Vibrations: communication through a morphic field Part 2: Simulations Results and Video by Ralph H. Abraham, abraham@vismath.org and Peter Broadwell, peter@meer.net Visual Math Institute, http://www.vismath.org POB 7920, Santa Cruz, CA, USA-95061-7920 Abstract. This is a further progress report on the computer simulation of a mathematical model for a morphic field. The model is a two-dimensional lattice of oscillators derived from the d'Alembertian wave equation by spatial discretization. The communication is between two clamped objects inserted into the field. A change of shape in one of them sets off a transient wave which perturbs the boundary field of the other one after a brief delay. Unlike radio propagation, this is a static monopole transmission. In this second simulation, we clamp the field at the edges of a rectangular region. This note is an explanation of the companion video, which is a record of the experiment. CONTENTS 1. Introduction 2. The model 3. Visual representation 4. The actors 5. The play 6. The retina 7. The written record 8. The experiments 9. Conclusion Acknowledgments Bibliography 1. Introduction. This note is a supplement to our MS#86 of the same title, and is intended to explain the companion video. Its most obvious features are: · the model, the mathematical object which rules the simulation, · the field, a wildly vibrating color pattern in a square frame, · actor H, a black square in the field, · actor D, a black triangle in the field, · the play, in which the actors play their roles, · the retina, a broad vertical line alongside actor D, · the retinal recording, a vertical column with a web-like pattern, · the color bar, which indicates the mapping from numerical values to colors, and · the time (iteration) counter, in the lower right corner of the screen. We now describe these features, and the sequence of the companion video. 2. The model. Our model for the morphic field is derived from an electromagnetic field in two dimensions. The field may be conceptualized as the vertical displacement of a horizontal drumhead on a square drum. However, there is no damping. Thus, once the membrane is struck, it vibrates endlessly. The displacement is restricted to a range of minus 1 to plus 1 unit, relative to the equilibrium (flat, horizontal) configuration, in our computer simulations. The bounding square is clamped at value 0. Further, the membrane is discretised as a square array of harmonic oscillators connected by springs, in the classical style of the Bernoullis and d'Alembert. We used an array of size 88 by 88. The coupling is Laplacean, as always in the discretization of the wave equations of mathematical physics. 3. Visual representation. The displacement is represented on the computer screen as a pattern of colors superimposed upon the square membrane. The range -5 to +5 units is represented by spectral hues from blue to red. With each discrete increment of time, typically 0.01 units, the displacement (and thus the color) at each point of the array is updated, creating a new pattern of color on the square array. These successive patterns are then recorded to videotape at the rate of 10 frames per second. 4. The actors. We use this model to simulate the effect upon the field (membrane) due to two actors. One, D, will be the receiver of a communication transmitted by the other, H. In imitation of the senses of sight, hearing, and smell, we imagine that H sends the message by a motion or change of shape, like ringing a bell. An oscillatory transient then propagates through the scalar field, a vibration like a sound wave. As the transient arrives at the boundary of the receiver, D, a perception occurs. In this simulation, the actors are represented by grounded (that is, immovable) objects: D is a triangle, while H by a square. These figures hold down the membrane, that is, they clamp the filed or displacement at the zero value. 5. The play. In these experiments, the actors are directed to play out a scenario of communication as follows. First, the stage (the square array or membrane) is empty. Then D takes his place on the stage, creating an excited state of the membrane. We wait a few minutes. This is a state of active silence or background noise. Then, H is appears upon the stage. She affects the active silence as by a heavy weight upon a drum head. The entire vibrating field is effected. The effect near the receiver, D, is subtle, but perceptible. 6. The retina. The receiver (D, triangle) is supposed to be sensitive to motion of the field nearby. We model this supposition by inserting a retina into the field near the receiver. This may be interpreted more as an organ of Corti, which senses the vibration of the basilar membrane in the perception of sound, or a sort of microphone. However, as we are representing the field visually as a pattern of spectral colors, we are going to think of the organ of perception as a (one-dimensional) retina. It sees, instant by instant, the row of colors written upon it by the membrane motion. 7. The written record. It is difficult to remember the moving patterns of color near the actors, so we create a written record of the retinal images by stacking up the rows of color into a two-dimensional pattern, a memory engram as it were. This is shown alongside the square array in the video. The rows of color are written from the bottom up. The upward gliding black line is the recording head. When it reaches the top of the screen, it reappears at the bottom, and writes over the previous records. These space-time patterns comprise our experimental results. 8. The experiments. Two plays of three acts each are recorded on the video, which lasts about 20 minutes altogether. The two plays differ primarily in the stiffness of the membrane (a parameter in the model). The frequency of vibration is higher in the second simulation, and the wavelengths correspondingly shorter. First play, act 1. At iteration 0, D appears alone on the stage, and sits still for five minutes (about 3574 iterations). This soon settles down to a steady vibration. Note the left-right symmetry in the retinal recording as it gradually develops. Act 2. Then H is placed on the stage, and we watch for another five minutes. Notice a slowly increasing left-right asymmetry in the retinal recording, and a tightening of the web. The weight of H on the drumhead skews the vibrating field, and increases the frequency. Act 3. Around iteration 7200, H is removed from the field. The retinal recording relaxes to its equilibrium state in Act 1, as the shadow of the square fades from the field, by the end of play, iteration 14,000. In the second play, we see the same scenario in three acts, with all the action accelerated, and completed in about 5000 iterations. 9. Conclusion. In these experiments, we clearly show that the space-time record of perception by actor D of the vibrating field is significantly changed by the sudden arrival of actor H on the stage . There is skewing and tightening of the web of excitation in the retinal recording of D's perception when H is in the field. Acknowledgments. We are grateful to Rupert Sheldrake for his support of this work, and deeply in debt to the Mathematically Illuminated Musical Instrument Project for use of its software, MIMI. Bibliography Abraham, Ralph H., Vibrations: communication through a morphic field, preprint, 1996. Abraham, Ralph, and Peter Broadwell, Morphic Resonance #1(10 minute video), Santa Cruz. CA: Visual Math Institute, 1996. Abraham, Ralph, and Peter Broadwell, Morphic Resonance #2(20 minute video), Santa Cruz. CA: Visual Math Institute, 1997. Sheldrake, Rupert, Seven experiments That Could Change the World: a do-it-yourself guide to revolutionary science, London: Fourth Estate, 1994; New York: Riverhead Books, 1995.