Shape |
David Beale © 1995 -- 2014 |
The pictures on this page show a way to form silhouettes by plotting a time-domain waveform with a fixed value rather than with the usual method of matching the sine/whatever waveform with a cosine wave.
The plots show circular graphs of the amount of something happening, plotted relative to time in a circle rather than from left-to-right.
The pictures here don't cross a sinusoidal waveform with a cosine-wave — the pictures here just use the time-domain samples as radii of a circle — so you'll see the sine-wave, here, doesn't translate into a circle as it would have been if graphed relative to a cosine wave. The sinewave has been graphed relative to a straight line, a constant level. These pictures show outlines as being describeable as plots of sequential numbers.
Lissajous patterns are usually shown as "time domain" in their form as two waveforms with difference in timing (phase). A special case for Lissajous patterns is the circle, when the two sinusoidal waves have a sine-cosine relationship in phase (their timing is 90° out of phase)
Pictures on this page were made by plotting relative strengths of the "Y" axis with a fixed central point and rotating the "X" instead of it being left-right.
The usual way, to describe a silhouette outline, is to have one axis plotted as a circle (as a cosine wave, if the resulting graph of two waves is to be a circle). These pictures here show shapes from oscillations in an unusual way, by keeping one value constant.
A circle is made from two sinusoidal waves (a sine and a cosine) when both waves are exactly the same frequency (number of cycles per second) and are timed perfectly with one delayed by a quarter of their wavelength, or time for one period. To show a circle on an oscilloscope use XY plotting with two sine-wave sources; and adjust the waves until they are are timed with a difference of 90degrees. If the two waveform sources are synchronized then the circle with be perfect; otherwise it will change into an ellipse. The shapes can be made to wobble by changing the frequency of either waveform. Addint harmonics, to a waveform, cand produce shapes. Any shape can be made using a waveform with harmonics and a cosine wave, or two waveforms with harmonics, plotted on two axes.
When one of the sine-waves (in the usual cosine way of showing interaction between two waves producing outlines) is twice the frequency of the other (for an XY graph/plot/picture) then the sine-wave pair form an Infinity sign or an "8" at some angle.
In these pictures, however, a co-sine wave is not used; instead a fixed-value for the "X" axis (usually horizontal left-right) causes only a change of position of that value, rotated.
An equivalent can be comprised using Fourier mathematics, for the Periodic Table of Elements. Multi-dimensional Fourier transforms could describe everything, at least for one nominal cycle.
Some consider/visualise orbiting 2D analiges as being real descriptions of force-field interactions who have their reality in many more dimensions than the main three. Predictions hit a ceiling of relevance if based only on 2D relationships and sometimes adding time as if time is another dimension.
A Fourier series has to have a fundamental, a nominal lowest "frequency" for the series to describe a series of events or one layer of a 3D shape as a silhouette. Some events are not cyclic; but if they can be considered to be a finite series, then for a Fourier-type description the fundamental can be their largest specification for a co-ordinate or axis. One-off events can be described according to how long they go on for and the complications of their progress as harmonics described as ratios of frequency (relative to the fundamental) and phase (timing --- where they cross zero).
FFT-type concentric-expansion-contraction, between various extra-to-3D-dimensions, in patterns, is an alternative for visualising, by analogy and sometimes tangible fact, energy, matter and movement events. I dislike the word "spin". I doubt very much whether there's many functions that orbit each other, to equalize exclusion-forces between events forming patterns within a universal exclusion force field.
A friend many years ago didn't believe me on the theory of using FFT to describe silhouettes. He actually graphed in the computer the result of crossing a repeated function (need 360degrees i.e. a full set of samples, to make a silhouette) with a cosine wave......and agreed with the results --- a way of storing shapes using FFT. His first success was to describe a triangle as a Fast Fourier Transform calculated list of value for one axis and a cosine wave for the other. Similarly any 2D shape can be described according to a fundamental and some harmonics in one set and a cosine wave for the other axis (or two complex series of numbers one for X and one for Y.
Similarly, slice by slice, any 3D shape can be described in 3D, provided a fourth axis, or factor, is added, to allow for space within a 3D shape. Add a fifth dimension to describe the positions of the layers and add a sixth dimension (time) and the shape can move.
A person running could be described using six sets of Fourier series interacting, plus a sixth system to describe variables...and then one has to allow for the countless other factors that intrude on the predictable series, according to external forces/shapes intruding.
A universe can be described, for any moment, according to its collection of Fourier series
FFT (Fast Fourier Transform) can describe any repeated function, of any complexity. The accuracy is up to whatever details are wanted........like there's not such thing as a truly perfect circle until one gets infinite accuracy (and anyway if you take a small enough sample of a perfect circle or any other curve then the result is a straight line for anything except a circle of infinite size and even then if the sample is infinitely small then the result is a straight line).
FFT is the "Fast" of "Fourier Transform" (after a French mathematician, Fourier). He publicly claimed that any waveform (repeated or one-off) can be described by the circular (sinusoidal) waves comprising it. The various types of "Fourier" series address the practicality/difficulty of just saying "take the fundamental and describe the harmonics relative to that". Real-world figures go from infinitely slow to infinitely fast, depending on the definition being attempted, for anything --- even sound has wind and some winds change direction very slowly, for example. Practical series allow that in Nature a perfectly-square start and stop to a waveform is impossible; so a reasonable limit is found for the number of harmonics that need to be described (infinite for a real infinitely-sharp-edged theoretical waveform such as a square wave). Maths does have the language but electronic machines tend to compromise in the complexity of their descriptions of events --- especially when dealing with infinity and zero --- including one-offs (single complex pulses) by limiting the number of harmonics practically.
Cosine descriptions are used, in Fourier maths, because cosines can include permanent levels above zero. The cosine of DC is 1 at 0° (the beginning of rotation of a circular pattern) whereas sine of DC starts at zero regardless of the level it will attain at 90°.
The figures here show some shapes that happen in nature, like patterns on butterfly wings and some shapes that appeal to people in a way similar to the "Golden Mean" (1.6180339etc:1..... or 0.6180339etc:1). The series goes forever on its exponent, with or without the "1". I worked out an algorithm that can calculate Phi indefinitely. It checks whether 1:.... inverted has the same exponent digit in the same position from the decimal point, and adjusts the digit value, one digit in the exponent at a time, indefinitely.
Figure 3
The golden-mean ratio has use extremely in nature; for the way many natural functions and constructions are sized relative to other parts of themselves and relative to time. The series goes forever when defining the equivalence of 1/0.6180339... compared to .6180339...with the accuracy depending on how many digits of definition everything being the same except the "1" or "0" at the beginning before the decimal point. Like pi (3.14159265358979323846 etc) the definition details go on to infinity.
The image above is four examples of how a linear graph/wave can represent a silhouette (a shape). The height of a moving event, in time, is shown. The event is something such as the amount of presence of a waveform, in 3D space, vs it being "elsewhere" (in dimensions other than the normal 3D we observe). The height is shown, relative to time, above a nominal level; and these levels are then displayed with the base-line taken as a centre and the levels represented at a regular number of degrees of rotation. These examples show fairly simple waveform (although the square-wave and the triangle wave have more harmonics than other pictures on this page. Considering that any complex waveform (graph) can be represented by a series of numbers that represent the component parts of the waveform, then any 3D shape can be made by starting with a sphere then building to anything, using harmonics--- such as the shape of a person's head.
The graph (Figure 2) above shows a wave made by adding two cycles of a fundamental (lowest) frequency plus three higher frequencies (harmonics). Vibrations have character that can be described, in groups, as relationships between fundamentals and their harmonics. The harmonics here are 2, 4 and 8 times the fundamental. The top lines are the result. The waves are in phase for their peaks.
Any vibration can be described by knowing all about its fundamental and harmonics. Whole-number harmonics (2x, 4x and 8x) are chosen here for convenience. Most vibrations are complex; the most-simple can be graphed as one circle or one sinusoidal wave from zero to 360 degrees. The whole of Creation could be described as "one sound"... The Word. One Cycle perhaps might describe something as relatively simple as a universe and it might be repeated with infinity --- and blended with countless other universes in countless dimensions --- to the limits of understanding that we have for Infinity; however the whole of creation is beyond our ability to form concepts such that the total implications of our present understanding of the concept of infinity would be vastly less, relatively, than was the forming the concept of zero that so facilitated our mathematics. Concepts beyond the concept of Infinity are ordinary in places we, as humans, may eventually develop into with understandings infinitely greater than the idea of concepts being different from tangibles. The definition of infinity allows that our ongoing understandings of creation may be compatible with those understandings we may grow into but that our ability to see the whole picture will always remain a numbered percentage of infinity.
Figure 4
A French mathematician (Fourier) developed maths to describe relationships in groups of vibrations. The same approach may be used for waves and shapes in three and more dimensions — as fundamentals plus harmonics' relative rates, timing-relationships (phases), sizes and (extra to Fourier's original vision) their percentage and type of use within each of their selection of co-ordinates.
This picture (Figure 3) puts the four sine-waves in phase --- all starting at the same time. It shows, like the first picture, both the result and, at the top, a mirrored image. Better than basing structure on imaginary orbits (like planets around a sun) here is a basis for describing fields that can oscillate concentrically (between various dimensions rather than just in 3D space)....although the graphs here are 2D linear.
A shapes' frequencies can have mathematical harmony, or any relationship. In this drawing the sine-wave cycles are all the same peak-to-peak amplitude(size) and they all start together. By comparison, the waves in the previous diagram have been phased to coincide their positive peaks. Waves (that constitute matter and energy) usually have a mix of places, types, frequencies, sizes and timing (relative to each other).
Waves may have back-and-forth motion mixed with concentric-motion, changing position, dimensions....They may be changed and moved by other waves. They exclude other waves from their space, with a variety of reactions to their efforts, depending on how many are sharing centres, when, where and the timing of competition.
Force-fields form shapes by interacting in stable patterns. Whether a photon, electron, atom, molecule or any other theory, it holds together because of concentric oscillations interacting with fields moving in other modes. This chapter, Shape, is pointing to these similarities to be found in every phenomena we can observe...light, energy, matter........
Figure 4 represents, in two ways, one cycle of the complex wave such as Figures 2 and 3. The top shape is a circular plot of the same set of levels as in the bottom graph.
Illustrating how a two-dimensional shape can be described, using a wave-form, one logical step further shows how a shape may be described in more than two dimensions.
A 2D linear graph is plotted in this circular way by using a fixed zero point. A 3D surface can be described as a 2D perimeter plus, in the third co-ordinate, a fundamental and whatever harmonics are also in the 3rd co-ordinate.
The zero level is set to avoid negatives that would confuse the plot and because I like showing how everything is just present, that is, how there is no negativethat opposes a positive. There are just low levels and high levels, relative to a nominal zero level (same idea as "closeness-vs-separateness-from-God" being an alternative description to "good and evil" or love/isolation .... and similar analogies).
When looking at more than three dimensions, one may consider everything being present in some amount in all places, from infinitely close to zero to infinitely everywhere. Relative to the observation's dimensions, waves can be from infinitely present to infinitely far away.
Matter, energy and fields are oscillations. Shapes can be communicated as waves.
Everything tangible is waves within a medium. Oscillations make much of our reality.
Figure 5 is just a larger version of one of the pictures.
With regard to figures 2 and 3 especially, another page has this way to say a main idea here:
"
The shape of anything can be described by a multi-dimensional version of Fourier's maths theory. Fourier's maths can describe level-variations according to time (by describing level-variations according to the individual frequency components). Variations of any complexity can be resolved as being a collection of pure sinusoidal components and their timing and size relative to the slowest frequency being considered in the set. {Cosines are calculated instead of sine because the standing level of the cosine is an actual level whereas if sine was used the value would be zero.]
The difficulties of getting an algorithm, for a practical of Fourier maths transformation, begin with for example the question of "what is the lowest frequency, to which the others shall be related in frequency, phase and amplitude?".... I believe every breath of wind is related, in its waxing and waning, to the finest details of the music in our lives. Harmony is in everything. Complexity is infinite; so harmony hides in complexity. Getting a Fourier series out of a sample of a noisy bit of pop music would be relatively easy (especially if the drums are perfectly tuned --- like electronic!). In Creation there are events that, although apparently irrelevant to each other according to our understandings, are perfectly in harmony with everything that exists. So how complex would the maths be, for a Creation that has oscillations exchanging between an infinite number of dimensions and phase; a Creation that uses time as if the same as space? Fourier might be, in some future/parallel time, in absolute heaven, working out the algorithms for the Fourier Series for the Whole of Creation!
By extending the maths theory (ascribed to Fourier) into a multi-dimensional system, then anything can be said to comprise a collection of concentric simple (sinusoidal) variations in presence in the nominated three dimensions. These multi-dimensional variations exchange places in whatever collection of dimensions they are getting excluded to." (David Beale circa 2000)
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