View source for Monograph:Experiments with vectors



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[[File:Handwiki_hand.svg|80px|HandWiki datascience encyclopedia]]
<font size="+2">Experiments with vectors</font> <br> 
<font size="+1">Wolfhard Hövel (Wolfhard.Hoevel@th-nuernberg.de)</font> 
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A digital computer is a wonderful tool for performing mathematical experiments. I would like to
introduce an algorithm that spontaneously generates particles that move and interact in Euclidean
space. This only happens with the help of position and displacement vectors.
First of all I have to define some rules according to which the algorithm should run. These rules are
not entirely random and when experimenting quickly show whether the rules produce something
interesting. I believe that the proposed algorithm actually gives amazing results.
The program describes a discrete dynamical system. In a loop calculation, the states of the system
change in equidistant time steps. With each iteration, the states are always recalculated according to
specified procedures. A general rule is introduced that a quantity <math>G</math> is made up of the integer
multiples of a constant base quantity <math>G_0</math> (<math>G = n \cdot G_0</math>, n is a natural number). All quantities are finite.

Now I can define an important quantity in this way: time. Each cycle in the calculation loop specifies the shortest time step <math>dt_0</math>. The time is then granular and corresponds to the number n of
iterations <math>t = n \cdot dt_0</math>. You also need objects with which something happens. Let's start with a point.
The point has a certain position in space, which is determined by the position vector <math>\mathbf{r}</math> (vectors are
printed in bold). This position vector points from a freely chosen zero vector to the specified point.
The point is by definition always in motion, I achieve this with a displacement vector <math>\mathbf{dr}</math>, which is
added to the position of the point with every iteration in the calculation loop. Since <math>\mathbf{dr}</math> has by
definition a constant length, the step width of the point is constant and is equal to the magnitude of
the displacement vector <math>|\mathbf{dr}| = dr_0</math>. The point now moves straight ahead and has the constant speed
<math>\mathbf{v} = \mathbf{dr} / dt_0</math>. According to Newton's laws, this is a physical statement.

After a while the point disappears from the screen and that would be a boring event. I have to
constrain the movement of the point to a defined area. I achieve this by providing the point with a
second point as a partner. These two points are supposed to dance around each other. For this, the
dancers need certain rules. The first rule is: move with the velocities <math>\mathbf{v}</math> and <math>\mathbf{-v}</math> respectively exactly
in opposite direction on two parallel lines with the distance <math>s</math>. The second rule is: If your distance
becomes longer than the elongation <math>e</math>, turn around and run on two new parallel lines with the
distance <math>s</math>. This distance s and the direction of rotation remaining always the same. A third rule
should take into account the principle that all quantities can only change step by step, including the
parameters <math>s</math> and <math>e</math>. They are integer multiples of the step width <math>dr_0</math>. The two points form a system
whose center of mass remains constant. The angular momentum of the points is constant and given
by <math>v</math> and <math>s</math>. In addition, a maximum distance is set, which the dancing points should not exceed if
possible. They don't quite succeed because they move step by step. The limit is not sharp, it can be
exceeded by a maximum of <math>dr_0</math>.


There is still something missing to make the algorithm physical: it is the mass. So I assign a
smallest possible mass <math>m_0</math> to each point. This assumption must not appear artificial or forced. A
common center of mass can also be specified for a point cloud. Now the constant momentum for a
point is given by <math>\mathbf{p} = m_0 \cdot \mathbf{v}</math> and the kinetic energy by <math>E = ½ m_0 v²</math>.
The described pair of points ("bivector") is the most important basic building block of the
calculation model. It is the smallest possible particle of the simulation. From a geometric point of
view, a parallelogram is spanned in space by the distance vector <math>\mathbf{r_{12} = r_2 - r_1}</math> between the two points
and the displacement vector <math>\mathbf{dr}</math>, whose area is equal to <math>s · dr_0</math>. The parallelogram is an oriented
plane and has a sense of rotation. The bivector exists in every dimension > 1. The bouncing points
describe a circular disc, the edge of which is not very sharp. For example, if <math>e >> s</math> the points are
oscillating in an almost star shape around the circle with the diameter <math>s</math> and for the special case
<math>e = s</math> the points are moving unsharp on a circle with the diameter <math>e</math>. The bivector contains two
points and therefore its mass is <math>2\cdot m_0</math>.

How can the bivector be represented in a program? At the beginning of each iteration, it is checked
whether the distance between the points exeeds <math>e</math>. If this is not the case, the points can continue to
move freely in a straight line. If the permitted limit is exceeded, the points must be reflected so that
their distance is reduced again. The rules given above must be observed. From a geometric point of
view, this is easy to illustrate. The distance between the points is halved and a circle with the
diameter <math>s</math> is drawn around the center found, which is also the center of mass of the bivector. Then,
starting from the two points, tangents are placed on the circle, whereby the sense of rotation must be
the same as before. The points continue to move on these newly found tangents. One can imagine
that the two points are connected by a chain with the length <math>e</math>. As long as the chain is still loose, the
points can move freely. However, if the chain is suddenly tensioned after a step, the points must
reverse abruptly. The chain can be spanned by a maximum of one chain link. This methode of
reflection has the advantage that only internal quantities of the bivector are used and no impulse is
exchanged to the outside. The bivectors can be moved freely in the space. This fact will become
important later.

Now we add more points. Each point must be at least a member of one bivector to give it a
momentum. However, one point can be a partner of several bivectors and then the situation quickly
becomes confusing. Therefore, it makes sense to represent the relationship between the points using
a graph of graph theory. The points are the vertices and the bivectors are the edges of the graph. If
the point cloud consists of <math>N</math> points, a maximum of <math>N(N-1)/2</math> bivectors can be inserted between the
points. If a point is involved in several bivectors, there is a strong coupling between these linked
bivectors.
This should be illustrated using a simple example. Suppose we connect three points <math>\mathbf{P_1}</math>,<math>\mathbf{P_2}</math> and <math>\mathbf{P_3}</math>
with two bivectors with the pairing (1-2) and (2-3). With each iteration, each bivector is treated as
before, regardless of the others. The associated displacement vectors <math>\mathbf{dr_{12}}</math> and <math>\mathbf{dr_{23}}</math> result from this.
Finally, the displacements are superposed. Point <math>\mathbf{P_1}</math> shifts by <math>\mathbf{dr_{12}}</math>, point <math>\mathbf{P_2}</math> by <math>\mathbf{(-dr_{12}+dr_{23})}</math> and
point <math>\mathbf{P_3}</math> by <math>\mathbf{-dr_{23}}</math>. Imagine that dancer <math>\mathbf{P_2}</math> has to perform the steps of two dance pairs at the same
time! The two dance couples will have to dance completely new figures, they are closely linked.
This example describes another possible particle. Since it contains three points, its mass is <math>3\cdot m_0</math>. If
you look at the energy situation, it gets a little more complicated. Two displacement vectors <math>\mathbf{-dr_{12}}</math>
and <math>\mathbf{dr_{23}}</math> are added to point <math>\mathbf{P_2}</math>. In the case that both vectors have the same direction, the point <math>\mathbf{P_2}</math> has
the speed <math>2\cdot v</math>. Its kinetic energy is therefore <math>2 m_0 v²</math>. On the other hand, if the two displacement
vectors are exactly opposite, the kinetic energy is zero. So that the theorem of energy conservation
remains valid for the simulation model, the energy of point <math>\mathbf{P_2}</math> could now be interpreted as potential
energy. As the directions of the displacement vectors will change, the potential energy is converted
back into kinetic energy. Nothing is lost, the number of displacement vectors is constant. This
interplay is also known from a tensioned and relaxed metal spring.

If you experiment with the calculation model described, you will experience a surprise. Under
certain conditions, self-organization creates stable structures in the space. These attractors can
change spontaneously, they can attract and repel, they behave like particles with mass and obey the
momentum conservation law.
To do this, however, the algorithm still needs to be expanded. A weak interaction must be added.
Let's start again with a point cloud. The points are linked by bivectors. When points are members of
several bivectors, there are strongly coupled sets of points, these are our particles. Several isolated
graphs and thus particles can be defined. The particles do not yet interact, their graphs are isolated
networks in space. How can I couple the isolated particles? To do this, I assign a plus or minus sign
to some points. The additional rule is: if the points have opposite signs, the points move iteratively
radially towards each other with the step size <math>dr_L</math>, if they are like-signed, they repel each other.
These shifts are added to the other displacements. The momentum conservation law is still
observed, the sum of the momentum vectors remains equal to zero. With <math>dr_L</math> you can control the
weak coupling between the particles, often I set <math>dr_L = dr_0 / 150</math>. The algorithm can be programmed
on the basis of real or integer numbers.

External links:

https://opus4.kobv.de/opus4-ohm/frontdoor/index/index/docId/253
    
https://opus4.kobv.de/opus4-ohm/frontdoor/index/index/docId/126
    
https://opus4.kobv.de/opus4-ohm/frontdoor/index/index/docId/34
    
https://drive.google.com/file/d/1u3lOz7ydiSStSIm_FsuUGXNnbtSUUWay/view?usp=sharing
   
https://drive.google.com/file/d/1JjF1Z4kmJaEV5BU2P9j0m9IP-BtaosOo/view?usp=sharing
   
https://drive.google.com/open?id=1I82uduHWlwbVNCdFygdsM_B6TqtEwKSM 
   
https://opus4.kobv.de/opus4-ohm/frontdoor/index/index/docId/481
  
https://drive.google.com/file/d/1pSUXVzf1uNchKwMq1tk1P-_TclCDjSs8/view?usp=sharing

https://opus4.kobv.de/opus4-ohm/files/253/Particle.jar 
  


'''Attachment'''

You can make a bivector geometrically clear with a drawing. Draw two points <math>P_1</math> and <math>P_2</math> 16 cm
apart. Connect the two points with a straight line and halve them. The point M found in this way is
the center of mass of the pair of points. Draw two circles around M with the radius s/2 = 2 cm and
e/2 = 4 cm. Place two parallel tangent lines starting from points <math>P_1</math> and <math>P_2</math> to the s-circle. Starting
from point <math>P_1</math>, move <math>P_1</math> step by step with the step width <math>dr_0</math> = 2 cm along the tangent line. The point
now jumps to the e-circle and runs through it. As soon as <math>P_1</math> has left the e-circle, a new tangent line
from <math>P_1</math> is placed to the s-circle, keeping the direction of rotation, and the game is repeated. The
point <Math>P_1</math> is now captured and circles the s-circle. Point <math>P_2</math> is always moved in the opposite direction
at the same time. This system is a bivector, it is characterized by a constant angular momentum (~ s)
and an elongation e. The bivector can be moved freely in space, its position is only determined by
the two points <math>P_1</math> and <math>P_2</math>.