A Gardner Like That One

I THINK that I shall never see
A poem lovely as a tree.
A tree that looks at God all day,
And lifts her leafy arms to pray;
Poems are made by fools like me,
But only God can make a tree.
--American Poet, Joyce Kilmer. 1886–1918

Trees, branches, fractals are some examples of dendrites and 2 key parameters characteristic of dendritic growth are the tip radius and tip velocity. 

"Dendritic growth also provides an archetypal problem in morphogenesis, where a complex pattern evolves from simple starting conditions. As one of the simplest nontrivial examples of spontaneous pattern formation, the physical understanding and mathematical description of dendrites remains of interest to mathematicians, scientists, and engineers."—from the article, Dendritic Growth Tip Velocities and Radii of Curvature in Microgravity, by Koss LaCombe, et.al.

Other naturally ocurring phenomena which display dendrite formation/dendritic growth are the shape of EMF, magnetic flux...

Tesla's 'Dooms Day Machine'

...organic and inorganic formations such as icicles and neural networks

Examples of Naturally occuring Dendrites.
Both organic and inorganic include (clockwise from top left);
 metal alloys, water icicles, fly eye neurons, brain cells/networks.

The fine structure constant, α ~ 1/137 is a dimensionless fundamental physical constant that surfaces in both Mechanistic and Anthropic considerations. (See Sifting Through The Rubble for more about what else may be couched in the fine grain structure constant).

SpaceTime as described in a Scientific American article.
According to 'spin foam' and how the 'spin network' changes with Time, is remeniscent of Dendritic growth.

3-D to 2-D Reduction of the Universe near the Planck length.
Similar charts found elsewhere at apageinthelife,
 including 'Building a Mystery'.
The image is not very clear but it reveals
the geometry at small scales is similar to dendrite patterns
The Evolution of Geometry in Time, shown above is taken from the Scientific American article Atoms of Space and Time (Lee Smolin, Scientific American Sp 15, 56 - 65 (2006) doi:10.1038/scientificamerican0206-56sp) depicts the geometry of spacetime from the evolution of 'spin foam', ie., the 'spin network' when it includes Time as the 4th dimension:
“The (spacetime) geometry changes discontinuously, becoming a single (1) quantum of volume and 3 quanta of surface area, as shown in the last frame. In this way, time as defined by a spin foam evolves by a series of abrupt, descrete moves, not by a continuous flow…"
(Note: This contradicts the 'Time as a Line' premise (see wolframalpha definition discussed below), in that the very definition of a line implies continuitity since between any 2 points on a Line there is an Infinite number of Points.)

"Although speaking of such sequences as frames of a movie is helpful for visualization, the more correct way to understand the evolution of the geometry is as discrete ticks of a clock. At one tick the orange quantum of area is present; at the next tick it is gone—in fact, the disappearance of the orange quantum of area defines the tick. The difference in time from one tick to the next is approximately the Planck time, 10^-43 second. But time does not exist in between the ticks; there is no “in between.” In the same way that there is no water in between two adjacent molecules of water…”
(Note:While the 'Time vanishes between Tics' or 'Pops In and Out of Existence between Tics' premise is one way of explaining Time in the geometry/math of the evolving 'spin foam', it is this very behaviour of Time near the Planck scale that would make it Suspect as a Higher Dimensional Object manifesting/Transitting? a Lower Dimension, (my take on this spacetime geometry, remember what fun Emily is having with Bob...Oh, and poor Fred the Flatlander!)

In contrast to the author, Lee Smolin's,  explanation of 'quantum tic Time' in this 2006 Scientific American article, the "conventional" concept of Time from wolframalpha.com/(keyword fractal dimension) follows:
"The term "fractal dimension" is sometimes used to refer to what is more commonly called the capacity dimension of a fractal (which is, roughly speaking, the exponent in the expression , where is the minimum number of open sets of diameter needed to cover the set). However, it can more generally refer to any of the dimensions commonly used to characterize fractals...

The notion of dimension is important in mathematics because it gives a precise parameterization of the conceptual or visual complexity of any geometric object. In fact, the concept can even be applied to abstract objects which cannot be directly visualized. For example, the notion of time can be considered as one-dimensional, since it can be thought of as consisting of only "now," "before" and "after." Since "before" and "after," regardless of how far back or how far into the future they are, are extensions, time is like a line, one-dimensional object..."
(Hmmm...'Time is a One-Dimensional Object Like a Line?'...Says the Observer from the 4-D SpaceTime Universe, refer to Flatlands--You mean to tell me it's a Line and Not a Complex Geometric Object? Not a Fractal? That it doesn't/cannot Loop Back Around?... And do other funny things (to my hair  /hare    /neverwhere)?)

Physicists Renate Loll and Dejan Stojkovic are working independently to discern the geometry of the 'early universe' and answer the question 'Did the early universe have just one spatial dimension?'

Studies suggest that the early universe — which expanded from a single point and was very, very small at first — was one-dimensional (like a straight line--here we go again with the linear thinking) before expanding to include two dimensions (2D, like a plane) and then three (like the world in which we live today).{Really, 3- and not 4-D is the world in which we live today?--That would mean my goose is cooked in No Time At All!} 

The theory of evolving dimensions presents a paradigm shift from the way we previously thought the cosmos came to BE.

The core idea is that the dimensionality of space depends on the size of the space we’re observing, with smaller spaces associated with fewer dimensions.

That means that a fourth dimension will open up (unfold)— if it hasn’t already — as the universe continues to expand. The theory also suggests that space has fewer dimensions at very high energies of the kind associated with the early, post-big bang universe.

According to Stojkovic, “What we’re proposing here is a shift in paradigm,...Physicists have struggled with the same problems for 10, 20, 30 years, and straight-forward extensions of the existing ideas are unlikely to solve them...We have to take into account the possibility that something is systematically wrong with our ideas,” he continued. “We need something radical and new, and this is something radical and new.”

Loll works on developing a theory of quantum gravity, reconciling the beautiful geometric description of space and time laid out in Einstein's theory of General Relativity with the insight that all of physics at its most fundamental level must be described by quantum laws of motion.

She is one of the pioneers of a new approach to the nonperturbative quantization of gravity, that of Causal Dynamical Triangulations which recently has produced a number of remarkable results. These include a dynamical derivation of the fact that space-time is four-dimensional (something that can be taken for granted only in classical gravity) and that it has the shape of a de Sitter Universe (like our own universe in the absence of matter), and of the so-called wave function of the universe which plays an important role in understanding the quantum behaviour of the very early universe.

Remarkably, the dimensionality of spacetime reduces smoothly to two (2-D) at short distances, indicative of a highly nonclassical behaviour of spacetime geometry near the Planck scale. These results are obtained by superposing elementary quantum excitations of geometry which have a notion of causality ("cause preceding effect") built into them at the very smallest scale.

Ink on Paper modeling 2-D Universe near Planck length, as seen on one episode of Through the Wormhole with Morgan Freeman featuring Dr. Renate Loll (the shape is reminiscent of dendrites).
{While this is a reduced spatial dimension, it corresponds to a higher energy state).
The 'Universal Mesh' modeling how the 2-D Dendritic type Universe near Planck scale (see previous) expands to higher 3- and 4-Dimensions as it grows.
(See references to 'interwoven firmament', 'heaven rife with interwoven paths,' ie., Quran Sign 51:7)

Shown: 4-D Julia Set
Interesting in that it is very similar to results from Computer-generated superpositions of causal geometries that show the emergence of an extended quantum groundstate;
 the geometric properties of which provide strong evidence that quantum spacetime indeed behaves macroscopically like a four-dimensional universe.

According to Loll it reveals a fractal structure on slices of constant time, and I posit that due to the 'Law of Similars' (ie, 'as above, so below'), Time itself is more than likely fractal in nature, like other natural phenomena and probably displays the same fractal dimensions as Rivers (see reference to Rivers of Time in 'Is the Sky a Hologram', elsewhere reference to 'tuples', quaternions/octads/octals).

Since all local curvature degrees of freedom of the geometry undergo large quantum fluctuations, a four-dimensional geometry can either crumple up to generate a geometry of an effectively higher dimension (like crumpling up a two-dimensional sheet of paper into a three-dimensional ball), or curl up to give a geometry of an effectively lower dimension (like rolling up a piece of paper into a thin tube, which will appear effectively one-dimensional at a scale much larger than the circumference of the tube). This is exactly what happens in nonperturbative Euclidean quantum gravity. It produces ground states of geometry that are either maximally crumpled with an infinite(!) effective dimension  or polymerized into thin and branched threads, with an effective dimension of two, neither of them promising candidates for the ultimate vacuum.

The main technical tool for constructing a causal nonperturbative path integral is the method of Causal Dynamical Triangulations, a Lorentzian version of dynamical triangulations , in which the new fundamental principle of microcausality is implemented by taking superpositions of only those geometries which have a well-defined causal structure down to the very smallest scales.

And the exciting news is that this idea seems to work!

Computer-generated superpositions of causal geometries show the emergence of an extended quantum ground state.

Recent work which analyzes some of its geometric properties provides strong evidence that this quantum spacetime indeed behaves macroscopically like a four-dimensional universe.

These very encouraging findings imply that an essential aspect of the classical limit, the dimension of spacetime, emerges correctly, a consistency check other approaches to nonperturbative quantum gravity have yet to pass.

This nontrivial result makes the model of causal dynamical triangulations a prime candidate for a theory of quantum gravity, although much work remains to be done to prove that it indeed is the correct theory.

Further research is under way to determine other classical and semiclassical properties of the "reconstructed universe", as well as its true quantum structure.

At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-D near the Planck scale, and reveals a fractal structure on slices of constant time. [http://en.wikipedia.org/wiki/Causal_dynamical_triangulation ]

Dendrite growth is imprinted in the Mechanistic world from the 'spin foam' to 'the universe' (i.e, the quantum level to the cosmic scale). (That covers the maxima and minima for The As Above ..So, Too, Below.)

For the medial, or Anthropic applications, there have been several studies to parametrically define and analyze dendritic growth in living cells to determine branching rules, morphological identifiers and the shape of their dendrite spanning fields for anything from fly eye neurons (i.e, lobula plate tangential cells or LPTCs, motion sensitive neurons) and cerebellar Purkinje cells.

A brief review of the literature turned up one article which mentions that some early approaches to describing and reconstructing dendrite branching in general for LPTCs had failed to take into account a major functional constraint governing dendrites: their need to reach specific input locations.

"More recent attempts to constructing dendrite morphology in relation to their function and the location of their inputs had led to dendrite structures of low complexity and accuracy in spite of high computational costs. However, circuitry and connectivity as well as simple wire packing issues are known to be determinants of dendrite morphology.

In addition, the specific organization and architecture of many parts of the brain helps to reduce wiring costs for the (brain/neural network) circuitry. It is therefore not surprising that such constraints can be used to describe dendrite branching in LPTCs and other cells.

Other planar space-filling cells, the cerebellar Purkinje cells, certainly follow a similar rule. However, the suggested approach is not restricted to planar dendrites and future analysis will cover all different neuron arborizations to clarify the ubiquity of the suggested branching rule. At the example of LPTCs, the usefulness of the approach presented here can be put forward: LPTC electrophysiology was studied in great depth (e.g. and precise models, so-called compartmental models, including the detailed anatomical structure were designed and are continuously being improved).

Understanding LPTC branching, these constraints can be directly put in relation with the optic flow processing occurring within their circuitry.

Assuming generality of principles, even the function of cells, which have not yet been reconstructed, can be inferred based on the contours of their dendrites alone. Moreover, the fly is the model animal in which the molecular components that determine neural growth are currently being unveiled, mainly through genetic tools.

Our framework therefore allows a quantitative study of the impact of gene modifications far beyond basic statistics.

In particular, molecular principles guiding neuronal self-avoidance during development and others can now be put in relation with the branching constraints presented here. Eventually, studying molecular factors shaping dendritic spanning fields separately from a specific branching rule within should elucidate a fundamental organizational element in the brain, i.e. the neuron's branching structure.

After 3D skeletonization and sparsening the carrier points, the remaining points were submitted to the same greedy algorithm (started at a user defined dendrite root location) as used for obtaining artificial dendrites Quadratic diameter decay was mapped on the resulting trees."
Iron Filings in A Jar
3-D Magnetic Field
The 'tendrils' branch from a central 'hub' or 'onion' (see Purkinje Cells)
Similar to Dendrite Growth/Networks found Elsewhere in Nature
Applying this vast body of information from the very large to the very small probably leads to inroads when similar dendrite growth/concepts are applied to developing, say, a monopole magnet?

[Monopole magnets may not exist, but bipolar magnets may be produced/constructed based on the shape of the magnetic field and modeling it such that the 'net effect' is a monopole magnet--much like a monopole antenna is essentially still a dipole antenna, but its net effect is to radiate in only 1/2 the space.]

An analogous case, if one were to model such an 'alien magnet,' would  be a monopole antenna with a perfectly conducting, infinite ground plane  which is identical to the top half of a dipole pattern, with its maximum radiation in the horizontal direction, perpendicular to the antenna.

The 1/4-wave monopole antenna is a single element antenna fed at one end, that behaves as a dipole antenna. It is formed by a conductor in length. It is fed in the lower end, which is near a conductive surface which works as a reflector (see Effect of ground).

The current in the reflected image has the same direction and phase as the current in the real antenna.

The quarter-wave conductor and its image together form a half-wave dipole that radiates only in the upper half of space.

In this upper side of space the emitted field has the same amplitude of the field radiated by a half-wave dipole fed with the same current.

Therefore, the total emitted power is one-half the emitted power of a half-wave dipole fed with the same current.

As the current is the same, the radiation resistance (real part of series impedance) will be one-half of the series impedance of a half-wave dipole.

As the reactive part is also divided by 2, the impedance of a quarter wave antenna is ohms (most likely purely capacitive reactance).

Since the above-ground fields are the same as for the dipole, but only half the power is applied, the gain is twice (3dB over) that for a half-wave dipole ( ), that is 5.14 dBi.

The earth can be used as ground plane, but it is a poor conductor: the reflected antenna image is only clear at glancing angles (far from the antenna).

At these glancing angles, electromagnetic fields and radiation patterns are thus the same as for a half-wave dipole.  Impedance of the earth is far inferior to that of a good conductor ground plane -- this can be improved (at cost) by laying a copper mesh.

Because it radiates only into the space above the ground plane, or half the space
of a dipole antenna, a monopole antenna will have a gain of twice (3 dBi over) the gain of a
similar dipole antenna, and a radiation resistance half that of a dipole.

Thus a quarter-wave monopole antenna, the most common type, will have a gain of 5.19 dBi and a radiation resistance ofabout 36.8 ohms if it is mounted above a good ground plane.
Commercially manufactured ladder line or "window line" (for this metaphor see Talk Like An Egyptian, above) is a type of transmission line similar to twin-lead for balanced connection of antennas.

Ladder line is constructed as a pair of evenly spaced wires with supportive plastic webbing holding the wires apart. The plastic webbing has windows cut in it to reduce its dielectric effect and reduce loss in the transmission line. The alternating webbing and windows gives ladder line its characteristic look and name.

Conceivably, due to similaries in dendrite growth/shape of electric and magnetic fields, and their physical properties already clearly understood through related parameters from the coupling factor (fine structure constant α) to Maxwell's equations (which unify electricity, light, and magnetic radiation), monopolar magnets may be engineered by properly ~weaving~ the magnetic domains to produce a net effect of monopolar mangnetism--an analog (dual/complement) to how monopole antennas radiate radio signals or akin to how metamaterials 'appear' to violate Snell's Law for light.

{Back in the day we used to call this 'reverse engineering' and we didn't need to down alien space craft to do it--it was simply a matter of cracking open some competitors' devices, like RCA InGaAsP transcievers---now I am forced to resort to just pounding out 'crib notes' on A Page... from my kitchen table while I wait for the veggies to parboil.}

--it's not that easy, and me not so smart, so Yous guys have first crack at it and work from my notes.
Oh...I forgot, someOne already figured all this out, I just edited the in-flight black box recording for posterity...

...no need to bust a gasket over it, I do believe You Know It!
...turns out Jeremy here knew the answer to a History question and not any of this other stuff.

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