0 are also in the 2-plane
1
-2 0 1
0 -1
0
0 -1 -1
0 0
0
0 -1 -1
0 0
X_3
= X_4, X_1 + X_2 + X_3 + X_4 = -1, X_i =0: i > 4. This is the
trapezium
(T*1).
We must consider subtriangles
0 -1
0 -2 -4
2
1 0
-1 -3 3
-3 a,c not both null
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
1 -2
-1 -7 -5
1
0 1
0 2
4 -2 a, a' not both null
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
35) -2
1 0 Now -1
0 are also in the 2-plane.
1
-2 0 0 -1
0
0 -1 0
0
0
0 -1 0
0
0
0 1 0
0
This
is a triangle with 2 interior points on one side.
Subtriangles
to consider are
-2 0
-1 0 0
-6
1
0 0 -1
1 1
0 -1
0 -2 -4
2 a,c not both null
0 -1
0 -2 -4
2
0 1
0 2 4
-2
0 -2 -1 -6
-6 0
0 1
0 2 4
-2 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
1 0
0 -1 1
1
0 -2
0 -4 -8
4
0 1
-1 0 6
-6
-1 0
0 1 -1
-1 ditto
-1 0
0 1 -1
-1
1 0
0 -1 1
1
0 -2
1 -2 -10
8
0 1
-2 -2 8
-10
-1 0
0 1 -1
-1 ditto
-1 0
0 1 -1
-1
1 0
0 -1 1 1
0 -1
0 -2 -4
2
0 0
-1 -2 2
-4
-1 0
0 1 -1
-1 ditto
-1 0
0 1 -1
-1
1 0
0 -1 1
1
***************
36) -2
0 -1 equivalent to 12)
1
1 1
0 -2
0
0
0 -1
***************
37) -2
0 -1 equivalent to 20)
1
1 0
0 -2
1
0
0 -1
***************
38) -2
0 1 now
0 -1 are in the same 2-plane
1
1 -1 -1 1
0 -2
0 1 -1
0
0 -1 -1 0
This is the trapezium (T*2).
Subtriangles to consider are
-2 0 -1
0 0 -6
1 -1
1 -1 -5
7 a,c not both null
0 1
-1 0 6
-6
0 -1
0 -2 -4
2
0 -2 -1 -6
-6 0
-1 1
1 5 1
1 ditto
1 0
-1 -3 3
-3
-1 0
0 1 -1
-1
0 1
-1 0 6
-6
-1 -1
1 1 -7
5 ditto
1 0
-1 -3 3
-3
-1 -1
0 -1 -5
1
-1 0
1 3 -3
3
1 -1
-1 -5 -1
-1 ditto
-1 1
0 3 3
-3
0 -1
-1 -4 -2
-2
-1 -2
0 -3 -9
3
1 1
-1 -1 7
-5 ditto
-1 0
1 3 -3
3
0 0
-1 -2 2
-4
***************
39) -2
0 -1 equivalent to 2)
1
1 0
0 -2
-1
0
0 1
***************
40) -2
0 -1 Now -1
0 are also in the 2-plane
1
1 -1 1
-1
0 -2
0 -1 -1
0
0 1 0
1
X_2
+ 2 X_4 =1, X_1 + X_2 + X_3 + X_4 = -1, X_i = 0 : i > 4.
This
is trapezium (T3). Subtriangles to consider are
-2 -1
0 0 -6
0
1 1
-1 -1 7
-5
0 -1
-1 -4 -2
-2 orthogonality fails
0 0
1 2 -2
4
-2 0
-1 0 0
-6
1 1
-1 -1 7
-5
0 -2
0 -4 -8
4 a,c not both null
0 0
1 2 -2
4
-1 0
-2 -3 3
-9
-1 -1
1 1 -7
5 ditto
0 -1
0 -2 -4
2
1 1
0 1 5
-1
-1 0
-1 -1 1
-5 orthogonality implies
-1 -1
1 1 -7
5 d_1 =1, so a' is not
0 -1
-1 -4 -2
-2 null
1 1
0 1 5
-1
-1 0
0 1 -1
-1
-1 -1
1 1 -7
5 a', c not both null
0
-1 -2 -6
0 -6
1 1
0 1 5
-1
-1 -2
0 -3 -9
3
-1 1
1 5 1
1 orthogonality fails
0 0
-2 -4 4
-8
1 0
0 -1 1
1
-1 -2
-1 -5 -7
-1
-1 1
1 5 1
1 ditto
0 0
-1 -2 2
-4
1 0 0
-1 1 1
-1 0
-1 -1 1
-5
-1 1
1 5 1
1 a,c not both null
0 -2
-1 -6 -6
0
1 0
0 -1 1
1
-1 -1
0 -1 -5
1 orthogonality implies
1 -1
-1 -5 -1
-1 d_1 =1, so a is not
-1 0
-1 -1 1
-5 null
0 1
1 4 2
2
-1 -2
-1 -5 -7
-1
1 1
-1 -1 7
-5 a,c not both null
-1 0
0 1 -1
-1
0 0
1 2 -2
4
-1 -2
0 -3 -9
3
1 1
-1 -1 7
-5
-1 0
-1 -1 1
-5 ditto
0 0
1 2 -2
4
***************
41) -2
0 0 Now -1
is also in the 2-plane.
1
1 1 1
0 -2
0 -1
0
0 -1 0
0
0 -1 0
This is a triangle with midpoint of one
edge present.
-2 0
0 2 -2
-2
1
1 1 3
3 3
0 -2
0 -4 -8
4 a,c not both null
0 0
-1 -2 2
-4
0 0
-1 -2 2
-4
-1 0
-2 -3
3 -9
1 1
1 3 3
3
-1 0
0 1 -1
-1 ditto
0 -1
0 -2 -4
2
0 -1
0 -2 -4
2
0 -2
0 -4 -8
4
1 1
1 3 3
3
0 0
-2 -4 4
-8 ditto
-1 0
0 1 -1
-1
-1 0 0
1 -1 -1
0
-2 -1 -6
-6 0
1 1
1 3 3
3
0 0
-1 -2 2
-4 this is (Tr12)
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
42 -2
0 1 Now
0 -1 are in the 2-plane also,
1
1 0 0
1
0 -2
0 1 -1
0
0 -1 -1 0
0
0 -1 -1 0
given
by X_4 = X_5, X_2 - X_5 = 1, X_1 + X_2 + X_3 + X_4 + X_5 =-1, X_i =0 :
i
> 5. This is the trapezium (T4).
-2 0
-1 0 0
-6
1 0
1 1 -1
5
0 1
-1 0 6
-6 a,c not both null
0
-1 0 -2
-4 2
0 -1
0 -2 -4
2
-2 1
-1 2 4
-8
1 0
1 1 -1
5
0 0
-1 -2 2
-4 ditto
0 -1
0 -2 -4
2
0 -1
0 -2 -4
2
-2 1
0 4 2
-4
1 0
1 1 -1
5
0 0
-2 -4 4
-8 a', c not both null
0 -1
0 -2 -4
2
0 -1
0 -2 -4
2
0 1
-2 -2 8
-10
0 0
1 2 -2
4
1
0 0 -1
1 1 ditto
-1 -1
0 -1 -5
1
-1 -1
0 -1 -5
1
0 1
-1 0 6
-6
0 0
1 2 -2 4
1 0
-1 -3 3
-3 ditto
-1 -1
0 -1 -5
1
-1 -1
0 -1 -5
1
0 1
0 2 4
-2
0 0
1 2 -2
4
1 0
-2 -5 5
-7 ditto
-1 -1
0 -1 -5
1
-1 -1
0 -1 -5
1
0 -2
-1 -6 -6
0
0 1 1
4 2 2
1 0
-1 -3 3
-3 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
0 -2
0 -4 -8
4
0 1
1 4 2
2
1 0
-2 -5 5
-7 orthogonality fails
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
0 -1
0 -2 -4
2
0 1
1 4 2
2
1 -1
-2 -7 1
-5 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
-1 0
1 3 -3
3
1 0
0 -1 1
1
-1 1
0 3 3
-3 a,c not both null
0 -1
-1 -4 -2
-2
0 -1
-1 -4
-2 -2
-1 -2
1 -1 -11
7
1 1
0 1 5
-1
-1 0
0 1 -1
-1 ditto
0 0
-1 -2 2
-4
0 0
-1 -2 2
-4
-1 -2
0 -3 -9
3
1 1
0 1 5
-1 ditto
-1 0
1 3 -3
3
0 0
-1 -2 2
-4
0
0 -1 -2
2 -4
***************
43) -2
0 -1 now
0 -1 are also in the 2-plane
1
1 0 0
1
0 -2 0
-1 -1
0
0 1 1
0
0
0 -1 -1
0
X_4
+ X_5 =0, X_2 + X_4 =1, X_1 + X_2 + X_3 = -1, X_i = 0 : i > 5.
This
is trapezium (T5). Subtriangles to consider are
-2
-1 0 0
-6 0
1
0 1 1
-1 5 a,c not both null
0
0 -2 -4
4 -8
0
1 0 2
4 -2
0
-1 0 -2
-4 2
-2
0 -1 0
0 -6
1
0 1 1
-1 5
0
-1 -1 -4
-2 -2 orthogonality fails
0
1 0 2
4 -2
0
-1 0 -2
-4 2
-1
0 -2 -3
3 -9
0
0 1 2
-2 4 a,c not both null
0
-1 0 -2
-4 2
1
1 0 1
5 -1
-1
-1 0 -1
-5 1
-1
0 -1 -1
1 -5
0
0 1 2
-2 4 orthogonality implies d_1 =1
0
-1 -1 -4
-2 -2 so a is not null
1
1 0 1
5 -1
-1
-1 0 -1
-5 1
-1
0 0 1
-1 -1
0
0 1 2
-2 4 a', c not both null
0
-1 -2 -6
0 -6
1
1 0 1
5 -1
-1
-1 0 -1
-5 1
-1
-2 -1 -5
-7 -1
0
1 1 4
2 2
0
0 -1 -2
2 -4 orthogonality fails
1
0 0 -1
1 1
-1
0 0 1
-1 -1
-1
-2 0 -3
-9 3
0
1 1 4
2 2
0
0 -2 -4
4 -8 ditto
1
0 0 -1
1 1
-1
0 0 1
-1 -1
-1
-1 0 -1
-5 1
0
1 1 4
2 2 a',c not both null
0
-1 -2 -6
0 -6
1
0 0 -1
1 1
-1
0 0 1
-1 -1
-1
-1 0 -1
-5 1 orthogonality implies
1
0 0 -1
1 1 d_1 =2, d_2 =1
-1
0 -1 -1
1 -5 so a not null
0
1 1 4
2 2
0
-1 -1 -4
-2 -2
-1
-2 -1 -5
-7 -1
1
1 0 1
5 -1 a,c not both null
-1
0 0 1
-1 -1
0
0 1 2
-2 4
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
1
1 0 1
5 -1
-1
0 -1 -1
1 -5 ditto
0
0 1
2 -2 4
0
0 -1 -2
2 -4
***************
44) -2
0 0 The vector -1 is also in 2-plane.
1
1 -1 1
0 -2 0 -1
0
0 1 0
0
0 -1 0
This is triangle with midpoint of one
edge.
-2
0 -1 0
0 -6 a,c not both null
1
-1 1 -1
-5 7
0
0 -1 -2
2 -4
0
1 0 2
4 -2
0
-1 0
-2 -4 2
-1
0 -2 -3
3 -9
1
-1 1 -1
-5 7 ditto
-1
0 0 1
-1 -1
0
1 0 2
4 -2
0
-1 0 -2
-4 2
0
-2 0 -4
-8 4
-1 1 1
5 1 1
orthogonality fails
0
0 -2 -4
4 -8
1
0 0 -1
1 1
-1
0 0 1
-1 -1
0
-2 -1 -6
-6 0
-1
1 1 5
1 1 a,c not both null
0
0 -1 -2
2 -4
1
0 0 -1
1 1
-1
0 0 1
-1 -1
***************
45) -2
0 0 -1
is also in 2-plane.
1
1 0 1
0
-2 0 -1
0
0 -1 0
0
0 -1 0
0
0 1 0
This
is a triangle with midpoint of one edge.
-2
0 -1 0
0 -6
1
0 1 1
-1 5
0
0 -1 -2
2 -4 a,c not both null
0
-1 0 -2
-4 2
0
-1 0 -2
-4 2
0
1 0 2
4 -2
-1
0 -2 -3
3 -9
1
0 1 1
-1 5
-1
0 0 1
-1 -1 ditto
0
-1 0 -2
-4 2
0
-1 0 -2
-4 2
0
1 0 2
4 -2
0
-2 -1 -6
-6 0
0
1 1 4
2 2
0
0 -1 -2
2 -4 ditto
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
1
0 0 -1
1 1
0
-2 0 -4
-8 4
0
1 1 4
2 2
0
0 -2 -4
4 -8
-1
0 0 1
-1 -1 This is (Tr 13)
-1
0 0 1
-1 -1
1
0 0 -1
1 1
***************
46) -2
0 -1 equivalent to 11)
1
-2 1
0
1 0
0
0 -1
***************
47) -2
0 -1 now
-1 is also in 2-plane
1
-2 0 -1
0
1 1 0
0
0 -1 1
This is the parallelogram (P4).
Subtriangles to consider are
-2
0 -1 0
0 -6
1
-2 0 -5
-7 5 orthogonality fails
0
1 1 4
2 2
0
0 -1 -2
2 -4
-2
0 -1 0
0 -6
1
-2 -1 -7
-5 1 ditto
0
1 0 2
4 -2
0
0 1 2
-2 4
0
-2 -1 -6
-6 0
-2
1 0 4
2 -4
1
0 1 1
-1 5 a,c not both null
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
-2
1 -1 2
4 -8 ditto
1
0 0 -1
1 1
0
0 1 2
-2 4
0
-1 -1 -4
-2 -2
-2
0 -1 0
0 -6 orthogonality fails
1
1 0 1
5 -1
0
-1 1 0
-6 6
-1
-2 0 -3
-9 3
0
1 -2 -2 8
-10 a,c not both null
1
0 1 1
-1 5
-1
0 0 1
-1 -1
-1
-2 -1 -5
-7 -1
0
1 -1 0
6 -6 ditto
1
0 0 -1
1 1
-1
0 1 3
-3 3
-1
0 -1 -1
1 -5
0
-2 -1 -6
-6 0 ditto
1
1 0 1
5 -1
-1 0
1 3 -3 3
-1
-2 0 -3
-9 3
-1
1 -2 -1 7
-11 ditto
0
0 1 2
-2 4
1
0 0 -1
1 1
-1
-2 -1 -5
-7 -1
-1
1 0 3
3 -3 ditto
0
0 1 2
-2 4
1
0 -1 -3
3 -3
-1
0 -1 -1
1 -5 orthogonality implies
-1
-2 0 -3
-9 3 d_1=1, d_2 =3 so a'
0
1 1 4
2 2 is not null
1
0 -1 -3
3 -3
***************
47a) Triangle
-2
0 -1 0
0 -6
1
-2 0 -5
-7 5
0
1 -1 0
6 -6 a,c not both null
0
0 1 2
-2 4
0
-2 -1 -6
-6 0
-2
1 0 4
2 -4 ditto
1
0 -1 -3
3 -3
0
0 1 2
-2 4
-1
-2 0 -3
-9 3
0
1 -2 -2
8 -10 ditto
-1
0 1 3
-3 3
-1
0 0 -1
1 1
***************
47b) Triangle
0
-2 0 -4
-8 4
-2
1 -1 2
4 -8
1
0 -1
-3 3 -3
a,c not both null
0
0 1 2
-2 4
0
-2 0 -4
-8 4
-1
1 -2 -1
7 -11
-1
0 1 3
-3 3
ditto
1
0 0 -1
1 1
***************
48) Triangle
-2
0 1 4
-4 2
1
-2 -1 -7
-5 1 orthogonality fails
0
1 0 2
4 -2
0
0 -1 -2
2 -4
1
-2 0 -5
-7 5
-1
1 -2 -1
7 -11 a,c not both null
0
0 1 2
-2 4
-1
0 0 1
-1 -1
***************
49) -2
0 0
1
-2 -1 equivalent to 3)
0
1 1
0
0 -1
***************
50) Triangle
-2
0 1 4
-4 2
1
-2 0
-5 -7 5
a,c not both null
0
1 -1 0
6 -6
0
0 -1 -2
2 -4
0
-2 1 -2
-10 8
-2
1 0 4
2 -4 a',c not both null
1
0 -1 -3
3 -3
0
0 -1 -2
2 -4
1
-2 0 -5
-7 5
0
1 -2 -2
8 -10 a,c not both null
-1
0 1 3
-3 3
-1
0 0 1
-1 -1
***************
51) Triangle
0
-2 0 -4
-8 4
-2
1 1 6
0 0 orthogonality fails
1
0 -1 -3
3 -3
0
0 -1
-2 2 -4
0
-2 0 -4
-8 4
1
1 -2 -3
9 -9 a,c not both null
-1
0 1 3
-3 3
-1
0 0 1
-1 -1
***************
52) Triangle
-2
0 -1 0
0 -6
1
-2 0 -5
-7 5 a,c not both null
0
1 0 2
4 -2
0
0 -1 -2
2 -4
0
0 1 2
-2 4
-1
0 -2 -3
3 -9
0
-2 1 -2
-10 8 ditto
0
1 0 2
4 -2
-1
0 0 1
-1 -1
1
0 0 -1
1 1
***************
53) Triangle
0
-2 0 -4
-8 4
-1
1 -2 -1
7 -11
0
0 1 2
-2 4 a,c not both null
1
0 0 -1
1 1
-1
0 0 1
-1 -1
***************
54) Triangle
0
-2 0 -4
-8 4
0
1 -2 -2
8 -10
-1
0 1 -3
3 -3 a,c not both null
-1
0 0 1
-1 -1
1
0 0 -1
1 1
0
-2 0 -4
-8 4
-2
1 0 4
2 -4
1
0 -1 -3
3 -3 a', c not both null
0
0 -1 -2
2 -4
0
0 1 2
-2 4
***************
55) Triangle
-2
0 1 4
-4 2
1
-2 0 -5
-7 5
0
1 0 2
4 -2
0
0 -1 -2
2 -4 a,c not both null
0
0 -1 -2
2 -4
1
0 -2 -5
5 -7
0
-2 1 -2
-10 8
0
1 0 2 4
-2 ditto
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
56) Triangle
0
-2 0 -4
-8 4
-2
1 1 6
0 0
1
0 0 -1
1 1 orthogonality fails
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
0
-2 0 -4
-8 4
1
1 -2 -3
9 -9
0
0 1 2
-2 4 a,c not both null
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
57) Triangle
0
-2 0 -4
-8 4
-2 1 0
4 2 -4
1 0
1 1 -1
5 a', c not both null
0 0
-1 -2 2
-4
0 0
-1 -2 2
-4
0
-2 0 -4
-8 4
0
1 -2
-2 8 -10
1 0
1 1 -1
5 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
58) Triangle
0
-2 0 -4
-8 4
0 1
-2 -2 8
-10
0 0
1 2 -2
4 a,c not both null
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
1 0
0 -1
1 1
***************
59) -2
0 -1
1
0 1 equivalent to 2)
0 -2
-1
0
1 0
***************
60) Triangle
-2 0
-1 0 0
-6
1 0
-1 -3 3
-3 a,c not both null
0
-2 1 -2
-10 8
0 1
0 2 4
-2
-1 0
-2 -3 3
-9
-1 0
1 3 -3
3
1
-2 0 -5
-7 5 ditto
0 1
0 2 4
-2
***************
61) Triangle
-2 0
-1 0 0
-6
1 0
-1 -3 3
-3 orthogonality implies
0
-2 0 -4
-8 4 d_2 =1, now
0 1
1 4 2
2 a' is not null
-1 0
-2 -3 3
-9
-1 0
1 3 -3
3 a,c not both null
0
-2 0 -4
-8 4
1 1
0 1 5
-1
***************
62) -2
0 -1
1
0 1 equivalent to 3)
0
-2 0
0
1 -1
***************
63) -2
0 -1 Now
1 also lies in face but the
parallelogram
1
0 0 -1
0
-2 1 -1
0
1 -1 0
formed is not a face. Subtriangles are
-2
-1 1 2
-8 4
1
0 -1 -3
3 -3 a, c not both null
0
1 -1 0
6 -6
0
-1 0 -2
-4 2
-2
0 1 4
-4 2
1
0 -1 -3
3 -3 ditto
0
-2 -1 -6
-6 0
0
1 0 2
4 -2
1
-2 0 -5
-7 5
-1
1 0 3
3 -3 ditto
-1
0 -2 -3
3 -9
0
0 1 2
-2 4
1
-2 -1 -7
-5 1
-1
1 0 3
3 -3 ditto
-1
0 1 3
-3 3
0
0 -1 -2
2 -4
1
0 -1 -3
3 -3
-1
0 0 1
-1 -1 ditto
-1
-2 1 -1
-11 7
0
1 -1 0
6 -6
***************
64) Triangle
-2
0 1 4
-4 2
1
0 -1 -3
3 -3
0
-2 0 -4
-8 4 a,c not both null
0
1 -1 0
6 -6
1
0 -2 -5
5 -7
-1
0 1 3
-3 3 ditto
0
-2 0 -4
-8 4
-1
1 0 3
3 -3
***************
65) Triangle
-1
-2 0 -3
-9 3
0
1 0 2
4 -2 a,c
not both null
-1
0 -2 -3
3 -9
0
0 1 2
-2 4
1
0 0 -1
1 1
***************
66) Triangle
-2
0 -1 0
0 -6
1
0 -1
-3 3 -3
a,c not both null
0
-2 0 -4
-8 4
0
1 0 2
4 -2
0
0 1 2
-2 4
0
-1 -2 -6
0 -6
0
-1 1 0
-6 6
-2
0 0 2
-2 -2 a', c not both null
1
0 0 -1
1 1
0
1 0 2
4 -2
-1
-2 0 -3
-9 3
-1
1 0 3
3 -3 a,c not both null
0
0 -2 -4
4 -8
0
0 1 2
-2 4
1
0 0 -1
1 1
***************
67) Triangle
-1
-2 0 -3
-9 3
0 1
0 2 4
-2
0 0
-2 -4 4
-8 a,c not both null
-1 0
1 3 -3
3
1 0
0 -1
1 1
***************
68) Triangle
0
-2 0 -4
-8 4
-1 1
0 3 3
-3
0 0
-2 -4 4
-8 a,c not both null
-1 0
1 3 -3 3
1 0
0 -1 1
1
***************
69) Triangle
-1
-2 0 -3
-9 3
0 1
0 2 4
-2 a,c not both null
1 0
-2 -5 5
-7
0 0
1 2 -2
4
-1 0
0 1 -1
-1
***************
70) -2
0 -1
1
0 1 equivalent to 14)
0
-2 0
0
1 0
0
0 -1
***************
71) Triangle
-1
-2 0 -3
-9 3
0 1
0 2 4
-2
0 0
-2 -4 4
-8 a,c not both null
1 0
1 1 -1
5
-1 0
0 1 -1
-1
***************
72) Triangle
-2 0
1 4 -4
2
1 0
-1 -3 3
-3
0
-2 0 -4
-8 4 a,c not both null
0 1
0 2 4
-2
0 0
-1 -2 2
-4
0 1
-2 -2 8
-10
0
-1 1 0
-6 6
-2 0
0 2 -2
-2 ditto
1 0
0 -1 1 1
0
-1 0 -2
-4 2
1
-2 0 -5
-7 5
-1 1
0 3 3
-3
0 0
-2 -4 4
-8 ditto
0 0
1 2 -2 4
-1 0
0 1 -1
-1
***************
73) Triangle
1
-2 0 -5
-7 5
0 1
0 2 4
-2 a,c not both null
0 0
-2 -4 4
-8
-1
0 1 3
-3 3
-1 0
0 1 -1
-1
***************
74) Triangle
0
-2 0 -4
-8 4
1 1
0 1 5
-1
0 0
-2 -4 4
-8 ditto
-1 0
1 3 -3
3
-1 0
0 1 -1
-1
***************
75) Triangle
-2 0
0 2 -2
-2
1 0
0 -1 1
1
0
-2 -1 -6
-6 0 a,c not both null
0 1
0 2 4
-2
0 0
1 2 -2
4
0 0
-1 -2 2
-4
0
-2 0 -4
-8 4
0 1
0 2 4
-2
-1 0
-2 -3 3
-9 ditto
0 0
1 2 -2
4
1 0
0 -1 1
1
-1
0 0
1 -1 -1
***************
76) Triangle
-2 0
0 2 -2
-2
1 0
0 -1 1
1
0
-2 0 -4
-8 4 a,c not both null
0
1 -1 0
6 -6
0 0
1 2 -2
4
0 0
-1 -2 2
-4
0
-2 0 -4
-8 4
0 1
0 2 4
-2
0 0
-2 -4
4 -8 ditto
-1 0
1 3 -3
3
1 0
0 -1 1
1
-1 0
0 1 -1
-1
***************
77) Triangle
-2 0
0 2 -2 -2
1 0
0 -1 1
1
0
-2 0 -4
-8 4 this is (Tr4)
0 1
1 4 2
2
0 0
-1 -2 2
-4
0 0
-1 -2 2
-4
0 0
-2 -4 4
-8
0 0
1 2 -2
4
0
-2 0 -4
-8 4 a,c not both null
1 1
0 1 5
-1
-1 0
0 1 -1 -1
-1 0
0 1 -1
-1
***************
78) Triangle
-2 0
0 2 -2
-2
1 0
0 -1 1
1
0
-2 1 -2
-10 8 a,c not both null
0 1
0 2 4
-2
0 0
-1 -2 2
-4
0 0
-1 -2 2
-4
0 0
-2 -4 4
-8
0 0
1 2 -2
4
1
-2 0 -5
-7 5
0 1
0 2 4
-2 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
***************
79) Triangle
-2 0
0 2
-2 -2
1 0
0 -1 1
1
0
-2 0 -4
-8 4
0 1
0 2 4
-2 a,c not both null
0 0
-1 -2 2
-4
0 0
-1 -2
2 -4
0 0
1 2 -2
4
0 0
-2 -4 4
-8
0 0
1 2 -2
4
-2 0
0 2 -2
-2
1 0
0 -1 1
1 ditto
0
-1 0 -2
-4 2
0
-1 0 -2
-4 2
0 1
0 2 4
-2
0
-2 0 -4
-8 4
0 1
0 2 4
-2
0
0 -2 -4
4 -8
0 0
1 2 -2
4 ditto
-1 0
0 1 -1
-1
-1 0
0 1 -1
-1
1 0
0 -1 1
1
***************
Now
consider 2-planes including a type III, a type II and a type I
80) -2
-1 0 equivalent to 20)
1
1 0
0
-1 0
0
0 -1
***************
81) -2
1 0 2-plane also contains -1
1
-1 0 0
0
-1 0 -1
0
0 -1 1
and gives a parallelogram, but not a
face.
Subtriangles to consider are
-2 1
-1 2 4
-8
1 -1
0 -3 -3
3 orthogonality fails
0 -1
-1 -4 -2
-2
0 0
1 2 -2
4
-2 1
0 4 2
-4
1 -1
0 -3 -3
3 a',c not both null
0 -1
0 -2 -4
2
0 0
-1 -2 2
-4
1 -2
0 -5 -7
5
-1 1
0 3 3
-3
-1 0
0 1 -1
-1 a,c not both null
0 0
-1 -2 2
-4
1 -2
-1 -7 -5
1
-1 1
0 3 3
-3
-1 0
-1 -1 1
-5 ditto
0 0
1 2 -2
4
1 0
-1 -3 3
-3
-1
0 0 1
-1 -1
-1
0 -1 -1
1 -5 ditto
0
-1 1 0
-6 6
-1 -2
1 -1 -11
7
0 1
-1 0 6
-6 ditto
-1
0 -1 -1
1 -5
1 0
0 -1 1
1
-1
-2 0 -3
-9 3
0 1
0 2 4
-2 ditto
-1 0
0 1 -1 -1
1 0
-1 -3 3
-3
-1
1 0 3
3 -3
0
-1 0 -2
-4 2
-1
-1 0 -1
-5 1 ditto
1
0 -1
-3 3 -3
***************
82) Triangle
-2
-1 0 0
-6 0
1
-1 0 -3
-3 3 a,c not both null
0
1 0 2
4 -2
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
-1
1 0 3
3 -3 ditto
1
0 0 -1
1 1
0
0 -1 -2
2 -4
***************
83) -2
-1 0 equivalent to 81)
1
0 0
0
-1 0
0
1 -1
***************
84) -2
-1 0 equivalent to 20)
1
0 0
0
-1 -1
0
1 0
***************
85) -2
0 0
1
-1 0 equivalent to 16)
0
-1 -1
0
1 0
***************
86) Triangle
0
-2 0 -4
-8 4
-1
1 0 3
3 -3 a,c not both null
-1 0
0 1 -1
-1
1
0 -1 -3
3 -3
***************
87) -2
0 0
1
1 0 equivalent to 16)
0
-1 -1
0
-1 0
***************
88) -2
1 0
1
0 0 equivalent to 20)
0
-1 -1
0
-1 0
***************
89) Triangle
-2
0 0 2
-2 -2
1
0 0 -1
1 1
0
1 -1 0
6 -6 a,c not both null
0
-1 0 -2
-4 2
0
-1 0 -2
-4 2
0
-2 0 -4
-8 4
0
1 0 2
4 -2 ditto
1
0 -1 -3
3 -3
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
90) -2
0 0 0
is also in 2-plane.
1
0 0 0
0
1 0 -1
0 -1 -1 -1
0 -1
0 1
We have triangle with a midpoint of one
edge.
-2
0 0 2
-2 -2
1
0 0 -1
1 1
0
1 0 2
4 -2 a',c not both null
0
-1 -1 -4
-2 -2
0
-1 0 -2
-4 2
0 -2 0
-4 -8 4
0
1 0 2
4 -2 a,c not both null
1
0 0 -1
1 1
-1
0 -1 -1
1 -5
-1
0 0 1
-1 -1
0
-2 0 -4
-8 4
0
1 0 2
4 -2
1
0 -1 -3
3 -3 ditto
-1
0 -1 -1
1 -5
-1
0 1
3 -3 3
***************
90a) Triangle
-2
0 0 2
-2 -2
1
0 0 -1
1 1
0
1 0 2
4 -2 a,c not both null
0
-1 0 -2
-4 2
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
0
-2 0 -4
-8 4
0
1 0 2
4 -2 ditto
1
0 0 -1
1 1
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
**************
We
do not have to consider the situation with a type III and two type I
since
affine combinations of the type I
-1
0 give -2
1
0
-1 1 -2
and
so will have already been considered before.
Now
consider 2-planes including a type III and two type II.
Note
that configurations involving -2 -1
need not be considered
1 1
0 -1
. .
as
then 0 is also in the 2-plane so the
example will have already occurred.
1
-2
.
For
example
91) -2
-1 0 has 0
in the 2-plane so is equivalent to 12).
1
1 1 1
0
-1 -1 -2
0
0 -1 0
***************
92) -2 1
0 equivalent to 38)
1
-1 -1
0
-1 -1
0
0 1
***************
93) -2
1 -1 0
is in the 2-plane.
1
-1 -1 1
0
-1 0 -1
0
0 1 -1
This is a parallelogram, but not a face. Nor
can the subtriangles be
faces.
***************
94) -2
1 1 Triangle, but not face
1
-1 -1
0
-1 0
0
0 -1
***************
95) -2
1 -1
1
-1 0 equivalent to 81)
0
-1 -1
0
0 1
***************
96) -2
1 1
1
-1 0
0
-1 -1 equivalent to 26)
0
0 -1
***************
97) -2
1 -1
1
-1 1 equivalent to 38)
0
-1 0
0
0 -1
***************
98) -2
1 0
1
-1 1 equivalent to 93)
0
-1 -1
0
0 -1
***************
99) -2
1 0
1
-1 -1 triangle, not face
0
-1 1
0
0 -1
***************
100) -2
1 1 equivalent to 63)
1
-1 0
0
-1 0
0
0 -1
0
0 -1
***************
101) -2
1 1
1
-1 0
0
-1 0
triangle, not face
0
0 -1
0
0 -1
***************
102) -2
1 0
1
-1 1
0
-1 0 triangle, not face
0
0 -1
0
0 -1
***************
103) -2
1 0
1
-1 0
0
-1 1 triangle , not face
0
0 -1
0
0 -1
***************
104) -2
1 -1 now 0
is in the 2-plane.
1
-1 0 0
0
-1 0 -1
0
0 -1 1
0
0 1 -1
If the first two or last two are
present it is not a face , so no
subtriangle is a face.
***************
105) -2
1 0 triangle, not face
1
-1 -1
0
-1 0
0
0 -1
0
0 1
***************
106) -2
1 0 equivalent to 104)
1
-1 0
0
-1 -1
0
0 -1
0
0 1
***************
107) Triangle
-2
-1 -1 -2
-4 -4
1 -1 -1
-5 -1 -1
orthogonality implies
0
1 0 2
4 -2 (d_1,d_2)=(2,1), which
0
0 1 2
-2 4 contradicts nullity of c
-1
-2 -1 -5
-7 -1
-1
1 -1 1
5 -7
1
0 0 -1
1 1 a,c not both
null
0
0 1 2
-2 4
-1
-2 -1 -5
-7 -1
-1
1 -1 1
5 -7
a,c not both null
0
0 1 2
-2 4
1
0 0 -1
1 1
***************
108) -2
-1 -1 equivalent to 47)
1
-1 -1
0
1 -1
0
0 1
***************
109) Triangle
-1
-2 0 -3
-9 3
-1
1 -1 1
5 -7
1
0 -1 -3
3 -3 a,c not both null
0
0 1 2
-2 4
0
-2 -1 -6
-6 0
-1
1 -1 1
5 -7 ditto
-1
0 1 3
-3 3
1
0 0 -1
1 1
***************
110) -2
-1 -1 equivalent to 40)
1
-1 1
0
1 0
0
0 -1
***************
111) -2
-1 -1 equivalent to 24)
1
-1 0
0
1 1
0
0 -1
***************
112) -2
-1 0 equivalent to 110)
1
-1 -1
0
1 1
0
0 -1
***************
113) -2
-1 1 equivalent to 93)
1
-1 -1
0
1 0
0
0 -1
***************
114) -2
-1 0 equivalent to 113)
1
-1 1
0
1 -1
0
0 -1
***************
115) Triangle
-2
-1 1 2
-8 4
1
-1 0 -3
-3 3 a,c not both null
0
1 -1 0
6 -6
0
0 -1 -2
2 -4
-1
-2 1 -1
-11 7
-1
1 0 3
3 -3 ditto
1
0 -1 -3
3 -3
0
0 -1 -2
2 -4
1
-2 -1 -7
-5 1
0
1 -1 0
6 -6 a,a' not both null
-1
0 1 3
-3 3
-1
0 0 1
-1 -1
***************
116) Triangle
-1
-2 -1 -5
-7 -1
-1
1 0 3
3 -3
1
0 0 -1
1 1 a,c not both null
0
0 -1 -2
2 -4
0
0 1 2
-2 4
-1
-2 -1 -5
-7 -1
0
1 -1 0
6 -6
0
0 1 2
-2 4 ditto
-1
0 0 1
-1 -1
1
0 0 -1
1 1
***************
117) Triangle
-1
-2 0 -3 -9
3
-1
1 -1 1
5 -7
1
0 0 -1
1 1 a,c not both null
0
0 1 2
-2 4
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
-1
1 -1 1
5 -7
0
0 1 2
-2 4 ditto
1
0 0 -1
1 1
-1
0 0 1
-1 -1
***************
118) Triangle
-2
-1 0 0
-6 0
1
-1 0 -3
-3 3
0
1 -1 0
6 -6 a,c not both null
0
0 1 2
-2 -4
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
-1
1 0 3
3 -3
1
0 -1 -3
3 -3 ditto
0
0 1 2
-2 4
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
0
1 -1 0
6 -6 ditto
-1
0 1 3
-3 3
1
0 0 -1
1 1
-1
0 0 1
-1 -1
***************
119) Triangle
-2
-1 0 0
-6 0
1
-1 0 -3
-3 3 orthogonality implies
0
1 1 4
2 2 d_2 =1, so a not null
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
-1
-2 0
-3 -9 3
-1
1 0 3
3 -3
1
0 1 1
-1 5 a,c not both null
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
0
1 -1 0
6 -6
1
0 1 1
-1 5 ditto
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
120) Triangle
-2 -1 0
0 -6 0
1
-1 1 -1
-5 7
0
1 0 2
4 -2 a,c not both null
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
-1
1 1 5
1 1
1
0 0 -1
1 1 orthogonality fails
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
1
1 -1 -1
7 -5
0
0 1 2
-2 4 a,c not both null
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
121) Triangle
-2
-1 1
2 -8 4
1
-1 0 -3
-3 3
0
1 0 2
4 -2 a,c not both null
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
-1
-2 1 -1 -11
7
-1
1 0 3
3 -3 ditto
1
0 0 -1
1 1
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
1
-2 -1 -7
-5 1
0
1 -1 0
6 -6
0
0 1 2
-2 4 orthogonality fails
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
122) -2
-1 0 equivalent to 33)
1
0 -1
0
1 1
0
-1 -1
***************
123) -2
-1 0
1
0 -1 equivalent to 33)
0
1 -1
0
-1 1
***************
124) -2
-1 0 now
-1 is in 2-plane
1
0
-1 0
0
1 1 0
0
-1 0 1
0
0 -1 -1
This is parallelogram (P5).
-2
-1 0 0
-6 0
1
0 -1 -3
3 -3
0
1 1 4
2 2 orthogonality fails
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
-2
-1 -1 -2
-4 -4
1
0 0 -1
1 1
0
1 0 2
4 -2 a,c not both null
0
-1 1 0
-6 6
0
0 -1 -2
2 -4
-2
0 -1 0
0 -6
1
-1 0 -3
-3 3 orthogonality fails
0
1 0 2
4 -2
0
0 1 2
-2 4
0
-1 -1 -4
-2 -2
-1
-2 0 -3
-9 3
0
1 -1 0
6 -6 a,c not both null
1
0 1 1
-1 5
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
1
0 0 -1
1 1 ditto
-1
0 1 3
-3 3
0
0 -1 -2
2 -4
-1
0 -1 -1
1 -5
0
-1 0 -2
-4 2
1
1 0 1
5 -1 orthogonality implies {d_1,d_3}
-1
0 1 3
-3 3 = {1,2} so a not null
0
-1 -1 -4
-2 -2
0
-2 -1 -6
-6 0
-1
1 0 3
3 -3
1
0 0 -1
1 1 orthogonality fails
0
0 1 2
-2 4
-1
0 -1 -1
1 -5
0
-2 -1 -6
-6 0
-1
1 0 3
3 -3 orthogonality fails
1
0 1 1
-1 5
0
0 -1 -2
2 -4
-1
0 0 1
-1 -1
0
-1 -1 -4
-2 -2 orthogonality implies d_2=1, d_3=2
-1
0 0 1
-1 -1 so a' is not null
1
0 1 1
-1 5
0
1 -1 0
6 -6
-1
-1 0 -1
-5 1
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
0
0 1 2
-2 4 a,c not both null
1
0 -1 -3
3 -3
-1
0 0 1
-1 -1
-1
-2 0 -3
-9 3
0
1 -1 0
6 -6
0
0 1 2
-2 4
1
0 0 -1
1 1 ditto
-1
0 -1 -1
1 -5
-1
-1 0 -1
-5 1 orthogonality implies {d_1,d_5}
0
0 -1 -2
2 -4 = {1,2}, so a not null
0
1 1 4
2 2
1
-1 0 -3
-3 3
-1
0 -1 -1
1 -5
***************
125) -2
-1 0
now -1 is in 2-plane
1
0 -1 0
0
-1 -1 0
0
1 0 -1
0
0 1 1
This is parallelogram (P6)
-2
-1 0 0
-6 0
1
0 -1 -3
3 -3 orthogonality fails
0
-1 -1 -4
-2 -2
0
1 0 2
4 -2
0
0 1 2
-2 4
-2
-1 -1 -2
-4 -4
1
0 0 -1
1 1
0
-1 0 -2
-4 2 a,c not both null
0
1 -1 0
6 -6
0
0 1 2
-2 4
-2
0 -1 0
0 -6
1
-1 0 -3
-3 3 orthogonality fails
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
0
1 1 4
2 2
-1
-2 0 -3
-9 3
0
1 -1 0
6 -6
-1
0 -1 -1
1 -5 a,c
not both null
1
0 0 -1
1 1
0
0 1 2
-2 4
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
-1
0 0 1
-1 -1 a,c not both null
1
0 -1 -3
3 -3
0
0 1 2
-2 4
-1
0 -1 -1
1 -5 orthogonality
implies d_1=1
0
-1 0 -2
-4 2 d_3 = 2 and now a'
-1
-1 0 -1
-5 1 is not null
1
0 -1 -3
3 -3
0
1 1 4
2 2
0
-2 -1 -6
-6 0
-1
1 0 3
3 -3
-1
0 -1 -1
1 -5 orthogonality fails
0
0 1 2
-2 4
1
0 0 -1
1 1
0
-2 -1 -6
-6 0
-1
1 0 3
3 -3
-1
0 0 1
-1 -1 orthogonality fails
0
0 -1 -2
2 -4
1
0 1 1
-1 5
0
-1 -1 -4
-2 -2
-1
0 0 1
-1 -1 orthogonality
implies
-1
-1 0 -1
-5 1 d_3 =2, d_2 =1 so a is not null
0
1 -1 0
6 -6
1
0 1 1
-1 5
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
0
0 -1 -2
2 -4 a,c
not both null
-1
0 1 3
-3 3
1
0 0 -1
1 1
-1
-2 0 -3
-9 3
0
1 -1 0
6 -6 ditto
0
0 -1 -2
2 -4
-1
0 0 1
-1 -1
1
0 0 1
-1 5
-1
-1 0 -1
-5 1 orthogonality implies
0
0 -1 -2
2 -4 d_1 =1, d_5 = 2
0
-1 -1 -4
-2 -2 and a' is not null
-1
1 0 3
3 -3
1
0 1 1
-1 5
***************
126) Triangle
-2 -1 0
0 -6 0
1
0 -1 -3
3 -3 a,c not both null
0
-1 1 0
-6 6
0
1 0 2
4 -2
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
0
1 -1 0
6 -6 ditto
-1
0 1 3
-3 3
1
0 0 -1
1 1
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
-1
1 0 3
3 -3
1
0 -1 -3
3 -3 orthogonality fails
0
0 1 2
-2 4
-1
0 0 1
-1 -1
***************
127) Triangle
-2
-1 0 0
-6 0
1
0 -1 -3
3 -3
0
1 0 2
4 2 a,c not both null
0
-1 0 -2
-4 2
0
0 1 2
-2 4
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
0
1 -1 0
6 -6
1
0 0 -1
1 1 ditto
-1
0 0 1
-1 -1
0
0 1 2
-2 4
0
0 -1 -2 2 -4
0
-2 -1 -6
-6 0
-1
1 0 3
3 -3
0
0 1 2
-2 4
0
0 -1 -2
2 -4 orthogonality fails
1
0 0 -1
1 1
-1
0 0 1
-1 -1
***************
128) -2
1 0 equivalent to 34)
1
0 1
0
-1 -1
0
-1 -1
***************
129) -2
1 0 now
-1 is in 2-plane
1
0 1 0
0
-1 -1 0
0
0 -1 1
0
-1 1 -1
This is parallelogram (P7).
-2
1 0 4
2 -4
1
0 1 1
-1 5
0
-1 -1 -4
-2 -2 orthogonality fails
0
0 -1 -2
2 -4
0
-1 0 -2
-4 2
-2
-1 0 0
-6 0
1
0 1 1
-1 5
0
0 -1 -2
2 -4 a,c
not both null
0
1 -1 0
6 -6
0
-1 0 -2
-4 2
1
-2 0 -5
-7 5
0
1 1 4
2 2
-1
0 -1 -1
1 -5 orthgonality fails
0
0 -1 -2
2 -4
-1
0 0 1
-1 -1
1
-2 -1 -7
-5 1
0
1 0 2
4 -2
-1
0 0 1
-1 -1 orthogonality fails
0
0 1 2
-2 4
-1
0 -1 -1
1 -5
1
0 -1 -3
3 -3
0
1 0 2
4 -2
-1
-1 0 -1
-5 1 a,c not both null
0
-1 1 0
-6 6
-1
0 -1 -1
1 -5
0
-2 1 -2
-10 8
1
1 0 1
5 -1
-1
0 -1 -1
1 -5 ditto
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
1
1 0 1
5 -1
-1
0 0 1
-1 -1 ditto
-1
0 1 3
-3 3
0
0 -1 -2
2 -4
0
1 -1 0
6 -6
1
0 0 -1
1 1
-1
-1 0 -1
-5 1 a,a' not both null
-1
0 1 3
-3 3
0
-1 -1 -4
-2 -2
-1
-2 1 -1
-11 7
0
1 0 2
4 -2
0
0 -1 -2
2 -4 a,c not both null
1
0 0 -1
1 1
-1
0 0 -1
1 -5
-1
-2 0 -3
-9 3
0
1 1 4
2 2
0
0 -1
-2 2 -4
orthogonality fails
1
0 -1 -3
3 -3
-1
0 0 1
-1 -1
-1
1 0 3
3 -3
0
0 1 2
-2 4
0
-1 -1 -4
-2 -2 orthogonality fails
0
0 -1 -3
3 -3
-1
-1 0 -1
-5 1
***************
130) Triangle
-2
1 0 4
2 -4
1
0 1 1
-1 5 a', c not both null
0
-1 0 -2
-4 2
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
1
-2 0
-5 -7 5
0
1 1 4
2 2
-1
0 0 1
-1 -1 orthogonality fails
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
0
-2 1 -2
-10 8
1
1 0 1
5 -1
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4 a, c not both null
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
**************
131) -2
-1 0 equivalent to 47)
1
0 1
0
-1 -1
0
1 -1
***************
132) Triangle
-2
-1 0
0 -6 0
1
0 1 1
-1 5
0
-1 -1 -4
-2 -2 orthogonality fails
0
1 0 2
4 -2
0
0 -1 -2
2 -4
-1
-2 0
-3 -9 3
0
1 1 4
2 2
-1
0 -1 -1
1 -5 orthogonality fails
1
0 0 -1
1 1
0
0 -1 -2
2 -4
0
-2 -1 -6
-6 0
1
1 0 1
5 -1
-1
0 -1 -1
1 -5 a,c not both null
0
0 1 2
-2 4
-1
0 0 1
-1 -1
***************
133) -2
-1 0 equivalent to 129)
1
0 1
0
1 -1
0
0 -1
0
-1 0
***************
134) Triangle
-2
-1 0 0
-6 0
1
0 1 1
-1 5
0
1 0 2
4 -2 a,c not both null
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
0
0 -1 -2
2 -4
-1
-2 0 -3
-9 3
0
1 1 4
2 2
1
0 0 -1
1 1 orthogonality fails
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
0
0 -1
-2 2 -4
0
-2 1 -6
-6 0
1
1 0 1
5 -1
0
0 1 2
-2 4 a, c not both null
0
0 -1 -2
2 -4
-1
0 0 1
-1 -1
-1
0 0 1
-1 -1
***************
135) Triangle
-2
1 0 4
2 -4
1
0 -1 -3
3 -3 a,a' not both null
0
-1 -1 -4
-2 -2
0
-1 1 0
-6 6
1
-2 0 -5
-7 5
0
1 -1 0
6 -6 a,c not both null
-1
0 -1 -1
1 -5
-1
0 1 3
-3 3
0
-2 1 -2
-10 8
-1
1 0 3
3 -3 ditto
-1
0 -1 -1
1 -5
1
0 -1 -3
3 -3
***************
136) Triangle
-2
1 0 4
2 -4
1
0 -1 -3
3 -3
0
-1 -1 -4
-2 -2 orthogonality fails
0
-1 0 -2
-4 2
0
0 1 2
-2 4
1
-2 0 -5
-7 5
0
1 -1 0
6 -6
-1
0 -1 -1
1 -5 a,c not both null
-1
0 0 1
-1 -1
0
0 1 2
-2 4
0
-2 1 -2
-10 8
-1
1 0 3
3 -3
-1
0 -1 -1 1
-5 ditto
0
0 -1 -2
2 -4
1
0 0 -1
1 1
***************
137) Triangle
-2
1 0 4
2 -4
1
0 -1 -3
3 -3
0
-1 1 0
-6 6 a', c not both null
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
1
-2 0 -5
-7 5
0
1 -1
0 6 -6
-1
0 1 3
-3 3 a,c not both null
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
0
-2 1 -2
-10 8
-1
1 0 3
3 -3
1
0 -1 -3
3 -3 ditto
0
0 -1 -2
2 -4
-1
0 0 1
-1 -1
***************
138) Triangle
-2
1 0 4
2 -4
1
0 -1 -3
3 -3
0
-1 0 -2
-4 2 a',c not both null
0
-1 0 -2
-4 2
0
0 1 2
-2 4
0
0 -1 -2
2 -4
1
-2 0 -5
-7 5
0
1 -1 0
6 -6
-1
0 0 1
-1 -1 a,c not both null
-1
0 0 1
-1 -1
0
0 1 2
-2 4
0
0 -1 -2
2 -4
0
-2 1 -2
-10 8
-1
1 0 3
3 -3
0
0 -1 -2
2 -4 ditto
0
0 -1 -2
2 -4
1
0 0 -1
1 1
-1
0 0 1
-1 -1
***************
139) -2
-1 -1 equivalent to 33)
1
0 0
0
1 -1
0
-1 1
***************
140) Triangle
-2
-1 -1 -2
-4 -4
1
0 0 -1
1 1 orthogonality and nullity
0
1 1 4
2 2 eqns have no integral solution
0
-1 0 -2
-4 2
0
0 -1 -2
2 -4
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
1
0 1 1
-1 5 a,c not both null
-1
0 0 1
-1 -1
0
0 -1 -2
2 -4
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
1
0 1 1
-1 5 ditto
0
0 -1 -2
2 -4
-1
0 0 1
-1 -1
***************
141) Triangle
-2
-1 -1 -2 -4 -4
1
0 0 -1
1 1 orthogonality and nullity relations
0
1 0 2
4 -2 have no integer solution
0
-1 -1 -4
-2 -2
0
0 1 2
-2 4
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
1
0 0 -1
1 1 a,c not both null
-1
0 -1 -1
1 -5
0
0 1 2
-2 4
-1
-2 -1 -5
-7 -1
0
1 0 2
4 -2
0
0 1 2
-2 4 ditto
-1
0 -1 -1
1 -5
1
0 0 -1
1 1
***************
142) -2
-1 -1 equivalent to 125)
1
0 0
0
1 -1
0
-1 0
0
0 1
***************