A geometrical picture for the family structure of fundamental
particles is developed. It indicates that there
might be a relation between the family repetition structure
and the number of space dimensions.

Introduction

With the discovery of the top quark
all in all 12 fundamental
fermions (6 leptons and 6 quarks) are known today.
At least a decade ago it has become clear that these particles
can be organized into three "families"
each containing 2 quarks and 2 leptons.
These particle families (or generations) behave
identically under the electroweak and strong
interactions and do not differ by anything else than their
masses.
The number of generations will probably be
restricted to three forever, because it has been shown
experimentally that at most 3 species of light neutrinos
exist. A fourth family,if it exists at all,
would necessarily contain a heavy
neutrino and would therefore be different in nature
from the known families.

Over the last 20 years there has been one outstanding puzzle
in elementary particle physics. This is
the question whether the variety of "elementary" particles,
the quarks and leptons, can be derived from some more
fundamental principle. To answer this question is quite
difficult because up to now no experimental indications
exist of which might be the nature of this principle.
Present models usually lead to an inflation of new
particles (like supersymmetry) at higher energies and/or tend to
shift the basic problems to higher energy scales where they
reappear in slightly modified form (like technicolor).

Preon models avoid these deficiencies but have
severe problems of other kinds, like the smallness of fermion masses
as compared to the binding scale. Still, I want to follow in this
work a preon type idea,
that the quarks
and leptons have a spatial extension, and contain
sub--constitutents sitting on a cubic lattice.
My guideline will be that the spatial
dimensions correspond to a sort of shells which are
successively filled up by the generations.
The third shell -- corresponding in some sense to the third dimension --
becomes closed with the top quark.
I shall make use of some discrete, nonabelean subgroups of
O(3), in particular the symmetric group $S_4$ which is closely
related to the symmetry transformations of a cube.
I shall also address the question of how to understand the
vector bosons and the mass hierarchy within this model.

I would like to warn the reader that in this article I am mostly
doing simple minded geometry without really clarifying
the mathematics behind it.
There are all sorts of unanswered questions concerning the
dynamics of the model.
My hope is that I can motivate readers to do more refined work
on the basis of these suggestions.

The Family Repetition

The known quarks and leptons within one family are:
\begin{equation}
\nu_{L,R} \qquad e_{L,R} \qquad u^{1,2,3}_{L,R} \qquad d^{1,2,3}_{L,R}
\label{eq1}
\end{equation}
where the upper index denotes the quark color degrees of
freedom and the lower index the helicity. I have made the
reasonable assumption
a righthanded neutrinos $\nu_R$ exist.

There are $ 2_{spin} \times 2_{isospin} \times (1+3_{color})=16$
degrees of freedom in one family (32, if
antiparticles are counted as well) and altogether 48
degrees of freedom in all three generations.
This will be related to a property of 3--dimensional space,
namely that there are
48 symmetry transformations which leave the
3--dimensional cube invariant.

One might think that some information on the nature of the
fermions can be obtained from their measured mass spectrum.
However, the fermion masses are running, i.e. energy dependent,
and it is not really
known which dynamics governs this energy dependence.
In other words, their renormalization group
equations are not known precisely and therefore no complete
knowledge of the fermion mass spectrum exists.
For instance, the masses could be running according to some SUSY--GUT
theory. However, apart from the fact that the SUSY breaking scale
is not known precisely, new physics
may set in at some point and modify the RG equations.
Furthermore, the masses are extremely tiny as compared to the binding
scale, so that in any preon model they will arise as
remnants of cancellations between large numbers.
Therefore it is not to be expected that the fermion
mass spectrum gives direct insight into the dynamics
of the preons.
The fermion mass spectrum is at most a qualitative
guideline to understand the family structure.

t'Hooft has suggested to decree chiral invariance
as the principle which suppresses the fermion masses, and from this has
derived conditions on the anomaly structure of the preon model.
In the present work this is not a necessary condition. The point
is that the extension of the observed fermions is not fixed
by the binding energy of a superstrong force but
{\it by the structure of space}. In my model,
space is essentially discrete with massless preons
sitting on the sites of a cubic lattice. I shall stick
to the notion of 'binding energy', though, to mean the inverse extension
of the 'bound states'.
Actually, the binding energy is the scale at which the fermion
masses should be defined.

The Model

The basic assumption in this work goes as follows:
the fermions in the first family define a planar
structure in space composed of a
'shell' which is successively filled up when one goes from
the electron--neutrino to the up--quark.
The closed first family shell survives in the second family
where another shell/plane orthogonal to the first
is beginning to fill.
Finally, the third family fills 3-dimensional space
completely.
This qualitative picture is depicted in fig \ref{fig4}
where the 'first', 'second' and 'third' family plane are drawn.
To account for the rather large mass difference
between the overall family masses (factors $\sim 100$) one might
speculate that they arise from "exciting" the successive dimensions.

More precisely, my model rests on the symmetry
groups of cubes and tetrads, which
are discrete subgroups of O(3).
Of particular relevance will be the group of permutations
$S_4$ which is isomorphic to the symmetry group of the
tetrad.
The discussion will concentrate on the subgroups themselves
and not on their representations. This is a somewhat unusual
approach, because normally in physics particle multiplets
are identified with the representation spaces and not with the
symmetry groups themselves. In contrast, the philosophy
here is that by applying the symmetry transformations on the
ground state ($\nu_e$) one can generate all other fermion
states. This is only possible if there is a symmetry {\it breaking}
which distinguishes the generated states.
Such a symmetry
breaking will be realized by geometrical means in the following
section. To show how this can happen, I have visualized
in fig. \ref{figt} an element of $S_3$, the permutation symmetry
of the three sides
of an equilateral triangle in 3 dimensions, assuming the
existence of distinct preons A,B,C on the sites and distinct
binding forces on the links of the triangle. If the $S_3$--transformation
is applied to the sites but not to the links of the triangle,
a completely different state is generated. In this case there are 6 such
states corresponding to the 6 elements of $S_3$.

The symmetry
group of the cube is the octahedral group $O_h$. It has
a direct product form $O_h\equiv P \oplus S_4$ where $P$ is
the parity transformation and $S_4$ is the
group of symmetry transformations of the tetrad, sometimes
also called $T_h$ or $O$ (where the $O$ stands for the octahedron
whose transformations it also describes).
$O_h$ contains exactly the 48=2x24 elements needed for a one to one
correspondence with the particles of the 3 families.
The philosophy here is that by applying the 48 symmetry
transformations of the cube on one of the 48 fermion states
one can generate all the other 48 fermion states.
This indicates that the fermions should have some 3 dimensional
substructure which completely breaks the cubic symmetry.
Such a symmetry breaking can be realized in a variety of different
ways as will be shown below.

The symmetry group $S_4$ of a tetrad is isomorphic
to the group of permutations of 4 objects. This way the
24 symmetry tansformations on a tetrad can be viewed as the set of all
(directed, open) paths that connect
the 4 corner points 1,2,3,4 of the tetrad.
From the tetrad a cube can be generated by applying
the parity transformation. In fact there are two tetrads,
a 'lefthanded' (with corner points 1,2,3,4) and a 'righthanded'
(with corner points 1',2',3',4') embedded in a cube.
(cf. fig. \ref{fig5}) related by P.

From now on I will follow the philosophy that the spatial
structure of a fermion (quark or lepton) is that of a tetrad.
Furthermore, I assume that
by applying the 24 symmetry transformations of the tetrad state
one can create all
24 fermion states of the three families out of one of these states.
In order to guarantee that all of the symmetry transformations
yield different states, the tetrad cannot be completely symmetric.
For example, one may assume that there are 4 different preons
sitting on the 4 corners of the tetrad.
Another possibility is that the preons are identical, but the binding
forces between them are different. In the following we shall
pursue this latter option. More specifically, we shall
assume that the bindings between the 4 preons are given
according to the 24 permutations of the set {1,2,3,4}.
One of the 24 permutations
(namely the identity element) is visualized in fig. \ref{fig5}
by the 3 arrows $1 \rightarrow 2 \rightarrow 3 \rightarrow 4$.
Any other element $1234 \rightarrow abcd$ of $S_4$, denoted
by $\overline{abcd}$ in the following,
could be
drawn as the path $a \rightarrow b \rightarrow c \rightarrow d$
in fig. \ref{fig5}.

Starting with the 'lefthanded' tetrad,
one can construct
all the 24 lefthanded fermion states of the 3 generations.
By applying the parity transformation
$(x,y,z) \rightarrow (-x,-y,-z)$, righthanded fermions
can be obtained. Any such righthanded state
will be denoted by $\overline{a'b'c'd'}$ in the following.
%As an example, $\overline{3'4'2'1'}$
%is shown in fig. \ref{fig6}
%I have not undertaken to explicitly produce Dirac fermions out of
%this construction. So admittedly, at the moment, only qualtitativ.
As well known, a (Dirac) fermion $f$ has four degrees of freedom,
of which only two, $f_L$ and $f_R$, have been described so far.
The way to obtain antiparticles $\bar f_L$ and $\bar f_R$ is as follows:
$\bar f_L$ is a righthanded object
and its preons should therefore form a righthanded tetrad
$\overline{a'b'c'd'}$ with field values corresponding not to $f_R$ but
to the complex conjugate of $f_L$. Similarly,
$\bar f_R=\overline{abcd}$ with field values which are complex
conjugate to $f_R$.

As a side remark note that
there might be something in the tetrad's centre,
but this is not modified by $S_4$ nor parity transformations
(cf. footnote 1 below).

Let us now explicitly relate the elements of $S_4$ to the various
members of the 3 generations. The geometrical model fig. \ref{fig5}
naturally suggests a separation of the 24 permutations into 3 subsets.
To see this, look at the figures \ref{fig8}, \ref{fig9} and \ref{fig10},
where the 3 possible closed paths which connects the points 1,2,3 and 4
are shown.
These closed paths consist of 4 links.
Permutations lying on the path I (fig. \ref{fig8}) will be attributed
to the members of the first family, permutations on path II (fig. \ref{fig9})
to the second family
and path III (fig. \ref{fig10}) to the third family.
For example, look at the lefthanded states of the first family
\begin{eqnarray} \nonumber
\nu_L &=& \overline{1234} \qquad u^1_L= \overline{2341}
\qquad u^2_L= \overline{3412} \qquad
u^3_L= \overline{4123} \\
e_L &=& \overline{4321} \qquad d^1_L= \overline{1432}
\qquad d^2_L= \overline{2143} \qquad
d^3_L= \overline{3214}
\label{eq8}
\end{eqnarray}
More precisely, these fermion states correspond to the various (open) paths
consisting of 3 links which one can lay on the closed path fig.
\ref{fig8}. A typical example of an open path
(representing $u^1_L$) is shown in fig. \ref{fig55}.

Using the assignments eq. (\ref{eq8})
one sees that a weak isospin transformation
corresponds to reversing a permutation, i.e. reversing all 3 arrows
in a figure like fig. \ref{fig55}. This way weak isospin is not
any more a quantum number carried by a fundamental constituent
but is determined by the binding of the state.
Analogously, constructions for color and elctric charge
can be envisaged.

For completeness let us write down the $S_4$--assignment of the second
and third generation. They correspond to the various paths consisting of
3 links which one can draw into the closed loops depicted into
figs. \ref{fig9} and \ref{fig10}.
\begin{eqnarray} \nonumber
\nu_{\mu L} &=& \overline{2134}
\qquad c^1_L= \overline{1342} \qquad c^2_L= \overline{3421} \qquad
c^3_L= \overline{4213} \\
\mu_L &=& \overline{4312} \qquad s^1_L= \overline{2431}
\qquad s^2_L= \overline{1243} \qquad
s^3_L= \overline{3124}
\label{eq9}
\end{eqnarray}
\begin{eqnarray} \nonumber
\nu_{\tau L} &=& \overline{4231} \qquad
t^1_L= \overline{2314} \qquad t^2_L= \overline{3142} \qquad
t^3_L= \overline{1423} \\
\tau_L &=& \overline{1324} \qquad b^1_L= \overline{4132}
\qquad b^2_L= \overline{2413} \qquad
b^3_L= \overline{3241}
\label{eq10}
\end{eqnarray}
It should be noted that there is an intimate connection
between the 3 closed paths figs. \ref{fig8}, \ref{fig9} and \ref{fig10}
and the 3 planes in fig. \ref{fig4}, i.e. the dimensionality of
space. Fig. \ref{fig8} corresponds to the first family plane,
Fig. \ref{fig9} to the second family plane and
Fig. \ref{fig10} to the third family plane.
This can be seen easily by drawing the octahedron with corners
given by the middle points of the cube's face diagonals.

Vector Bosons

Now that we have constructed all states of the fermion generations
the most important question is how to understand their interactions.
As is well known the interactions of fermions proceed through
left-- and right--handed currents with the vector bosons, more
precisely the lefthanded currents $\bar{F}_L \times f_L$ : $=
\bar{F}_L \gamma_{\mu} f_L$ interact with the weak bosons
and the sum of left- and righthanded currents interact with photons
and gluons. The strength of the photonic and gluonic interaction
is given by the electric and color charge, respectively.

The picture to be developed is that of
the vector bosons as
a sort of fermion--antifermion bound state. However, it will
be constructed in such a way
that the vector bosons do not
'remember' the flavor of the fermion--antifermion pair
from which they were
originally formed.
The way to obtain antiparticles (and Dirac fermions) is as follows:
I have already shown that left-- and right--handed fields are
interpreted as permutations of corners 1,2,3,4 and 1',2',3',4'
in the cube (cf. fig. \ref{fig11}), for example
$e_L=\overline{4321}$ and $e_R=\overline{4'3'2'1'}$.
The antiparticle of a lefthanded fermion is a righthanded object
and its preons should form a righthanded tetrad.
Therefore,
the $\bar e_L$ is defined to live on the righthanded tetrad
($\overline{4'3'2'1'}$) but with field values corresponding
to the complex conjugate of $e_L$. Similarly
$\bar e_R=\overline{4321}$, with field values which are complex
conjugate to $e_R$.
Antifield configurations are denoted by open circles in
fig. \ref{fig11}.
\footnote{
I do not know whether the preons at the corners are
real or complex, or whether one should prefer the bindings
between them as the more fundamental objects.
There are various disadvantages as to the existence of antipreons,
both on
the conceptual and on the explanation side. On the conceptual
side the main disadvantage is that the preons themselves
become more complicated than
just real scalar pointlike particles without
any further property than their simple
superstrong interaction with
neighbouring preons.
On the explanation side I have found
it difficult to accomodate parity violation -- everything
is so unpleasantly P--symmetric in the pictures so far.
An alternative one may follow is to do without antipreons,
and to put the information of
a quark or lepton being particle or antiparticle into the centre of
the cube. More precisely, assume there is some nucleus $M_L$ at the centre
of the lefthanded tetrads $ e_L, \nu_L, d^1_L, \dots$
and $\overline M_L$ at the centre of the righthanded tetrads
$ \bar e_L, \bar \nu_L,\bar d^1_L, \dots$.
Note that we do not assume the existence of a nucleus $M_R$
for the righthanded states $ e_R, \nu_R, d^1_R, \dots$ nor for their
(lefthanded) antiparticles.
We might assume its existence but
for the sake of parity violation we must demand that $M_L$ and
$M_R$ behave differently. In the following I shall
assume for simplicity no $M_R$ at all.
When a lefthanded current $\bar f_L \times f_L$ is formed,
the $M_L$ and $\overline M_L$in the centre
of the corresponding cube either annihilate or
encircle each other. If they annihilate each other, a state
is formed which cannot be distinguished from the corresponding
right handed current $\bar f_R \times f_R$. That corresponds
to the formation of a photon or a gluon.
If they keep encircling each
other, a Z or a W is formed, decaying very quickly after their short
lifetime back to a fermion antifermion pair.
The probability by which all these processes happen, is dictated
by the various charges. A neutrino--antineutrino pair
cannot form a photon nor a gluon, because it does not have an electric
nor a color charge.}
%I can think of two
%ways to implement antiparticles in the present picture.
%One possibility is to assume the existence of antipreons (white circles
%in fig. \ref{fig11})
%in addition to the ordinary preons (black circles), so that
%the corresponding antiparticle states can be formed.
%The antiparticle of a lefthanded fermion is a righthanded object
%and its preons should therefore form a righthanded tetrad.
More precisely,
the combinations of a lefthanded fermion of the first family
and their righthanded
antiparticles are shown in fig. \ref{fig11}. This way
all the corners of the cube are filled.
% by this combination.
Fig. \ref{fig11} more or less represents how vector bosons
should be imagined in this model.

Fig. \ref{fig11} is a rather characteristic picture of a fermion--antifermion
bound state. The point is that the vector boson interactions
always take place within
one family, and fig. \ref{fig11} corresponds to interactions
within the first family.
One sees that the bindings between
the links join together to form bindings along the plaquettes.
Altogether, the bindings form an oriented closed circle of
plaquettes.
In the case of the second family interactions there is also a closed
circle and it lies in the second family plane (cf. fig. \ref{fig4})
and similarly for interactions between members of the third
family.
The three planes can be rotated into each other to make the
corresponding vector bosons identical. The difference to fermions
will be understood
better in a group representation approach to be discussed below.

In my model, vector bosons are superpositions
of fermion--antifermion states $\bar F \times f$
with the appropriate quantum numbers.
The $\bar F \times f$ binding
arises from interactions along
the four body diagonals of the cube defined by fermions (1234) and
antifermions (1'2'3'4'), i.e. interactions between the full and
open circles in fig. \ref{fig11}. I shall come back to
the body diagonals later.

For finite groups the number of irreducible representations (IR's)
is equal to the number of conjugacy classes. In the present case
the IR's are usually called $A_1$, $A_2$, $E$, $T_1$ and $T_2$
with dimensions 1, 1, 2, 3 and 3, respectively, and their
characters are shown in table 1. $A_1$ is the identity
representation. $A_2$ differs from $A_1$ by having a negative
value for odd permutations.
$T_1$ is the representation induced by
the permutations of the corner points 1,2,3,4 of a tetrad
in three dimensions. Its representation space is therefore
the three dimensional space, in which the fermions live,
i.e. {\it $T_1$ konstituiert den Anschauungsraum}.
$T_2$ is obtained from $T_1$ by changing the sign of the representation
matrices for the odd permutations. Finally, $E$ is induced by
a representation of $S_3$ on the corners of a triangle,
as discussed in connection with fig. \ref{figt}, for example
$E(\overline{2134})\bf{a}=\bf{b}$, $E(\overline{2134})\bf{b}=\bf{a}$,
$E(\overline{2134})\bf{c}=\bf{c}=-\bf{a}-\bf{b}$,
$E(\overline{1243})\bf{a}=\bf{b}$, $E(\overline{1243})\bf{b}=\bf{a}$,
$E(\overline{1243})\bf{c}=\bf{c}=-\bf{a}-\bf{b}$, etc.

In order to obtain the vector bosons $\bar F \times f$, one should
take the 9--dimensional product representation
\begin{equation}
T_1 \times T_1 =A_1+E+T_1+T_2
\label{eqzut}
\end{equation}
On the right hand side,
the term $T_1$ corresponds to arbitrary rotations of the closed loops
of plaquettes,
as claimed in connection with fig. \ref{fig11}.
%The interpretation of eq. (\ref{eqzut}) is as follows:
$A_1$ corresponds to the photon, the totally symmetric singlet
configuration, where all tetrad--antitetrad combinations
contribute in the same way.
$T_2$ is induced by 24 permutations
of some objects I,II,III,IV (much like the $T_1$ on the left hand side of
equation (\ref{eqzut}) was induced by
the 24 permutations of 1,2,3,4).
Finally,
$E$ is induced by the 6 permutations on the triangle (fig. \ref{figt}).

A possible interpretation of $T_2$ is as follows
\footnote
{Alternatively, $T_2$ could represent 24 gluons
which would then differ for the 3 families. The six
permutations of the triangle might be
the weak bosons $W^{1,2,3}$ and $W^{1,2,3}_{R}$.}
: By definition,
the different vector bosons correspond
to permutations of the cube's four body diagonals called I, II , III and IV,
which define another group $S_4$.
It is ordered not as in the case of fermions, equations
(\ref{eq8})--(\ref{eq10}), but according to its conjugacy classes.
In fact,
the 24 elements of $S_4$ can be
ordered in 5 conjugacy classes with 1, 3, 8, 6 and 6 elements.
They are given as follows:
\begin{itemize}
\item identity \\
$\overline{I,II,III,IV}$ \\
the U(1) gauge boson
\item 3 $C_2$ rotations by $\pi$ about the coordinates axes x, y and z \\
$\overline{II,I,IV,III}$, \qquad $\overline{III,IV,I,II}$, \qquad
$\overline{IV,III,II,I}$ \\
the SU(2) gauge bosons
\item 8 $C_3$ rotations by $\pm {2 \over 3}\pi$ about the cube diagonals
(like x=y=z) \\
$\overline{II,III,I,IV}$, \qquad $\overline{III,I,II,IV}$, \qquad
$\overline{II,IV,III,I}$, \qquad
$\overline{IV,I,III,II}$, \\
$\overline{III,II,IV,I}$, \qquad
$\overline{IV,II,I,III}$, \qquad
$\overline{I,III,IV,II}$, \qquad $\overline{I,IV,II,III}$ \\
the gluons
\item 6 $C_4$ rotations by $\pm {\pi \over 2}$ about the coordinate axes
\\
$\overline{II,I,III,IV}$, \qquad $\overline{III,II,I,IV}$,
\qquad $\overline{IV,II,III,I}$, \\
$\overline{I,III,II,IV}$, \qquad $\overline{I,IV,III,II}$,
\qquad $\overline{I,II,IV,III}$ \\
leptoquarks
\item 6 $C_2 '$ rotations by $\pi$ about axes parallel to the face
diagonals (like x=y, z=0) \\
$\overline{II,III,IV,I}$, \qquad $\overline{II,IV,I,III}$,
\qquad $\overline{III,IV,II,I}$, \\
$\overline{III,I,IV,II}$, \qquad $\overline{IV,III,I,II}$,
\qquad $\overline{IV,I,II,III}$ \\
leptoquarks
\end{itemize}
where reference is made to the cartesian coordinates x,y and z with
origin at the cube's centre.
This ordering is reminiscent of the ordering of gauge bosons in
grand unified theories where there are leptoquarks in addition
to the 8 gluons and the four electroweak gauge fields.
The elements of the first two classes form Klein's 4--group
(an abelean subgroup of $S_4$),
whereas the elements of the first three classes form the
nonabelean group of even permutations.

In summary, the interpretation of eq. (\ref{eqzut}) is as follows:
As discussed before, the fermions constitute ordinary three--dimensional
space. As soon as two fermions approach each other to form
a vector boson, space opens up to 9 dimensions. Three of them
correspond to ordinary space, whereas the remaining six decompose
into 1+2+3 dimensional representation spaces $A_1$, $E$ and $T_2$ of $S_4$.
They become fibers to ordinary space. It remains to be shown how the complex
structure of a $U(1)\times SU(2)\times SU(3)$ Lie algebra
arises.

Since parity violation is not present in these pictures,
I want to add an alternative related to the observation
that there are two 1--dimensional and two 3--dimensional,
but only one 2--dimensional IR of $S_4$.
One could relate parity transformations to even--odd
transitions between permutations by modifying the assignments
made in equations
(\ref{eq8})--(\ref{eq10}), namely
\begin{eqnarray} \nonumber
\nu_L &=& \overline{1234} \qquad u^1_R= \overline{2341}
\qquad u^2_L= \overline{3412} \qquad
u^3_R= \overline{4123} \\
e_L &=& \overline{4321} \qquad d^1_R= \overline{1432}
\qquad d^2_L= \overline{2143} \qquad
d^3_R= \overline{3214}
\label{eq8s}
\end{eqnarray}
i.e. assigning odd permutations to righthanded states.
According to the character table 1 the character of $E$ vanishes
for odd permutations. Therefore, there is no action of $E$ on
righthanded fermions. In contrast, the products $A_1\times A_2$
and $T_1\times T_2$ act like $-\gamma_5$ on left and
righthanded fermions.
{\bf Conclusions}

According to present ideas the elementary particles (leptons,
quarks and vector bosons) are pointlike and their mathematical
description follows this philosophy (Dirac theory, Yang--Mills theory).
They certainly receive an effective extension by means of quantum
effects, but these are fluctuations and do not affect the primary
idea of pointlike objects.

In contrast,
in my model the observed fermions naturally have an
extension right from the beginning.
This seems to be difficult to accomodate because their radius should
be of the order of their inverse masses. Following t'Hooft
one may assume that there is a symmetry principle which leaves
the masses small.

In its present stage, the model does not
allow to make quantitative predictions of fermion masses,
although
some qualitative statements about fermion masses have been
made in the previous sections.
As compared to the binding energy, all fermion masses
(including $m_t$)
are tiny perturbations which might be induced by some radiative mechanism
of the 'effective'
standard model interactions
leading to masses $\sim \alpha^F$ for the F--th family.
The 'textures' of those masses have been discussed by Ramond.
Within the present model one may argue that
the fermions are basically massless by some symmetry and
that there are small symmetry breaking effects within the
family planes
leading to different family masses.

Whatever this symmetry principle may be, there is still the question how
large the radius R of the quarks and leptons is. In principle, there
are three possibilities, it may be large ($\sim 1$ TeV$^{-1}$), small
($\sim M_{Planck}^{-1}$) or somewhere in between. In the first case
there will be experimental signals for compositeness very soon.
In the second case there will never be direct experimental
indications and it will be difficult to verify the preon idea.
Furthermore, in that case one would have the GUT theories as correct
effective theories whose particle content would have to be
explained. In addition, it may be necessary to modify the theory
of relativity. In fact, the superstring models are a realization
of this idea, the 'preons' being strings instead of point particles.

Personally, I like the scenario $R \sim M_{Planck}^{-1}$ reasonably well.
In the model presented in this paper, the preons are pointlike and
sitting on a cubic lattice. This lattice would have to fluctuate in
some sense to reconstitute Lorentz invariance. This certainly
raises many questions which go beyond the scope of this article.
For example, the renormalization of gravity would be modified
because high energies ($> M_{Planck}$) would be cut away by the
lattice spacing.

As for the third possibility 1 TeV $<< R << M_{Planck}$,
gravity and its problems
play no role and my models are just a more or less consistent
picture of particle physics phenomena. Since no attempt was
made to explicitly construct the states
of quarks and leptons in their known complex
representations they are at best a qualitative guideline for
understanding.
I did not write a Lagrangian for the preons and just speculated about
their interactions. The ultimate aim would be to construct a Lagrangian
and derive from it an effective interaction between Dirac fermions
and gauge fields.