1. THE GEOMETRY HIDDEN IN THE PHYSICAL LAWS.
By ELYSIO R. F. RUGGERI
Abstract
THE POLYADIC GEOMETRY.
I.1 - Polyadics and physical magnitudes.
The Polyadic Calculus (Ruggeri, 2000) defines polyadics and operations with polyadics;
through this operations we can express the physical (non relativistic) laws in a unified
manner. Indeed, we see the physical magnitudes mathematically represented by entities
called tensors, or free polyadics of different valences. The polyadics of valence zero are the
scalars; its space is 1-dimensional. The polyadics of valence one are the vectors or
monadics and can be sketched in its 3-dimensional space by an arrow with a single pointer
at its ending point. In general, the polyadics of valence H, shortly referred as H-adics – the
dyadics (H=2), triadics (H=3), tetradics (H=4) etc – can be sketched by hyper-arrows (or,
simply, arrows) in its 3H
-dimensional space with, respectively, double, triple etc pointer at
its ending points; the other extreme point of these arrows are the starting points (or origin).
The valence H of a polyadic defines the maximal number of ordinary and different
directions inherent to it. This can be viewed by the so called 3H-1
-nomial representation of
the H-adic: the symbolic sum of 3H-1
sets of H juxtaposed vectors (3H-1
H-ades). To a vector
is associated only one direction; to a dyadic (by the trinomial representation), as a sum of
three sets of two juxtaposed vectors (three dyads); to triadics, as a sum of nine sets of three
vectors (nine triads); to tetradics, twenty seven sets of four vectors (twenty seven tetrads)
etc.
The several physical laws dictates the relationship between the polyadics involved in a
certain phenomenon.
I.2 - Physics and geometry.
Some simple physical laws can be more easily understood evoking single geometric
concepts and figures of the 3-dimensional euclidian geometry, the like the arrows. This
treasure – the geometrical picturesque view of the physical laws, left overly known and still
conserved for vectors (in Mechanics, Electromagnetics etc.) - was lost in the time for
polyadics of major valence perhaps owing to difficulties. We shall show here that, with the
2. 2
Polyadic Calculus magic touch, we can to recover and to extend this procedure for the
geometrical understanding of the more complex physical laws, in the same euclidian
geometric manner, but now inside the abstract multi-dimensional spaces.
I.3 - The polyadic space.
The polyadic space is conceived in the same way as those of vectors (arrows). The
dimension of the H-adic space, 3H
, is the major number of linear independent H-adics we
can find in this space. Helped by the algebra, we can determine this number or check
weather a set really constitute a base. It is enough to apply the following theorem: the
necessary and sufficient condition for a given set of H-adics to be a base of its space is that
be non null its multiple mixed product. The H-adic space embraces all the spaces defined
by polyadics whose valence be minor than H; these ones are subspaces of the first. With the
vectors of a vector base we can construct a dyadic base of a dyadic space; with dyadic bases
we can construct a tetradic base for a tetradic space etc.
Polyadic coordinates.
The H-pointer arrows of a H-space, postulated to be free arrows, can be imagined
applied to a fixed point O of this space. This point can be taken as "origin" of the H-adics
of a base (perhaps with unit modulus), to each H-adic direction being associated a cartesian
axes; this configuration defines a cartesian system of reference for the H-adic space. With
respect to a H-adic base of a H-adic space, a H-adic has 3H
coordinates, the cartesian
coordinates of the end point of its arrow when this one is applied at the origin; this arrow is
its "positional arrow". The product of a certain coordinate by its correspondent H-adic of
base - a H-ade - is a component of this H-adic. Hence, with respect to a H-adic base, a H-
adic is the sum of its H H-ade components.
With respect to bases formed with polyadics of valence R minor than the valence H of a
certain polyadic, it is always possible to ordinate the coordinates (which are polyadics of
valence H-R) and to arrange then in a definite order in a square or in a rectangular matrix.
In practice we often work with vector and dyadic bases to which we refer vector, dyadic
and tetradic magnitudes.
With respect to a vector base: 1) – a vector has 3 scalar coordinates arranged in a
column (or in a row); 2)- a dyadic has 3 vector coordinates arranged in a column; if we
substitute each one of this vectors by the row of its scalar coordinates we obtain a 33
matrix with 9 dyadic scalar coordinates; 3)- a tetradic has 3 triadic coordinates arranged in a
column; if we substitute each one of these triadics by the line of its dyadic coordinates we
obtain a 33 matrix with the 9 dyadic coordinates of this tetradic; if, now, we substitute
each dyadic by the column of its 3 vector coordinates we obtain a 93 matrix (whose
elements are the vector coordinates of this tetradic); and, finally, if we substitute each one
of these vectors by the line of its coordinates we obtain the 81 tetradic scalar coordinates
disposed in a 99 matrix.
With respect to a dyadic base: 1)- a dyadic has 9 scalar coordinates arranged in a column;
3)- a tetradic has 9 dyadic coordinates arranged in a 33 matrix and 81 scalar coordinates
arranged in a 99 matrix. Given a H-adic by its matrix scalar coordinates in a vector base,
we can determine, through that precise ordering, the coordinates of all its R-adic
coordinates of minor orders (R<H).
The similarity with vectors.
3. 3
Starting from the definition of the multiple dot product of two H-adics, which result is a
scalar, we find the concepts of norm and modulus of a H-adic (some of its invariant) and
angle of two H-adics.
The angle of two H-adics is the ordinary angle of its arrows. The modulus of a H-adic is
a positive real number that defines its intensity. When referred to a orthogonal and unit H-
adic base – the called orthonormal bases – the norm of a polyadic is the sum of the squares
of its coordinates (and only in this case); its modulus is the positive square root of its norm
(the like the vectors). Adopting a certain scale we can draw the H-adic arrow, its length
being defined by its modulus.
I.4 – Constructing figures in the polyadic space.
I.4.1 - The main graphical property of the arrows: the plane hyper-trigonometry.
It is easy to see that we sum (or subtract) two H-arrows by the parallelogram rule.
Indeed, supposing that is acute the angle A formed by the arrows of the H-adics H
and H
and being H
its sum, we can write the norm of H
(the square of its modulus) in the form
| | ( ) ( )H H H H H H
2
. , to be, | | | | | | | || |H H H H H
cosA 2 2 2
2 .
This formula is exactly the so called "Carnot's formula" for the triangles and confirm the
parallelogram's law.
I.4.1 - A 3H
-angular pyramid as a natural reference system in a H-adic space.
In the 3-dimensional (3-dim) vector space, 4 is the maximal number of independent
points, that is, the position of any one fifth point in this space can be univocally determined
with respect to the triangular pyramid defined by the given four points. Taking one of
these 4 points as origin of vectors and the other three as ending points we define a vector
base in this space. In the 2-dim sub-spaces, the maximal number of independent points is 3;
in the 1-dim, it is 2. The figures in 1-dim can be constructed with segments; in 2-dim, with
triangles; in 3-dim, with triangular pyramids.
In the 3H
-dim H-adic space, the even number P=3H
+1 is the maximal number of
independent points; we shall say that they define a 3H
-angular pyramid, or a (P-1)-angular
pyramid. There exist a hyper-sphere which contain all these points. Every set of 3H
points
belongs to a (3H
-1)-dim subspace; there exist 3H
of then and they are contained by a hyper-
circle. For brevity the pyramids in this space will be named (3H
+1)-points, or simply, a P-
point. The points defining a P-point are said to be its vertex. In general, the sets of points
defined by 2k<P of the vertex of a P-point, which exist in number of CP
k , are said its k-
point. Particularly, the H-adics defined by any 2-point will be sometimes called the wedges
of the P-point and the modulus of this H-adic the length of these wedges; for k>2, a k-point
any (a sub-space) will be named also a face of the P-point. Hence, there are P-1=3H
wedges
and the same number of 3H
-faces from a given vertex.
Taking one of the vertex of a P-point as origin and the others as ending points of H-adic
arrows, we define a H-adic base in this space; if the P-point is any, this base is not
orthogonal in general. A variable (P+1)-th point in this space will describe all the space; it
4. 4
can be coincident with a vertex, belongs to a wedge, to a face or occupy a notable position.
This variable point is a fifth point in the vector space, a eleventh point in a dyadic space etc.
I.4.2 – The hyper-planes geometry and trigonometry.
We have seen that in a face (which dimension is P-1) of a P-point there exist P-1=3H
points which we will ordinate one time for all; under this condition we shall say that these
3H
points define a ordered 3H
-angle (or, simply, a 3H
-angle) of which the 3H
points are the
vertex. In this face we can define a set of ordered H-adics whose starting and ending points
of its arrows are those 3H
points in the chosen order such that vanish its sum; we shall say
that, under these conditions, to any 3H
-angle is associated a closed 3H
-angle. The modulus
of the H-adics associated to a closed 3H
-angle are said to be the sides of the 3H
-angle. The
angles of the 3H
H-adic arrows will be named the 3H
-angle internal angles; the
supplementary angles of the internal angles will be said the 3H
-angle exterior angles.
With these basic concepts and definitions, added to others like medians, bisector angles,
altitudes, circumscribed and inscribed circles etc. it is possible to extend all the classical
theorems about triangles to 3H
-angles.
In the same way all the classical plane trigonometric formulas for triangles are to be
valid (extended) to the 3H
-angles of the faces of a P-point; it could be called
I.4.3 - Constructing parallelepipeds in the polyadic space.
We have seen that there are P-1=3H
wedges and 3H
(3H
-1)-faces from a given vertex. In
each one of these faces we can determine the ending point of the (H-adic) sum of its H-
adics. We add 3H
=P-1 new points to the previous P. The ending point of the sum of all the
wedges is a further added point. This configuration defines a particular 2P-point which we
shall call the "hyper-parallelepiped" associated to the given P-point or to the (P-1)-angular
pyramid.
As the diagonal relative to a vertex of a parallelepiped (in the vector space) is the
modulus of the sum of the 3 vector edges co-initial in this vertex, the like the diagonal
relative to a vertex of a hyper-parallelepiped in a H-adic space is the modulus of the sum of
3H
H-adic edges co-initial in this vertex; etc..
I.4.4 - Some characteristic properties of a (P-1)-angular pyramid.
Many consequences can be obtained from the parallelogram law. For example: the H-
adic position of the midpoint (also named centroid or baric point) of two any points is
determined by the one half the sum of the H-adic which define the positions of these points;
to this point we associate a weight 2. In general, the centroid of a k-point is the average of
the H-adic position of the k given points and to it we assign the weight k.
Let us numerate the vertex of a P-point from 0 to 3H
in a arbitrary but fixed way. We
can divide the P-point in two sets of points having each one the same number (because P is
even); in this case the two sets are said to be opposite and they are
C P! (
P
2
!)P
P/2 2
in number, to be, 6 in vector space, 252 in the dyadic space etc. When the number of points
of the two sets are different they are said to be complementary. Thus, there are: CP
k pairs of
sets one of then containing k points and the other the resting, for all k=1,2,3, ..., P/2.
5. 5
Opposite and complementary sets have each one its centroid (with its correspondent
weight). For k=1 the modulus of the H-adic defined by a vertex and the centroid of the
opposite face will be called median; for k=2, the modulus of the H-adic defined by the two
centroids will be called 2-median etc. For the particular value k=P/2 the modulus of the H-
adic defined by the two centroids will be called bi-median (essentially different from the 2-
median). The total number of medians, 2-medians, ..., except bi-medians is
C C ...+C CP
1
P
2
P
P
2
P
P
2
1
.
The centroid of a k-point, H
, which weight is k, and the centroid of the resting points,
H
, which weight is P-k, admits also a centroid with the total weight, P. This last centroid
is evidently the centroid of the P-point, say H
. Hence we can write:
H H H H
i
P
k+(P-k)
k P k
P
1 1
1
[ ( ) ] ,
if H
i is the positional H-adic of the point i. The first two members of this equality show
that over the k-median (defined by the two set) the P-point centroid is situated at the (k/P)-
th part of this k-median measured from de centroid H
, or, what is the same, at the (P-
k)/P-th part of the k-median, measured from the centroid H
. Hence:
The P medians of a (P-1)-angular pyramid concur in the centroid of its
vertex to the the (1/P)-th part of each one measured from the vertex;
and
The CP
k k-medians of a (P-1)-angular pyramid concur in the centroid of
its vertex to the the (k/P)-th part of each one measured from the vertex.
For the bi-medians in particular, we enunciate:
The CP
P/2 bi-medians of a (P-1)-angular pyramid bisect they self at the
centroid of its vertex.
For P=4 (in the 3-space) pass by the centroid 7 lines (the supports of 4 medians and 3
bi-medians); for P=10 (in the 9-space), pass 737 lines (the supports of 10 medians, 45 2-
medians, 120 3-medians, 210 4-medians and 252 bi-medians; etc..
I.4.4 – The hyper-spherical trigonometry.
We have mentioned that there ever exist a sphere containing the vertex of a P-point and
a circle containing the points of a face (a P-1 point). The center of the sphere and P-2 points
between the P-1 of a face define a maximal hyper-circle of this sphere. The (curved) hyper-
polygon formed by the P-1 arch of maximal hyper-circles in a face will be called a
spherical (P-1)-angle, or spherical 3H
-angle: a spherical triangle for vector-space (H=1), a
spherical nine-angle for dyadic space (H=2) etc..
The sides of a spherical 3H
-angle can be calculated by the analogous methods of
spherical trigonometry; the set of these new methods and formulas will be called hiper-
spherical trigonometry; hence, the object of the hiper-spherical trigonometry is to calculate
the spherical 3H
-angles.
6. 6
2 – A GENERAL LINEAR PHYSICAL LAW.
2.1- Concepts and definitions.
At any point O of a field, the magnitudes are represented by polyadics, say R
of valence R,
H
of valence H etc. The relationship between these polyadics – defining a physical law –
are translated by the operations defined in the Polyadic Calculus. When this relationship is
linear it can be written as a multiple dot multiplication
R R+H H H
G . ,
the polyadic R+H
G being not dependent on R
or H
. This means that each R
coordinate is
proportional to all the coordinates of H
, each one of these later entering with different
weight; these weights are defined by R+H
G, perhaps as functions of time, temperature,
position etc. (but never as functions of the H
or R
coordinates). The polyadics R
and H
are the dependent and independent variables, respectively. The more simple examples of
this general law are the linear constitutive laws in Continuum Mechanics.
The aim theme of this paper will be those linear laws for which R=H, that is, for
proportional magnitudes of the same order; hence,
H 2H H H
G . , (2.01).
Thus, for H=1,we shall be dealing with vectors linked by a dyadic; for H=2, with dyadics
linked by a tetradic etc..
The equation (2.01) express many general laws in Physics. In Classical Mechanics we
can cite, for H=1: the third Newton's law f=ma with m=mI (I being the unit dyadic); the
law of the dynamic of rigid body, j=J.w, where j is the angular moment vector, J is the
inertia dyadic and w the angular velocity vector. In Elasticity, for H=1, we have the
Cauchy's law, t . n , where t is the vector stress on a element of area with unit normal
vector n ; for H=2 we have the generalized Hooke's law, 4 G : , where (epsilon) is
the (adimensional) strain dyadic and 4
G the Green's tetradic (also named Hooke's tetradic).
A lot of laws in Crystallography and many other branches of Physics could be cited.
In each particular law the three polyadics can assume a special form, being for example
(simple or multiple) symmetric, anti-symmetric etc; or to be of some special nature like a
tonic, a rotation, simple shearer, a complex shearer, a strictily triangular etc.
7. 7
2.2- The physical law is a mapping.
For practical purposes, the physical space has up to three dimensions. A sheet can be seen
as a 2-dimensional space and a cable as a one dimensional space. This realistic approach
brings us to conceive the polyadics of valence Q existing in euclidian spaces of dimension
G3Q
.
The variable and independent H-adic H
, applied in the fixed point O of its 3H
-space, is
the positional arrow of a "object point" E of this space and the proportionality 2H-adic 2H
G
can be seen as a linear operator that transforms E into the end point S of the positional H
(inside this same space); S is the "image point" of E. If, with respect to the point O, the
multiple mixed product (H
1
H
2... H
3H ) of the positional H-adics of the points E1, E2,
... E3H is non null, this H-adics constitute a base of this space; the 3H
+1 points O, E1, E2, ...
E3H are said to be independent. The 3H
hyper-planes defined by the sets of 3H
points
defines a hyper-polyhedra in the space.
If, with respect to a fixed O in the field, by any process or law of correspondence, we
know the image S1, S2, ..., S3H of the 3H
object points E1, E2, ... E3H of this space, which,
with O, define a independent set, the 2H-adic stays univocally determined and can be used
as a pre-factor in multiple H-dot multiplication by H-adics. We write:
H 2H H H
G . , for =1,2, ...,3H
and 2H H H
G
(2.02),
since we state that that in this last equality is established a sum in the index and that the
H-adics H
are the reciprocals of the positional H
. In particular, the positional H
. can
constitute a mutually orthogonal set, that is, they can be orthogonal two by two; in this case
the H-adic base are said to be orthogonal. If, besides a H-adic base to be orthogonal, their
H-adics are taken unitary (what is always possible), the base is said to be orthonormal.
If { , , }H H H
G..., 1 2 is an orthonormal base we can write
H
i
H
j ij . (i,j=1,2, ..., G),
where the ij are the deltas of Kronecker; and we can conclude that if a base in orthonormal
it is coincident with its reciprocal.
One notable particular case is that for H=2 in which the base dyadics are dyads formed with
vectors of a orthonormal vector base { }i j k , that is,
, , , , 1 9 ii ij ik ji kk....,2 3 4 ,
being, evidently,
||| || ( ... )
...
...
... ...
...
1 2
2
1G
ii:ii ii:ij ii:kk
ij:ii ij:ij ij:kk
kk:ii kk:ij kk:kk
.
8. 8
2.3- The 2H
G determination.
To complete the structure of the constitutive equation (2.01) we need first to determine the
proportionality polyadic since it is a characteristic of the media through which the variable
magnitudes are linked, be this media full or empty with matter. The second of expressions
(2.02) furnishes naturally the necessary and more general way (perhaps not ever simple
when it is possible) to get that determination. Laboratory devices and accurate
measurements fulfil the necessary conditions to construct the polyadics H
and H
(23H
in number), and hence the polyadic 2H
G, being enough to chose a convenient vector base
with respect to which the measures could be easily made.
We can choose 3H
states inside the same phenomenon (like to put a body in charge with
different efforts), and a point any or a set of points in the field (the body), under which the
variables to be measured, H
and H
, may assume simple forms (stresses and strains),
presenting the smallest number of coordinates. Practice, apprenticeship and lucky can help
in this choose. Since there is correspondence between each pair (H
, H
), since the G
measured H-adics of the independent variable (H
, say strains) are independent – and this
will be confirmed if non vanish the multiple mixed product of the 3H
measured H-adics (see
section 1) – we really have got a 2H
G measure.
2.4- A decisive and simplifier assumption.
If, in all the points of the field, the variable scalar 2W H H H H H H
. . exists
univocally determined and can be measured, then:
( )H H H 2H 2H H H H
: . .G G
0 , to be, 2H 2H H
G G
,
and only in this condition. In resume:
2W H H H H H H
. . 2H 2H H
G G
, (2.03).
Be Sij ...l and Ei'j' ...l' (for =1,2, ...,3H
) the measurable H
and H
coordinates,
respectively, with respect to an orthogonal and unit vector base { , , }e e e1 2 3 , in which case
the two sets of H indexis i,j, ...,l, and i',j',...l' assume the values 1,2,3. We can write, in
cartesian coordinates,
2H
ij ... l i' j' ... l' i j l i' j' l'S E ... ...G e e e e e e
and, hence, associate to 2H
G, in this vector base, the 3H
x3H
symmetric matrix [Gij ...l i'j' ...l']
product of the 3H
x3H
symmetric matrices [Sij ...l] and [Ei'j' ...l']. The column of order in
[Sij ...l] is formed with the H
coordinates; the coordinates of H
form the [Ei'j' ...l'] row of
order . As we see, to determine the (1+3H)3H/2 elements of [Gij ...l i'j' ...l'] we need to make
23H
3H
measures, to be, up to 18 for H=1 (vector magnitudes), up to 162 for H=2 (dyadic
magnitudes) etc. Troubles in experimental measures will be ever present.
Now, for H=2Q2 (or Q1), be Si,j, ... l and Ei',j',... l' the 2Q
and 2Q
coordinates,
respectively, with respect to an orthogonal and unitary dyadic base { , , } 1 2 9..., for the
Q index i,j, ...,l and the Q others i',j', ...,l' running now from 1 to 9. We can write, in
cartesian coordinates,
9. 9
4Q
ij ... l i' j' ... l' i j l i' j' l'S E ... ...G .
We can bring this problem in the same way we have brought for vector bases. Perhaps it
could present a bit less laboratory work to accomplish but we still can not to construct a
dyadic base in a way to determine polyadic coordinates in these bases; for while they are
useful only for handling and researching properties as we shall see.
Under this fault we will admit now on that we posses a 2H
G associated 3H
x3H
symmetric
matrix referred to some conveniently chosen orthonormal vector base, which we will
denote by [2H
G]. The 2H
G determinant, det2H
G, equals the determinant of the matrix [2H
G],
sometimes denoted by det[2H
G], is one of the 2H
G invariant. If this determinant is non null,
2H
G is named complete and indicates that it can be inverted; otherwise, 2H
G is named
incomplete and its inversion is not defined.
2.5- A convenient change of variables.
To simplify the mathematical handling we will introduce the new variables
H
H
H
| |
,
H
H
H
H
0
0
H H
2W=| | 2W and 2w=
2W
| |
2W
|| ||
| |
, , , (2.04),
in which case, besides the unchanged law (01), we have:
H H H H H H H
2 2
G G. . , (2.05),
2w H H H H H H H H 2H H H
. . . .G , (2.06),
2W0
H H H H H H
. . , (2.07),
H H H . 1, (2.08).
The magnitudes H
and 2W0 will be named "specific magnitudes", or "magnitudes by
unit of H intensity (modulus)" since | H | represents a quantity of the (independent
variable) magnitude H . Recalling isomorphic concepts with vectors we shall name H
( H
) the H-adic (specific) projection of 2H
G in the direction H . Similarly, 2W (2w) is
the scalar (specific) projection of H
( H
) in the direction H
; it will be named also the
radial (specific) value of 2H
G relative to the direction H .
For two different directions H and H
we will write (in accordance with
polyadic algebra) 2 w H H 2H H H H H 2H H H
. . . .G G , or
2 w H H H H H H
. . , (2.06').
Hence we have demonstrate the following theorem (of Betty):
The 2H
G projection relative to a direction H , H
, projected on a
second direction H , is equal to the 2H
G projection relative to this
direction H , H
, projected on the first direction H .
We will name the scalar 2w''' the 2H
G tangential value relative to the directions H
and
H
.
10. 10
It is interesting this change of variables because the independent variable can be
considered now with unit fixed norm or modulus. This permits a significant geometrical
interpretation: when H
varies with fixed origin O, assuming all positions about the point,
its end point, being always a unit distance from O, describes the (hyper)spherical surface
(08), centered at the point, with unit radius.
3 - THE QUADRICS ASSOCIATED TO THE PHYSICAL LAW.
Suppose that 2H
G is complete in the 3H
-space. From (05), considering (08), we can
deduce:
H H H H H
. .G2 2
1
, (3.01).
Similarly, from (2.06),
Q
2w
2w|
H H 2H H H
. .G
|
1, (3.02),
where H
is the H-adic parallel to the unitary H-adic H
whose modulus is the inverse of
|2w| square root, that is,
H H
2w
1 , (3.03).
Hence, while the H
end point describes the spherical surface, the H-adics H
and H
end points, P and Y – whose distance to O are the modulus of H
and H
- describe the
(hiper)surfaces (3.02) and (3.03), respectively. These are quadric surfaces centered at the
point O. The first – representing the 's variations – is the Lamé's ellipsoid. The second –
representing the 2w variations and named Cauchy's quadric or indicator quadric – could
be an ellipsoid or an hyperboloid (of one or two sheets) depending on the 2H
G scalars
invariant; by these reasons 2H
G will be named, in correspondence, elliptic and hyperbolic.
The indicator exists always independent on 2H
G to be complete.
Imagined outlined the quadrics (3.02) and (3.03) centered at the point O, |H
| and 2w,
related to a given H
, can be easy determined. Indeed, to calculate 2w it is enough to
determine the point Y where H
intercepts the indicator Q, in which case, according to
(3.03), |2w|=1/(OY)2
. To calculate |H
| it is enough to fix its direction in space and to write
|H
|=|OP.|
Representing by Q+ and Q- the indicator (3.03) correspondent to signals (+) and (–),
respectively, we can conclude: 1) - If Q+ is a real ellipsoid, Q- will not be sketched because
it is a imaginary ellipsoid. In this case 2w>0, whatever be H
and all the 2H
G eigenvalues
are positive. The angle between H
and H
is acute always; 2) - If Q- is a real ellipsoid, Q+
is imaginary and 2w<0; the angle between H
and H
is obtuse always; 3) - If Q+ is an
hyperboloid, Q- is its conjugate hyperboloid; both are separated in the space by the
common assyntoptic cone, with the equation
C =0H H 2H H H
. .
G .
The point Y, intersection of H
with Q, could be a proper point, situated on Q+ or on Q-
, or a improper point (if H
should be parallel to any generator of the cone). In the first
11. 11
case, the angle between H
and H
should be always acute and 2w>0; or obtuse, with
2w<0; in the second case, this angle is 90 and 2w=0.
The hipercurves intersection of the hipercone C and the hypersphere define over this
hipersphere the regions with respect to which correspond 2w>0 and 2w<0; their normal
projected upon the coordinate planes will be ellipses or arch of hyperbolas.
4 – THE PROPORTIONALITY POLYADIC CHARACTERISTIC ELEMENTS.
4.1- Concepts.
The real and symmetric proportionality polyadic 2H
G defines the maximal dimension of the
spaces to be considered: Gmax=32H
=9H
. This animates us to look for particular bases
concerning facilities.
It can be proved that a polyadic 2H
G, any, can transform some non null H-adic H
in a
H-adic parallel to H
, that is,
2H
.G XH H H
, (4.01),
where X is a scalar. This means that there exist solution for the equation
( )2 2H H
.G O X H H H
, (a), with X and the H–adic H
as unknowns; this equation is
named the 2H
G polyadic characteristic equation. The necessary and sufficient condition for
the existence of H
is that the 2H-adic between parentheses – the 2H
G characteristic
polyadic - be incomplete. Geometrically this means that if we represent this 2H-adic in a H-
adic base { , , }H H H..., H 1 2 3 , say in the form
2 2H H
G X F X) H H
( ,
then its 3H
H-adic coordinates F X) F X)H H
( , ( etc belongs to a same hyper-plane
to which the H-adic H
might be orthogonal. Algebraically this the same to say that the
2 2H HGX determinant – named the 2H
G characteristic determinant - might vanish.
With the 2H
G associated matrix (that one determined by experiments, see section 2) we can
solve the 3H
degree algebraic equation – named the 2H
G algebraic characteristic equation -
defined by this determinant, to each of its 3H
roots – named the 2H
G eigenvalues and
represented by G1, G2, ..., G3H - correspond a certain H-adic solution. This 3H
H-adics are
the 2H
G eigenH-adics and will be represented by H H H
3..., H 1 2, , .
The set of the 2H
G eigenvalues and correspondent 2H
G eigenH-adics are the 2H
G
characteristic elements, sometimes called the 2H
G eigensystems.
4.2- The 2H
G characteristic elements are all real.
Let us prove that for 2H
G single symmetric, 2H 2H H
G G
,
the eigenvalues must to be all real.
There must exist a possibly complex H-adic H
satisfying (4.01) with X complex scalar,
hence complex H
, H
being its conjugate. Hence we write, evidently:
12. 12
H H 2H H H H
X . . .G , (b). Taking complex conjugate from (4.01) it comes:
H H 2H H H H H 2HX . .G G
since : (2H H 2H H H H H 2H H
, ). .
and 2H
G is real and symmetric. Hence, H H 2H H H H
X . . .G , (c). Now,
comparing (b) and (c) we infer that (X X 0H H H
) . . But for every non null H
,
H H H
0 . ; hence X X . This implies that the eigenvalues are all real.
If the 2H
G eigenvalues – which now on will be denoted by G1, G2, ..., G3H - are all real,
the correspondent eigenH-adics – taken unitary and denoted by H H
2
H
3, ..., H , 1 - are
also real and
To each pair of different 2H
G eigenvalues correspond a pair of orthogonal unitary
eigenH-adics.
Indeed, if G1 and G2 are different, we can write from (a) :
2
1 1
H H H
1
H
GG . , and 2
2
H H H
2
H
2GG . , (d)
or, since 2H
G is symmetric,
H H H H
1
H H H H
G = 1
2
1 1
2
. .G G
, (e).
Hence, by double dot pre-multiplication of the second of equations (d) by H 1 , post-
multiplication in (e) by H
2 and consequent subtraction member by member, we have:
(G G1 2
H H H
) 1 2 0. .
As G1G2 we get H H H H H H 1 2 1 2 0. . , that is, H 1 is perpendicular to H 2 .
If all the 3H 2H
G eigenvalues are to be different, we have 3H
different unitary eigenH-
adics orthogonal two by two, that is,
G G ... G1 2 3
H
i
H H
j ijH . (i,j=1,2, ..., 3H
), (4.02).
The metric matrix associated to this set is the 3H
3H
unit matrix whose determinant equals
one. Hence, the set constitute a orthonormal base in the space, being coincident with its
reciprocal. So, we can write:
2H H H
GG
(=1,2, ..., 3H
), (4.03),
where is established a sum in . The form (4.03) to represent 2H
G is named the tonic form.
The H-adic base { , }H H
2
H
3, ..., H 1 is the "principal base" of the field.
4.3- Vanishing and multiples eigenvalues.
Let us suppose that 2H
G has, for instance, a triple eigenvalue, say G G G3 3 3H H H 2 1 .
Using multiple cross product, put
H
3 -1
H H H
3 -3
H
3 -2H H H..., , 1 2 and H
3
H H H
3 3
H
3 2
H
3 -1H H H H..., , 1 2 .
13. 13
The H-adics H
3 -1H and H
3H belongs to the 3H
-1 and 3H
spaces, respectively, are unitary
and perpendicular. Besides, both are perpendicular to all the eigenH-adics; hence they
constitute with then a orthonormal base of the hole space. Then we can write, resolving 2H
G
in this base:
2H
i
H
i
H
i
H
3 -1
H
3 -1
H
3
H
3G Y ZH H H HG for i=1,2, ..., 3H
-2,
becoming obvious that H
3 -1H and H
3H are two 2H
G eigenH-adics to which correspond
the eigenvalues Y and Z. Hence Y=Z= G3H 2
because, by hypothesis, 2H
G has only 3H
-2
different eigenvalues and G1G2 ... G3H 2 . What we have deduced for a triple
eigenvalue we can also deduce for a R-ple eigenvalue since R3H
. Hence:
If, in a 3H
-space, a 2H
G has a R-ple eigenvalue, the multiple cross product of all the 3H
-R+1
different 2H
G eigenH-adics, a H-adic of the (3H
-R+2)-space, is also a 2H
G eigenH-adic with
respect to that R-ple eigenvalue; the multiple cross product of this new eigenH-adic and the
precedings is also a 2H
G eigenH-adic with respect to that R-ple eigenvalue and belongs to
the (3H
-R+3)-space, and so on, each multiple cross product belonging to a space one
dimension higher than the anterior.
This property can be applied to all groups of equal eigenvalues (in particular to that
group for which the eigenvalues are single). When all the eigenvalues are equal, say to G, it
is a spherical 2H-adic, G 2H
I, to whom every H-adic of the space is an eigenH-adic.
If R of the 3H
eigenvalues vanish we say that the polyadic exist in G=3H
-R subspace, the
correspondent eigenH-adic being determined like proved before.
4.4- The Cauchy and Lamé's quadric reduced equation.
The quadrics associated to the 2H
G-adic at the point O can be represented in a more simple
form – in the reduced form – if the field is referred to its principal H-adic base. In this case,
(3.02) and (3.03) are written in the respective forms
(
)
H H H
G
.
2
1 and ( )H H H
G .
2
1 ,
in whose first members is established a sum on . Denoting by S and Y the H
and H
coordinates we can write:
( )
S
G
2 1 and (
|
)
Y
|G
1/
12 , (4.04),
the signal of each piece in the first member of the second equation being the G signal. In
the reduced form it is more easy to classify the indicator quadric.
Let us consider that the polyadic 2H
G is variable with some variable other than H
and
H
. If, in a particular stage, 2H
G has some eigenvalue tendind to zero, say G1, the first piece
in the second of equations (4.03) tends also to zero. This means that the quadric is
orthogonal projected (into quadrics) on the space defined by H H
3..., H , 2 . By (3.03) we
14. 14
see that the distance OP tends to infinite for directions parallel to H 1 . Hence the indicator
quadric (in the 3H
-space) tends to a hyper-cylindrical surface whose generators are parallel
to that direction. If vanish more than one eigenvalues we can deduce similar results; there
are occurrence of new degeneration. It could not vanish all the eigenvalues without 2H
G to
be the null 2H-adic.
If some eigenvalue is double, triple etc, we say that the quadrics are of revolution with
respect to two, three etc axis.
5 – THE STATIONARY PROPORTIONALITY POLYADIC RADIAL VALUE.
The 2w stationary value at the point O is a linked extreme because H might satisfy (2.08).
If there exist a direction by O that makes 2w stationary, in this direction will be d(2w)=0.
Differentiating (2.06) it comes (in accordance with polyadic analysis):
2 2 0dw
w
d dH
H H H H 2H H H
. . .G .
Being H H H
d .
0, we conclude that H and H 2H H H
G .
are orthogonal to
d H , that is, orthogonal to the same hyper-plane tangent to the spherical surface
H H H .
1. This means that these two H-adics must be parallels.
From the two first members of (2.06) it comes1:
2w=| | cos( , )H H H
, (5.01),
from where we deduce that the 2w stationary value is |H
| if the H-adics H
and H
are
parallels (the maximum corresponding to the null angle and the minimum to 180). The
parallel condition may be expressed in the form XH H H H H
2
G .
, (1.03).
Remembering the section 4 we conclude:
The 2H
G radial value, 2w, given by (2.06), is stationary at the point O
of the 3H
- space for directions H lined by O and parallels to the 2H
G
eigenH-adics.
From the general expression we deduce also immediately:
H H 2H H H 1 2
0. .
G = H H 2H H H 1 3. .
G .....= H H 2H H H
= ... 2 3. .
G , (5.02),
that is:
The tangencial value of 2H
G relative to any two different principal
directions at a point are always null.
If we represent by Eu the projections (coordinates) of H on the principal base, and
remembering that Su are those of H
, that is, if we put
1 - It is valid for multiple dot multiplication of polyadics the same concepts valid for scalar multiplication of
vectors.
15. 15
H H H
u uE . and H H H
u uS .
, (5.03),
then the law (01) is equivalent to the system
S G E
S G E
S G E
1 1 1
2 2 2
G G G
...
, (5.04).
We conclude:
When, in the vicinity of a point O of a field, the space is referred to the
principal base of this point, the ratio of the same name proportional
magnitude coordinates is equal to the correspondent 2H-adic
proportionality eigenvalue.
Substitution of (1.04) into (03,Intr.) gives:
2w=( ) G
( ) G ( ) G ...,
H H H
u
2
u
H H H
1
2
1
H H H
2
2
2
.
. .
(u=1,2, ..., G) (1.05),
from where we conclude:
Each one 2H
G eigenvalue is a stationary value of its radial value, 2w,
in the point O of the G-space, which occur for the correspondent 2H
G
eigenH-adic direction.
6 - THE PROJECTION NORM AND OCTAHEDRAL DIRECTIONS.
For an arbitrary direction H considered by the point O of the G-space we can write:
H H H H
u
H
u
( ) . , for running from 1 to 3H
, with
( )H H H
u
2
G
.
1
1 , (6.01),
because H H H .
1. The numbers H H H
u
.
are the 3H
principal director cosines of
the direction; in general, they are all different but for a particular direction they can be all
equal. For a given and ordered set of 3H
squares, whose sum is equal to one, there are 23H
directions (that is, all the arrangements with repetition of the signs + and – taken 3H
by 3H
with the modulus of the director cosines, (AR)2
3 3H H
2 ) whose director cosines have the
same modulus.
We shall call octahedral directions, or octahedral H-adics of a field in a point – and
will denote then by H
( )oct - the unitary H-adics equally inclined to the principal
directions of the point. For a general octahedral direction we can write
H
oct
H
oct
H H
u u
( ) . ; and from (6.01), since the cosines (cos oct) are all equal:
16. 16
for (u=1,2, ..., 3H
) H
oct
H H
u oct
H
H
H
cos
3
3
3
.
1
, (6.02).
So, in 2-space, oct
45
, in 3-space oct
54 44
' , in 4-space oct
60
etc..
For any octahedral direction, to which correspond w=woct, we have
2w = G
G
Goct
H
oct
H 2H H H
oct
H
oct
H H
u u u
1
G
( ) . . .G 2 1
, (6.03),
that is:
In any octahedral direction, the 2H
G radial value, 2woct, is equal to its
eigenvalues average invariant.
For a direction any of the space we can write the norm of the correspondent 2H
G
projection as
|| || H H H 2H H 4 H H H H 2H H H
= . . . . .G G G2 , being 2H H 2H 2H
=G G G.
2
, (6.04).
But from (4.03) we write also 2H
u
H
u
H
u
=(GG
) 2 2
; from where we deduce
2H
E
2
u
3
(G
H
G
) 2
1
, (6.05).
Hence: || || ( ) ( )H H H H
u u
2
G .
2 , or, writing in full:
|| || ( ) ( ) ( ) ( ) ( ) ( ) H H H
1 1
H H H
2 2
H H H
3 3G G ... GH H. . .
2 2 2 2 2 2 , (6.06).
The Gu are invariant, that is, they don't depend on the H . Hence, ||H
|| varies with the
square of the 3H
director cosines ( )H H H
u
.
2
whose sum, in conformity with (6.01), is
equal to one. Thus, when these factors are all equal, that is, for all octahedral directions,
representing by H
oct the 2H
G projection correspondent to any one of the octahedral
directions, we have:
|| || ( )
H
oct H u
3
H
H
E
2
3
G
3
H
1 12
1
2 G , (6.07),
in which case ||H
||max is a invariant. We conclude:
The norms of the 2H
G octahedral projections at a point are all equal to
the 2H
G eigenvalue squares average (or equal to the 3H
-th part of the
2H
G dot square scalar).
17. 17
7- SOME RELATIONSHIP BETWEEN EIGENVALUES.
Let us calculate now the difference between the 2H
G square eigenvalues average and the
square of the 2H
G eigenvalues average, that is,
1 1 12 2 2 2 2
3 3 3
G
G
3H
H
E
2
H
H
E H u
u
H
G G
( ) ( ) ( )
.
We have, in developing the squares:
1 1
32 2 2 2 2 2
3
G
G
3H (3 )
G G ... G G G ... GH u
u
H 2
H
1 2 3 1 2 3H H( ) ( ) { [( ) ( ) ( ) ] ( ) }
1
1 22 2 2
(3 )
G G G ... G G G G G G GH 2 1 2 3 1 2 1 3 + ... 1 3 +H H{( )[( ) ( ) ( ) ] (
G G G G ... G G G G G G G G G G }2 3 2 4 2 3 3 4 3 5 + ... 3 3 + ... 3 3 1)H H H H , (7.01).
Grouping pieces conveniently and noting the presence of new perfect squares we have:
1 2 2
3
G
G
3H u
u
H
( ( ))
1 2 2 2
(3 )
G G G G ... G GH 2 1 2 1 3 1 3H[( ) ( ) ( )
( ) ( ) ( ) ( ) ]G G G G ... G G ... G G2 3 2 4 2 3 3 3H H H
2 2 2
1
2
.
As we have C3
2
H squares of differences inside the brackets, we write:
1 1
2
2 2
3
G
G
3
3
3
G)
CH u
u
H
H
H
2
3
2
H
( ( ))
, (7.02),
where with ( )G 2
we are representing the sum of squares of the eigenvalues differences
taken two by two. Interpreting (7.02) we enunciate:
The 2H
G eigenvalues square average is equal to the square of its
average summed to the ( )/3 1 2H H3 of the average of its squares
difference two by two.
*
The 3H
principal 2H
G invariant are the coefficients of its characteristic equation
X X X ...3 2H
E
1 3 2H
E
2 3H H H
G G
~ ~
1 2
X + X 02H
E
3 2 2H
E
3 2H
3
H H
H
G G G(
~
) (
~
)2 1
, (7.03),
where:
18. 18
2 2H
E
H
E
1
G G
~
is the scalar of the (3H
-1)-th 2H
G adjunct determinant, that is, the
sum of the diagonal minors of degree 1 of its determinant, which is equal to the sum of the
eigenvalues;
2H
E
2
G
~
, is the scalar of the (3H
-2)-th 2H
G adjunct determinant, that is, the sum of
the diagonal minors of degree 2 of its determinant, which is equal to the sum of the two by
two products of the eigenvalues;
... etc.;
2H
E
H
E
(3H
G G
~ ~
)
2 1
is the scalar of the (first) 2H
G adjunct, this is, the sum of the
diagonal minors of degree 3H
-1 of its determinant, which is equal to the sum of the 3H
-1 by
3H
-1 products of the eigenvalues;
2H
G
2H
E
G
G G
~
is the sum of the diagonal minors of degree 3H
of its determinant,
this is, the proper determinant.
Considering the first and the second 2H
G invariant we write the expression (7.01) in the
form
2 2 21
2
H
E
2 H
E
2 H
E
2
G G G
~
( ) , (7.04).
Hence:
The sum of the 2H
G two by two eigenvalues products is equal to one
half the difference between the square of its sum and the sum of its
squares.
8 - THE PROPORTIONALITY POLYADIC TRANSVERSAL VALUE.
From (2.06) we see that : || || (2w)H H 2 . Hence, it does always exist a positive
number, say t2
, which complement (2w)2
to perform ||H
||. We can write:
: | | =(2w) tH H 2 2 2 , (8.01) (1.13).
This pitagoric relationship – used to calculate the square of a vector solved in two
perpendicular directions – suggests to name |t| the 2H
G transversal value relative to the H
direction.
In Elasticity, for H=1, t2
is the square of the tangential stress vector on a plane defined
by a normal unit vector n where acts the stress vector p. If is the normal stress vector,
then (8.01) writes p2
=2
+2
. For H=2, t2
is a always existing positive number which
summed to the square of the specific energy 2w performs the norm of the specific stress
dyadic; and we write (8.01) as t2
+(2w)2
=||2
=||||.
Applying (7.02) to the case of octahedral direction and considering (6.07) and ( ) we
can enunciate:
19. 19
The 2H
G square transversal value relative to octahedral directions,
toct
2
, is equal to the ( )/3 2 3H H 1 of the average of its square
difference two by two:
t
3
3
G)
C
oct
2
H
H
2
3
2
H
1
2
, (1.131).
We could also write (1.132) in the form
t
3
G)oct H
2
1
, (1.132),
that is:
The 2H
G transversal value relative to octahedral directions, toct , is
equal to the 3H
-th part of the square root of the sum of its square
difference two by two.
From (1.132) we see that the limiting case toct=0 occurs for (G)2
=0, and vice-versa.
This implies that the 2H
G eigenvalues are all equal, to be, 2H
G is a scalar polyadic. We
write:
t G with G=G G ... and || Goct
2H 2H
1 2
H
oct
2
0 G || , (1.133).
If at least two of the 2H
G eigenvalues are different, toct0.
*
The 2H
G transversal value extreme.
Let us search the directions with respect to which the square of 2H
G transversal value, t2
,
is a maximum (since its minimum is zero).
In accordance with the Lagrangian multipliers method (to find stationary values of a
multivariable function) we must extreme the function
F t L2 H H H
. , (1.14),
(with the conditional equation H H H .
1) as F would be a free extreme, L being a
constant.
From (1.14), considering (1.13), we write (recalling polyadic analysis):
F
L =
|| || w
LH
2
H
H
H
H H
H H
2 2 , (1.142),
where H
0 is the null H-adic.
From (1.08) we have
|| ||H
H
H H H
2 2 2
G . , (A),
20. 20
or, from (1.09), with respect to the base formed by the 2H
G eigenH-adic:
|
| ||
( ) ( )
|
G
H
H u
2 H H H
u
2 2
. , (A').
Being
w
w
w
H H
( )
2 2 ,
hence, considering (03,Intr.) we derive:
ˆ2)w2(2
ˆ
w HHH2
H .G
,
or, with respect to the 2H
G eigenH-adic base:
w
w GH u
H H H
u
H
u
( ) ( )
2 2 2 . , (B').
Since, evidently, H H H 2H .
, and considering (A) and (B), we write (1.141)
in compact polyadic notation:
F
w) -L =H
H 2 2H 2H H H H
[ ( ]
2 2 22 G G . , (1.142);
or, with respect to the 2H
G eigenH-adic base, now considering (A') and (B'):
2 2 22
[( ) ( ) ]( ) G w G Lu u
H H H
u
H
u
H
. , (1.143).
The linear combination (1.143) implies the nullity of all eigenH-adic (H u ) factors
(because these H-adics form a base). Hence
[( ) ( ) ]( )G w G Lu u
H H H
u
2
2 2 0 . (u=1,2, ..., 3H
), (1.15).
The expression (1.15) represents the following system up to 3H
equations whose will be
independent for different eigenvalues:
[( ) ( ) ]( )
[( ) ( ) ]( )
...
[( ) ( ) ]( ) ,
G w G L
G w G L
G w G L
1 1
H H H
1
2 2
H H H
2
3 3
H H H
3H H H
2
2
2
2 2 0
2 2 0
2 2 0
.
.
.
(1.16).
21. 21
Theorem 1:
To each pair of single eigenvalues of a symmetric 2H-adic there
corresponds a direction H that leave stationary its transversal value
and is perpendicular to at least one of its eigenH-adics different from
those correspondent to the single eigenvalues.
Let us consider 2H
G with two single eigenvalues, say G1 and G2, to which correspond
the unitary and orthogonal eigenH-adics H H
2
and 1
. If H
- the unitary H-adic that
makes stationary the 2H
G transversal value square,t2
– is not orthogonal to any 2H
G eigenH-
adic, from system (1.16) it comes
( ) ( ) ( ) ( ) ( ) ( )G w G G w G ... G w G L1 1 2 2 3 3H H
2 2 22 2 2 2 2 2 , (1.161),
because we could divide both members of each equation in (1.16) by the correspondent non
null double dot H
.
H H
u
0 and explicit the value of L. As G1 and G2 are single values,
G G G G ..., G1 2 3 4 3H , , .
The first two members of (1.161) gives G +G w1 2 4 ; the first and the third gives
G +G w1 3 4 , and so on. But this is an absurd because we might accept that G2=G3=G4=
...= G3H .
Hence H
must be orthogonal to at least one 2H
G eigenH-adic. If H
was perpendicular
to H
1
, the system (1.161) would be reduced to utmost 3H
-1 equations and we could write
( ) ( ) ( ) ( )G w G ... G w G L2 2 3 3H H
2 22 2 2 2 ,
from where we could deduce G +G w2 3 4 , G +G w2 4 4 , ..., to be G2=G4=...= G3H . But
this is an absurd (G2 is a single eigenvalue). By the same reason H
can not be
perpendicular to H
2
.
Theorem 2:
In the 2-space defined by a pair of (unitary and orthogonal) eigenH-
adics of a symmetric 2H-adic there exists two other H-adics, unitary
and orthogonal between itself, which, bisecting the supplementary
angles of the two first, makes its transversal value, |t|, a maximum.
If in the anterior theorem, H
should be perpendicular to two, three ... up to 3H
-3 of the
eigenH-adics (between which couldn't exist H
1
nor H
2
) we should yet find out absurd
because the correspondent conditions would imply the equality of G1 and G2.
But if H should be perpendicular to H H H
G
.., , , . 3 4 - in which case H should
belong to the 2-principal space defined by H 1 and H 2 and still would make statyionary
22. 22
the square of the 2H
G transversal value - the system (1.161) would stay reduced to the
following two equations
( ) ( ) ( ) ( )G w G G w G L1 1 2 2
2 2
2 2 2 2 ,
from where we could deduce G +G w1 2 4 , or, taking into account the correspondent
expression (1.05) of the 2H
G radial value, 2w:
G +G G G1 2
H H H
1
H H H
2
2 1
2
2
2
[( ) ( ) ] . .
.
Remembering that, in the 2-space,
H H H H
1
H
1
H H H
2
H
2
( ) ( ) . .
with ( ) ( )H H H
1
H H H
2
. .
2 2
1 ,
it comes:
G +G G ]G1 2
H H H
1
H H H
2
2 11
2
1
2
{( ) [ ( ) } . .
,
or, simplifying and remembering that G1–G20: ( ) /H H H
1
.
2
1 2 . Hence:
H H H
1
H H H
2
. .
2
2
.
Related to H H H
1
/ .
2 2 we get the solution H H
(+)
under which H
(+) makes
the angle of 45 with H
1
; related to H H H
1
/ .
2 2 , the solution H
(-) makes the
angle of 135 with H
1
. Analogously with respect to H
2
. Hence H
(+) and H
(-) bisect the
supplementary principal directions defined by H
1
and H
2
; evidently they are
perpendicular to each other.
Corollary 1:
If the proportionality symmetric 2H
G-adic has N single
eigenvalues (1N 3H
), there exists CN
2
H-adics that leaves
stationary the square of its transversal value,|t|, each H-adic
belonging to a 2-principal space defined by a pair of its
eigenH-adics.
For, to each pair of single eigenvalues, there exists a H-adic bisector of the
supplementary angles defined by the correspondent eigenH-adics; and if N is the number of
single eigenvalues, these exist in number of CN
2
.
Theorem 3:
The radial value of the proportionality symmetric 2H-adic, 2w,
relative to any bisector direction is equal to one half the sum of the
23. 23
eigenvalues related to the correspondent bisected principal
directions.
Indeed, to the bisector H H H ( ) / 1 2
2 2 for example, in the 2-space defined by
H
1
and H
2
, (03,Intr.) gives the correspondent 2H
G radial value, 2w12:
2w12
H H H 2H H H H
1
2 1 2 1 2( ) ( ) . .G .
Substituting 2H
G for (1.04), operating and considering (1.041) and (1.042), we get
2w G G12 1 2
1
2
( ) .
In general, then, we write for the principal directions H
u
and H
v
:
2w G Guv u v
1
2
( ) , (u,v=1,2, ...,3H
) (1.17).
Theorem 4:
The transversal value of a symmetric 2H-adic, |t|, relative to each
bisector direction at a point, is equal to one half the modulus of the
difference of the eigenvalues related to the correspondent bisected
principal directions.
We can calculate t2
for the particular case of the (orthogonal) bisector directions
considered in the demonstration of the Theorem 3, that is, H H H ( ) / 1 2
2 2 . Noting
that, from (1.09),
|| || ( ) ( ) [ ( ) ] ( )H H H H
u u
H H H
u uG G . .
2 2
1 2
2 21
2
,
we have, developing the sum in u:
|| || [( ) ( ) ]H
2G G
1
2 1
2 2 .
Hence, using (1.13) and considering (1.17) we write:
t G G G G
G G2
2 2
2
1
2
1
4 21
2 2
1
2 1 2
[( ) ( ) ] ( ) ( ) ,
that is,
|t|
|G G
1 2
2
|
.
24. 24
Certainly we should obtain the same result if the considered bisector direction was
H H H ( ) / 1 2
2 2 .
In general, then, we write for the principal directions H
u
and H
v
:
|t |
|G G
uv
u v
|
2
, (u,v=1,2, ..., G) (1.18).
Note:
By (1.18) we can calculate the highest |t| in the 6-space, which is (G6-G1)/2. But if
the eigenvalues are all positive or all negative, this value is not the |t|max because
this maximum is |G6|/2 and occurs in the 9-space (the zero eigenvalue must now be
considered).
9 - THE MOHR PLANE REPRESENTATION.
The simultaneous laws (01,02 and 03, Intr.) under study were transformed into the
simultaneous system of scalar equations
2W=
W T
H H 2H H H
2 H H 2H H H
H H H
( )
. .
. .
.
G
G2
1
2 2
enclosing the 2H
G radial and transversal value, 2w and |t|, correspondent to the variable H-
adic H
.
This system, in the 2H
G eigenH-adic (unitary and orthogonal) base, can be written
as
2w=( G G G ... G
w t ( (G (G ... (H
H H H
u u
H H H
1 1
H H H
2 2
H H H
G G
2 H H H
u u
H H H
1 1
H H H
G G
H H H H H H
1
) ( ) ( ) ( )
( ) ) ) ( ) ) ( ) )
(
. . . .
. . .
. .
2 2 2 2
2 2 2 2 2 2 2
2
1
) ( ) ( ) ,2 2
H H H
2
H H H
G
2... . .
where, without loss of generality, we can suppose
G G G ... G1 2 3 G
, (1.23).
These three equations involves G+2 variable letters: 2w, |t| and the G square coordinates of
H
. Representing these last ones in the form H H H
u u
E .
for all u=1,2, ..., G, we
rewrite the system in the form
25. 25
2w=(E G (E G ... (E G
w t (E (G (E (G ... (E (H
(E (E ... (E
1 1 2 2 G G
2
1 1 2 2 G G
1 2 G
) ) )
( ) ) ) ) ) ) )
) ) )
2 2 2
2 2 2 2 2 2 2
2 2 2
2
1
, (1.24).
Interest us to discuss this system when H
varies, to be, when the Eu's are variable.
9.1 - H
varies in a 3-space.
Let us imagine initially that the H-adic H
varies in the 3-space defined by
H H
2
H
3
and , 1 , in which case E4=E5= ... = EG
=0 and the system (1.24) is reduced to
2w=(E G (E G (E G
w t (E (G (E (G (E (G
(E (E (E
1 1 2 2 3 3
2
1 1 2 2 3 3
1 2 3
) ) )
( ) ) ) ) ) ) )
) ) )
2 2 2
2 2 2 2 2 2 2
2 2 2
2
1
(1.25).
In a coordinate plane 2wx|t| the current point (2w,|t|) traces a certain portion of
surface as H
varies (because this point varies with two independent parameters). The
analytical calculation of this area can be done, of course, by the system (1.25) which is
linear in (E1)2
, (E2)2
and (E3)2
. Solving (1.25) we get, remembering that G1G2 G3:
( )
( )( )
) )
( )
( )( )
) )
( )
( )( )
) )
E
w-G w-G t
(G -G (G -G
E
w-G w-G t
(G -G (G -G
E
w-G w-G t
(G -G (G -G
1
2 3
3 1 2 1
2
3 1
2 3 2 1
3
1 2
3 1 3 2
2
2
2
2
2
2
2 2
2 2
2 2
(1.26).
As the first members must be all positive we might have
( )( )
( )( )
( )( )
2 2 0
2 2 0
2 2 0
2
2
2
w-G w-G t
w-G w-G t
w-G w-G t
2 3
3 1
1 2
(1.27).
The first equation in (1.27) can be written also in the form
( )( ) ( ) ( )2 2
2 2
2 2 2
w-G w-G t
G -G G -G
2 3
3 2 3 2
or, transforming the first member:
t w
G +G G -G3 2 3 22 2 2
2
2 2
( ) ( ) , (1.28).
26. 26
The inequality (1.28) represents points not interior to the semicircle centered in
C G G23 2 3
2 0 (( ) / , ) with radius 2/)GG(R 2323 .
Similar interpretation we could do to the other two inequalities in (1.27), the second
representing points not exterior to the semicircle centered at )0,2/)GG((C 3113 with
radius 2/)GG(R 1313 and the third, points not interior to the semicircle centered at
)0,2/)GG((C 1212 with radius 2/)GG(R 1212 .
As the pairs (2w,|t|) must satisfy system (1.27) its images in the plane 2wxt will be
points of the dashed area shown in Fig. 1 (draw for the particular case of all G>0).
This graphical representation on the 2H
G radial and transversal values variation will be
named Mohr's representation because of its analogous in Elasticity; the border circles ,
Mohr's circles and the plane 2wx|t|, Mohr's plane.
We shall show now how to determine in the Mohr plane the point N correspondent to a
given direction in a 3-space without calculating 2w and |t|. Any direction H
can be defined
for example by the angles 1 and 3 it defines with H
3
. Let us look for the locus of points
equally inclined over H
3
; with other words, we ask: when N describes the parallel of co-
latitude 3 of the spherical surface H H H .
1, what curve will N describe in the Mohr
plane?
The equation of this curve is obtained by eliminating E1 and E2 in the system (1.25); we
have:
t w
G G G G
cos G G G G2 2
( ) ( ) ( )( )2
2 2
1 2 2 2 1 2
3 3 1 3 2 , (1.29).
This is the equation of a circle centered at C12 with radius equal to the square root of its
second member. In the Mohr plane, as shown in Fig .2, let us draw by (G3 ,0) the line r3
inclined 3 over the |t| axes, which cuts the circles (C13,R13) and (C23,R23) at A2 e A1 ,
respectively. Being G2A1 e G1A2 perpendicular to the same line r3 (because they project the
extremes of the diameters of the circles C13 e C23 over r3) they will be parallel between
itself and parallels to the mediatrice (perpendicular bisector) of A1A2 which contains
necessarily C12. Hence we deduce (helped by the Fig. 2, draw for G1<0):
27. 27
C A C G G
G G
12 12
2
12 3
2 2
3 3
cos ( ) cos , 1 2 2 2
32
( ) ( ) sen ( ) ( cos ),
A A G G G G1 2 2 1 2 2 2
3
2 1 2 2
32 2 2
1
C A C A
A A G G
G G G G12 1
2
12 12
2 1 2 2 2 1 2
1 3 2 3
2
32 2
( ) ( ) ( )( ) cos , (1.30).
Hence the radius of the circle (1.28) is C A12 1 as we can conclude by comparing the second
members 0f (1.29) and (1.30).
Let us see now between what limits will vary the radius C A12 1:
for 3 0 , C A
G G
G G G G G
G G
C G12 1
2 1 2
1 3 2 3 3
1 2
12 32 2
( ) ( )( ) ;
for 3 2 / , C A
G G
G
G G
C G12 1
2 1
2
1 2
12 22 2
.
With these results we glimpse the possibility to quote in 3 the circle (C23,R23) for example,
to locate immediately the arch of circle (1.30), as we show in Fig .3.
We will proceed in analogous way with respect to the inclination 1 of H
over H
1
,
but now we must have 1 32 / to be possible the third equation in (1.25). The locus
of the points, on the Mohr plane, whose represent 2w with respect to equal inclined
directions over H
1
is the circle
t w
G G G G
cos G G G G C B22 2 3 2 3 2 2
1 3 1 2 1 23 2
2
2
2 2
( ) ( ) ( )( ) , (1.31),
Quoting the semicircle (C12,R12) in 1, in the same way as we have done before, it will
be possible to locate immediately the point N whose coordinates are 2w and |t| (Fig. 3).
It is evident that the points of the semicircle centered at C13 with radius R13 are related
to directions with 2 2 / rd, this is, with direction perpendicular to the plane defined by
directions H
1
and H
3
, to be, the directions with respect to which 2W are extreme. It is
evident, also, that the extreme value of |t| is R13, this is,
28. 28
| |t R
G G
max 13
3 1
2
, (1.32).
There is also a third circle passing through the point N, not represented in the figures,
with center at C13; its equation is
t w cos22 3 1 2 3 1 2
2 3 2 1 22
2 2
( ) ( ) ( )( )
G G G G
G G G G , (1.311).
and the square of its radius is the second member of (1.311). As in the anterior cases this
radius can be detected graphically.
Note:
We must to notice that, in Elasticity, for H=1, the stress dyadic radial
value, 2w, represents normal stress and |t| tangential stress; for H=2,
the Gren's tetradic radial value, 2W, for a certain unitary strain
dyadic , represents stored energy and |t|, a new variable which we
could name complementary energy since 2t|2w|max . Hence, in this
science we can apply Mohr's circle theory in the diagram energy x
complementary energy. As the minimum energy is zero, the
proportionality polyadic eigenvalues are non negative and the origin
must belong to the valid area, to be, the Mohr circle relative to the
null eigenvalue must be considered.
9.2 - H
varies in a 4-space, 5-space ....
Let us imagine now that the H-adic H
varies in the 4-space defined by
H H
2
H
3
H
4
, and , 1 , in which case E5= ... = EG =0 and the system (1.24) is reduced to
2w=(E G (E G (E G (E G
w t (E (G (E (G (E (G (E (G
(E (E (E (E
1 1 2 2 3 3 4 4
2
1 1 2 2 3 3 4 4
1 2 3 4
) ) ) )
( ) ) ) ) ) ) ) ) )
) ) ) )
2 2 2 2
2 2 2 2 2 2 2 2 2
2 2 2 2
2
1
(1.33).
29. 29
Taking (E4)2
from the last equation, substituting in the first two and grouping pieces we
write the new equivalent system
2w-G =(E (G -G (E (G -G (E (G -G
w G t (E [(G (G (E [ (G (G (E [(G (G
(E (E (E (E
4 1 1 4 2 2 4 3 3 4
4
2
1 1 4 2 2 4 3 3 4
4 1 2 3
) ) ) ) ) )
( ) ( ) ) ) ) ] ) ) ) ] ) ) ) ]
) ) ) ) .
2 2 2
2 2 2 2 2 2 2 2 2 2 2
2 2 2 2
2
1
The determinant of this system in the unknown E1
2
, E2
2
and E3
2
is the positive number
( )( )( )G G G G G G2 1 3 1 3 2
0;
solving it (as a function of E4
2
) we find:
( )
( )[ )] ( )( )[ ) ]
) )
( )
( )[ )] ( )( )[ ) ]
) )
( )
( )[ )] ( )( )[ ) ]
)
E
w-G w-(G +G -G t G -G G -G 1-(E
(G -G (G -G
E
w-G w-(G +G -G t G -G G -G 1-(E
(G -G (G -G
E
w-G w-(G +G -G t G -G G -G 1-(E
(G -G (G
1
4 2 3 4 4 2 4 3 4
3 1 2 1
2
4 1 3 4 4 1 4 3 4
3 2 2 1
3
4 1 2 4 4 1 4 2 4
3 1 3
2
2 2
2
2 2
2
2 2
2 2
2 2
2 2
-G2 )
(1.34),
to be
( )[ )] ( )( )[ ) ]
( )[ )] ( )( )[ ) ]
( )[ )] ( )( )[ ) ]
2 2 0
2 2 0
2 2 0
2 2
2 2
2 2
w-G w-(G +G -G t G -G G -G 1-(E
w-G w-(G +G -G t G -G G -G 1-(E
w-G w-(G +G -G t G -G G -G 1-(E
4 2 3 4 4 2 4 3 4
4 1 3 4 4 1 4 3 4
4 1 2 4 4 1 4 2 4
(1.35),
Summing up [( ) / ]G G G2 3 4
2
2 to both members of the first inequality in (1.35),
operating, simplifying and transposing pieces we obtain:
t w
G G G G
G G G G E2 2 3 2 3 2 2
4 2 4 3 4
2
2
2 2
( ) ( ) ( )( )( ) .
As 0(E4)2
1 we can write
( ) ( ) ( )
G G
t w
G G
G
G G3 2 2 2 2 3 2
4
2 3 2
2
2
2 2
, (1.351).
From the second and the third inequalities we obtain also two other inequalities; a second,
similar to (1.351):
30. 30
( ) ( ) ( )
G G
t w
G G
G
G G2 1 2 2 1 2 2
4
1 2 2
2
2
2 2
, (1.352),
and a third, slightly different,
t w
G G
G
G G2 1 3 2
4
1 3 2
2
2 2
( ) ( ) , (1.353).
The meaning of these inequalities are obvious: (1.351), for example, represent the set of
points not interior to the circle centered at ( (( ) / , )G G2 3
2 0 ) with radius ( ) /G G3 2
2
and not exterior to the circle with the same center and radius G G G4 2 3
2 ( ) / .
Hence, in this 4-space the set of points in the 2wx|t| plane that satisfies the system (1.35)
involves all those points not exterior to the circle centered at ( (( ) / , )G G2 3
2 0 ) with
radius G G G4 2 3
2 ( ) / and those not interior to the following two circles: a first
centered at (( ) / , )G G1 2
2 0 with radius ( ) /G G2 1
2 and a second centered at
(( ) / , )G G2 3
2 0 with radius ( ) /G G3 2
2 . As the 3-spaces defined by
( H H H , , 1 2 3
), ( H H H , , 2 3 4
) etc. are subspaces of the considered 4-space, its
correspondent areas could be valid areas; hence we need to exclude the points not interior
to the semicircle centered at (( ) / , )G G3 4
2 0 with radius ( ) /G G4 3
2 .
If we had taken (E1)2
from the last equation (1.33) in place of (E4)2
we would obtain
inequalities in much appeared with (1.351), (1.352) and (1.353); they would have G1 and G4
changed in place.
Discussing all possibilities we can conclude that the interested points will be those not
interior to the semicircles (C12,R12), (C13,R13), (C34,R24) and not exterior to the semicircle
(C14,R14).
It is easy to extend these conclusions to 5-space, 6-space, ..., G-space.
*
We show now how to determine in the Mohr plane of a 4-space the point N
correspondent to a given direction with director cosines cos1, cos2 etc. It is convenient to
observe that the sum of two any direction angles that satisfies the third equation (1.33) must
be always not inferior to 90 . Indeed, for two any the sum of the squares of the respective
cosines must be less than one, say cos cos2 2
1 2
1 , to be
cos sen cos2 2 2
1 2 2
90 ( )
, or 1 2
90
.
Let us consider initially the directions inclined of the given angle i
over the reference
H-adic H
i
and m
over H
m
. When N describes the spherical surface H H H
.
1 (with
fixed i
and m
) what curve would describe the correspondent N in the Mohr plane ?.
31. 31
As in the later case, the equation of this curve can be obtained by eliminating the
cosines Ej and Ek in the system (1.33). So proceeding we can find the equation
t (2w
G G
2
) R2 j k 2
jk
2
, (1.36),
with
R (
G G
2
) (G G )(G G )cos (G G )(G G )cosjk
2 k j 2
k i j i
2
i k m j m
2
m
, (1.361).
Suppose now given four angles satisfying the third equation of the system (1.33). We
can eliminate in 6 different ways pairs of angles. This is the same to say that from (1.36),
running j and k from 1 to 4 (with jk), we can obtain the equations of the six circles
centered each one at the Mohr's circle centers with radius R*
jk. But this six circles will have
necessarily a common point, precisely the point of the Mohr plane to which correspond the
direction specified by the given angles. One way to confirm this statement is presented in
Appendix I.
9.3 - Some properties of Mohr's circles.
Some particular properties of this circles can be signed.
For example: for 4
90
, this is, for a direction perpendicular to the base H-adic H
4
,
this direction belongs to the 3-space of base H H H
1 2 3
, , . If we make j=1, k=2 and i=3 in
(1.36) and (1.361) we obtain the equation (1.29); if we make j=2. K=3 and i=1 we obtain
equation (1.31). In this case the geometrical meaning are the same as in section 1.6.1.
*
For i m
90
the considered direction belong to the 2-space of base H
j
H
k
, .
Hence the correspondent Mohr's circle is (Cjk,Rjk), what is evident from (1.36) and (1.361).
*
The normal to the 2w-axes draw by G2 and G3 cuts the semicircles (C13,R13) and
(C24,R24) at fixed points F4 and F1, respectively, equidistant from C14. Indeed, we can write:
C F C G14 4 14 2
2 2
2 4
2
G F , and C F C G14 1 14 3
2 2
2 1
2
G F (1.37).
But
C G
G G
14 2
2 1 4 2
2
( )G2 , and C G
G G
14 3
2 1 4 2
2
( )G3 (1.38).
Then, remembering that the perpendicular from a point of a circle over a diameter is
proportional media between the segments determined by its foot,
G F G G G G G G G G2 4
2
1 2 2 3 2 1 3 2
( )( ) , (1.39),
and
G F G G G G G G G G3 1
2
3 4 2 3 4 3 3 2
( )( ) , (1.391).
32. 32
Substituting (1.38), (1.39) and (1.391) into the equalities (1.37) we conclude:
C F C F
G G
G G G G G G14 4
2
14 1
2 1 4 2
3 2 1 3 2 42
( ) , (1.40).
*
For octahedral directions we must consider
cos cos cos cos2
1
2
2
2
3
2
4
1
4
.
Representing the radii of the correspondent circles by Rjkoct we write from (1.361):
4 4
2
2 2
R
G G
G G G G G G G Gjkoct
k j
k i j i m k m j
( ) ( )( ) ( )( ) .
Developing the second member, adding pieces conveniently and representing by R2
the
sum of the squares of the radii of the Mor's circles, that is,
R
G G G G G G
G G G G G G
2 4 1 2 4 2 2 4 3 2
3 2 2 3 1 2 2 1 2
2 2 2
2 2 2
( ) ( ) ( )
( ) ( ) ( ) ,
(1.41),
we obtain,
4
2 2
2 2 2 2
R R
G G G G
jkoct
k j m i
( ) ( ) , (1.42).
Summing up the six expressions obtained from (1.42) giving to the index the values 1,
2, 3 and 4 we deduce that
R R R R R R Roct oct oct oct oct oct12 34
2
13 24 14 23
1
2
2 2 2 2 2 2
, (1.43).
*
By the Mohr plane representation it is evident that if all the 4
G eigenvalues are positive,
then all of its radial values are positive; to each point of the valid area correspond one and
only one point of the correspondent indicator ellipsoid (see section 1.5). If there are some
negative eigenvalues – in which case the indicator is a hyperboloid - then to the segment of
the |t| axes, interior to the valid area, correspond the (twisted) hipercurves intersection of
the cone with the sphere. With other words we can say also that to the points belonging to
that segment corresponds directions parallels to the cone generators. Yet in this case, to
each point of the valid area corresponds a point of the hyperboloid indicator.
9.4 - A criterion of proportionality.
As we have emphasized (see Introduction) the 2H-adic characterizes the medium with
respect to the phenomenon governed by the law of proportionality. In the previous
33. 33
paragraphs we have shown that to each pair (H
, H
) corresponds a point P in the valid area
of the Mohr's plane with coordinates (2w, t), see Fig. .
Fig. .......
A necessary and sufficient condition for the point P to be not exterior to the valid area –
that is, to exist the proportionality established by the law (021,Intr.) - is that its distance d to
center of the circle with the greatest radius be minor than this radius, to be, minor than the
greatest t, the invariant tmax which is a characteristic of the medium.
Hence, supposing that the eigenvalues are positive and negative, tmax=|Gmax-Gmin|/2
that is, the double of tmax is the modulus of the difference between the greatest and the
minor of the 4
G eigenvalues (zero included). By other side, we can write
d [|2w| G t )] t (t )2
max max
2 2
max
2
( .
Hence,
|| ||+G G
G +G
|2w|
H
max min
max min
, (1.44).
From this inequality we deduce a "criterion of proportionality" between those magnitudes:
If in the current point of the field of a certain physical phenomenon, a
dependent variable magnitude H
is proportional to an other variable
magnitude H
through a constant polyadic magnitude 2H
G whose
extremes eigenvalues are Gmax and Gmin, then the quantity of H
by
unit of intensity of H
, (the polyadic) H
, might verify (1.44) where 2w
is the 2H
G radial value relative to the unit of H
.
Remembering (2.04), we write (1.44) as
|| || || || ( )| |H
max min
H
max minG G G G W 2 0 (1.441).
If in the current point of the field of a certain physical phenomenon, a
dependent variable magnitude H
is proportional to an other variable
34. 34
magnitude H
through a (constant) magnitude 2H
G whose extremes
eigenvalues are Gmax and Gmin, then it might be verified (1.441) where
2W0 is the 2H
G radial value relative to H
.
In Elasticity for example, where, for H=2, the 2w values are non negative and Gmin=0,
the inequality (1.44) assumes the form | | |2G | |w|2
max and the proportionality criterion
is so enunciated whatever be the elastic material:
If in the current point of a charged elastic solid, the stress dyadic is
proportional to the correspondent strain dyadic through the Green's
tetradic 2H
G of this solid, which maximal (always positive) eigenvalue
is Gmax then the norm of the quantity of by unit of intensity of - the
norm of the dyadic - do not surpass the double product of Gmax by
the specific density stored energy w (energy by unit of volume and by
square of intensity).
Putting (1.441) in the form | | |2G | |W |2
max 0 we can also enunciate:
If in the current point of a charged elastic solid, the stress dyadic is
proportional to the correspondent strain dyadic through the Green's
tetradic 2H
G of this solid, which maximal (always positive) eigenvalue
is Gmax then the dyadic stress norm do not surpass the double product
of Gmax by the density stored energy W0 (energy by unit of volume).
APPENDIX I
Common point to circles centered on the same axis.
Let us search the necessary condition for the circles
Y X A R
Y X A R
Y X A RN N
2
1
2
1
2
2
2
2
2
2
2 2 2
( )
( )
...
( )
, (App.1),
centered on the X-axes, have a common point.
This means that the above system is verified for some pair (X,Y) and we can
eliminate these two letters with all the equations. Resting from the first equation the other
N-1, factorizing in each new equation and dividing both members for the non null
differences between the A's, we obtain the new system of N-1 equations
35. 35
2
2
2
1 2 1
2
2
2
1 2
1 3 1
2
3
2
1 3
1 1
2 2
1
X A A R R A A
X A A R R A A
X A A R R A AN N N
( ) ( )/( )
( ) ( )/( )
...
( ) ( )/( )
involving the X coordinate of the point. Resting again from the first equation the other N-2
and operating in the new equations formed, we eliminate X and obtain the system
( )( ) ( )( ) ( )( )( )
( )( ) ( )( ) ( )( )( )
...
( )( ) ( )( ) ( )( )( )
R R A A R R A A A A A A A A
R R A A R R A A A A A A A A
R R A A R R A A A A A A A AN N N N
2
2
1
2
3 1 3
2
1
2
2 1 2 1 3 2 3 1
2
2
1
2
4 1 4
2
1
2
2 1 2 1 4 2 4 1
2
2
1
2
1
2
1
2
2 1 2 1 2 1
(App.1.1),
which can also be written in the form of determinants if desired to facilitate memorization:
1 1 1 1 1 1
1 1 1 1 1 1
1 2 3
1
2
2
2
3
2
1 2 3
1
2
2
2
3
2
1 2 4
1
2
2
2
4
2
1 2 4
1
2
2
2
4
2
A A A
R R R
A A A
A A A
A A A
R R R
A A A
A A A
...
, (App. 1.2).
For N=3 (in the 3-space) the system (App. 1) is reduced to the three first equations and
(App.1.2) stay reduced to the first equation. In the 3-space (1,2,3) we have
A G G
G G
G G G G1 2 3
2 3 2 2
1 3 1 2
2
1
1
2 2
( ), ( ) ( )( )R cos1
A G G
G G
G G G G2 3 1
2 1 3 2
2 3 2 1
2
2
1
2 2
( ), ( ) ( )( )R cos2
A G G
G G
G G G G3 1 2
2 1 2 2
3 1 3 2
2
3
1
2 2
( ), ( ) ( )( )R cos3
,
and
cos cos cos2
1
2
2
2
3
1 ,
expressions that verifies identically the system (App.1.2). Hence the circles have a common
point.
For N=4 (in a 4-space) the system (App.1) is reduced to the first four equations and
(App.1.2) is reduced to the first two. The expression of the radii are more complicate now
36. 36
and can be derived from (1.361). We must attribute in (App. 1) to the four (different) Ai any
value of the set ( )/G Gj k 2 and to the four Ri any value of the set defined by (1.361), for
all j,k=1,2,3,4. This verification is not complicate but fatiguing.
APPENDIX II
Elimination of an angle in a system of functions.
The problem consists in the elimination of the angle is the system
A B C L R
A B
sen2
sen2
cos ( ) ( )
cos
2
2 0
where A', A", B', B", R are constants and C() is a given function of , or even a constant if
L()=0.
Multiplying the first for B", the second for B' and resting the second from the first we
obtain
( )sen A B A B CB2 .
Similarly we can obtain
( )cos A B A B A C2 .
Hence, squaring these two equations and summing member by member we get the equation
C L R
A B A B
A B
K A B( ) [ ( ) ]
( )
( sen cos ) 2 2
2
2 2
2 2
2 2
, (App.2.1).
If L()=0, then C()=R and the problem is finished: was eliminated, and the
eliminating equation is (App.2.1) for C()2
=K2
=R2
.
If C() is still a function of we can write now the given system in the form
C L R A B A B A B K
A B
( ) ( ) ( ) /
cos ,
2 2 2
2 0sen2
or
L R K
A B
( ) ( )
cos .
2 2
2 0
sen2
where K is the number defined by (App.2.1) and L() is a given function (which can be
null).
37. 37
Particularly, if L() is a linear function of sen and cos, we must eliminate in the
system
( cos ) ( )
cos
A sen B R K
A B
2 2
2 0
sen2
Developing the square in the first equation, remembering the classical trigonometric
identities
2 1 2sen2
cos and 2 1 2cos2
cos
the system is transformed in
2 2 2
2 0
2 2 2 2 2
AB sen2 B A R K A B
A B
( cos ( ) ( )
cos
)
sen2
But this system is similar to the system considered in the beginning for C'()=constant, in
which case the eliminating equation is (App.2.1) for
C R K A B 2 2 2 2
( ) ( ) , A AB2 and B B A2 2
.
We have
[ ( ) ( )]
[ ( )]
( )
2
22 2 2 2
2 2 2
2 2
R K A B
ABB A B A
A B
, (App.2.2),
With other words, we can say that the eliminating equation of the system
A B R
A B
sen2
sen2
cos
cos
2
2 0
is the equation (App.2.1) for L()=0 with K=R. The eliminating equation of the system
A B A B R
A B
sen cos cos
cos
sen2
sen2
2
2 0
is the equation (App.2.2) for K given by (App.2.1).
Certainly (App.2.2) can be reduced to (App.2.1) for A=B=0. Indeed, from (App.2.2) we
have R K 0, what is coherent with (App.2.1) for L()=0.
38. 38
THE POLYADIC GEOMETRY........................................................................................1
I.1 - Polyadics and physical magnitudes. ...............................................................................1
I.2 - Physics and geometry. ....................................................................................................1
I.3 - The polyadic space. ........................................................................................................2
Polyadic coordinates. .....................................................................................................2
The similarity with vectors.............................................................................................2
I.4 – Constructing figures in the polyadic space. ...................................................................3
I.4.1 - The main graphical property of the arrows: the plane hyper-trigonometry. ................3
I.4.1 - A 3H
-angular pyramid as a natural reference system in a H-adic space.......................3
I.4.2 – The hyper-planes geometry and trigonometry. ...........................................................4
I.4.3 - Constructing parallelepipeds in the polyadic space. ....................................................4
I.4.4 - Some characteristic properties of a (P-1)-angular pyramid. ........................................4
I.4.4 – The hyper-spherical trigonometry...............................................................................5
2 – A GENERAL LINEAR PHYSICAL LAW. ................................................................6
2.1- Concepts and definitions.................................................................................................6
2.2- The physical law is a mapping........................................................................................7
2.3- The 2H
G determination. ...................................................................................................8
2.4- A decisive and simplifier assumption. ............................................................................8
2.5- A convenient change of variables. ..................................................................................9
3 - THE QUADRICS ASSOCIATED TO THE PHYSICAL LAW. ...............................10
4 – THE PROPORTIONALITY POLYADIC CHARACTERISTIC ELEMENTS.........11
4.1- Concepts. ......................................................................................................................11
4.2- The 2H
G characteristic elements are all real. .................................................................11
4.3- Vanishing and multiples eigenvalues............................................................................12
4.4- The Cauchy and Lamé's quadric reduced equation.......................................................13
5 – THE STATIONARY PROPORTIONALITY POLYADIC RADIAL VALUE. .......14
6 - THE PROJECTION NORM AND OCTAHEDRAL DIRECTIONS.........................15
7- SOME RELATIONSHIP BETWEEN EIGENVALUES. ...........................................17
8 - THE PROPORTIONALITY POLYADIC TRANSVERSAL VALUE......................18
The 2H
G transversal value extreme...............................................................................19
9 - THE MOHR PLANE REPRESENTATION. .............................................................24
9.1 - H
varies in a 3-space. ................................................................................................25
9.2 - H
varies in a 4-space, 5-space .... ..............................................................................28
9.3 - Some properties of Mohr's circles................................................................................31
9.4 - A criterion of proportionality.......................................................................................32
APPENDIX I................................................................................................................34
Common point to circles centered on the same axis. ...................................................34
APPENDIX II ..............................................................................................................36
Elimination of an angle in a system of functions. ........................................................36
