Fractal Home
Ideas & experiments
on self-organization of vectors and the EightPoints Machine.
At this location, you'll find changing texts with background information on Hövel's system.
Experiments with complete data sets are found within the dotted frames. You can copy the whole set and
paste it directly into the command line Cmd of the EightPoints-Maschine on the previous page, followed by
the enter key. The attractor is animated with the command run or clicking the play button.
Oscillators
There are news on this subject. Visitors of the page have discussed about oscillators. The bivectors are oscillators like lossless billiard balls. For example, if two bivectors are coupled by electric charge, this influences the phases and the oscillations may synchronize. Here is a demonstration with the EightPoints-Machine:
Experiment 1
read k01=1; e01=201; s01=101; k02=0; e02=1; s02=1; k03=0; e03=1; s03=1; k04=0; e04=1; s04=1; k05=0; e05=1; s05=1; k06=0; e06=1; s06=1; k07=0; e07=1; s07=1; k12=0; e12=128; s12=64; k13=0; e13=1; s13=1; k14=0; e14=1; s14=1; k15=0; e15=1; s15=1; k16=0; e16=1; s16=1; k17=0; e17=1; s17=1; k23=1; e23=201; s23=101; k24=0; e24=1; s24=1; k25=0; e25=1; s25=1; k26=0; e26=1; s26=1; k27=0; e27=1; s27=1; k34=0; e34=128; s34=64; k35=0; e35=1; s35=1; k36=0; e36=1; s36=1; k37=0; e37=1; s37=1; k45=0; e45=141; s45=100; k46=0; e46=1; s46=1; k47=0; e47=1; s47=1; k56=0; e56=128; s56=64; k57=0; e57=1; s57=1; k67=0; e67=141; s67=100; q[0]=1; q[1]=0; q[2]=-1; q[3]=0; q[4]=0; q[5]=0; q[6]=0; q[7]=0; zoom=0.008; scale=1; pull=0.0073; dim=3; tStop=2000; vor=1; a=10;
The setting e01=202 (simply written into the command line Cmd of the machine) stops
the synchronization of the bivectors, but the phases are "pulled along" sometimes.
This could be measured numerical by counting the number of iterations during two reflections.
Coupled Oscillators are an important topic, for example described by
the Kuramoto model.
There are comments that the bivector model has a similarity to
the Vicsek model.
The bivector points are active and show a kind of swarm behavior. But there is an important difference: In the Vicsek model
the center of gravity isn't stationary, in opposite to the bivector model.
The bivector model has another characteristic: It represents a multibody system. The bivectors interact and
fulfill the law of conservation of energy, mass, momentum and electric charge. Actually the bivector model algorithm works
with a system of coupled differential equations. By discretization the equation system is transferred into discrete counterparts.
The change of the position vectors (dx, dy, dz, ...) has been replaced by the applied unit vector at each step.
The change of time dt is simply an iteration step.
Experiment 2
The second example demonstrates a complete graph. Clicking New you'll see many different attractors. Some are periodically, some are nearly chaotic.
read k01=1; e01=6; s01=4; k02=1; e02=6; s02=4; k03=1; e03=6; s03=4; k04=1; e04=6; s04=4; k05=1; e05=6; s05=4; k06=1; e06=6; s06=4; k07=1; e07=6; s07=4; k12=1; e12=6; s12=4; k13=1; e13=6; s13=4; k14=1; e14=6; s14=4; k15=1; e15=6; s15=4; k16=1; e16=6; s16=4; k17=1; e17=6; s17=4; k23=1; e23=6; s23=4; k24=1; e24=6; s24=4; k25=1; e25=6; s25=4; k26=1; e26=6; s26=4; k27=1; e27=6; s27=4; k34=1; e34=6; s34=4; k35=1; e35=6; s35=4; k36=1; e36=6; s36=4; k37=1; e37=6; s37=4; k45=1; e45=6; s45=4; k46=1; e46=6; s46=4; k47=1; e47=6; s47=4; k56=1; e56=6; s56=4; k57=1; e57=6; s57=4; k67=1; e67=6; s67=4; q[0]=1; q[1]=1; q[2]=1; q[3]=1; q[4]=1; q[5]=1; q[6]=1; q[7]=1; zoom=0.1; scale=1; pull=0.01; dim=4; tStop=50000; vor=1; a=1;