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Suit Yourself™ International Magazine #38: Archetype - Symmetry

  

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Suit Yourself™ International Magazine  #38: Archetype - Symmetry

 

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ARCHETYPE - SYMMETRY

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This is the 38th in our articles series and I hope this information is helpful.

All previous articles in the series can be found in our Library:
https://suityourself.international/libraryindex.html
and in the Magazine Archives:
https://suityourself.international/appanage/index.php?_a=newsletter
If you are experiencing problems viewing this newsletter in email, please use one of these links.

Upon request, reprint permission and an addendum of substantiating resources are available for all magazine articles. When requesting reprint permission or addenda, please include the issue date and full issue title. All magazine articles are copyright © Debra Spencer, Suit Yourself ™ International. All rights reserved. ISSN 2474-820X.

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     The Argo approaching the Symplegades, the crushing rocks, on the way to Colchis, at The End Of The World, where lies The Golden Fleece.

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SPECTRUM 

This is a dense section so I'm apologizing in advance - extra credit and Golden Fleece to those of you who finish this one. 

To review: 

Things are interesting with only two points, one radii, and two diameters. Nature, however, starts getting really interesting when there is more than one circle. 

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       Polarity Y-Axis radiation Poincare hyperbolic disk; it's more interesting with more than one circle.

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Now, we have gears.

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       An Antikythera mechanism model, front panel, 2007 recreation.

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         The Antikythera mechanism's front face, animated.

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Spectrum is the noun we use for ordered sequences of manifestation, typically energy manifesting as a group sharing some characteristic or sharing a group of characteristics, and measurable in some way as a related sequence. In physics, a spectrum is defined as the distribution of a characteristic of a physical system or phenomenon. The term Spectrum includes groups of otherwise appearing unrelated energy that none the less share something in common, including but not limited to related qualities, ideas, activities, entities, energies, velocities, masses, capabilities, emotions, moods, movements, wavelength, frequency, dispersion patterns, charge distributions, continuous actions, varying opinions on a common subject, sharing a continuous series or sequence, family, fraternity, or other characteristics.

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    Essuie-glaces, Scheibenwischer, car windshield wipers in the rain, a scene from the Alfred Hitchcock movie Psycho

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Spectrum can indicate, for example, one shape appearing differently depending on how it's seen from different points of view, or one point of view seeing different shapes of something moving.

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   Rotation around the X axis.

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A flat circle can be "drawn" in different ways, as an active or passive "transformation", in either of two directions, clockwise or counterclockwise, and when drawn, there is nothing to indicate which methods was used to draw it.  Thus, there's a spectrum of various ways a circle can be drawn, known in physics as active and passive symmetry group transformations. For example, a circle can be drawn in a clockwise or counterclockwise direction, onto a stable vertical or horizontal or tilted surface, such as a chalk board, white board, or piece of paper. Alternately, a circle can be drawn by keeping a pencil or pen point stationary on a piece of paper, and instead of moving the pencil, rotating the paper instead (or rotating whatever surface is used), thus drawing the circle by rotating the paper in either a clockwise or counterclockwise direction. Another alternative is to use a compass.

 

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Left: Alexander Overwijk draws a circle on a chalkboard, clockwise, from the bottom.

Right: Actress Sarah Michelle Gellar draws a circle on a white board, counterclockwise, from the top.

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A circle can be drawn by rotating a piece beneath a stable pencil point held by a stable hand and arm.

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   Antique engraving showing a hand using a compass to create a circle.

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Active and passive are two ways to consider any transformation. In the active transformation, the axes remain fixed, while the object moves (a vector v becomes vector v' when rotated about the origin through an angle θ.) On the other hand, pun intended, with the passive transformation,  the object remains fixed, while the coordinate axes rotate through an angle θ. Note that the sense of the rotation is opposite in these two cases, but the effect on the coordinates describing the object is the same. The circle has inherent "polarity". 

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      Active and Passive Rotation matrix, Dr. J. B. Calvert,  http://mysite.du.edu/~jcalvert/index.htm

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Archetypal spectrums also manifest as active and passive symmetry group transformations. There is an entire range, a continuous change, over which will vary some phenomenon attributable to the archetype, some measurable, demonstrable, repeating property or quality of it, delineating it as a manifesting system. An archetype, and its' effects, appear in myriads of forms. 

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        Right and left chiral; the diameters matter.

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By analogy (or rather I should say "fraternity" as it's more accurate) it is possible to say that The Thirty-Six Dramatic Situations (of Polti, Carlo Gozzi, Goethe, von Schiller, et al.) can be referred to in mathematics as a group theory point set isomorphic to the unit sphere, the unit circle in three dimensions plus time. This finite 36-situation point set of infinite variation represents the transformations of Elementargedanke energy into the local Volkergedanke forms specific to their manifestation time, place, events, and persons. So do the Greek Gods, and so does the Mahabharata. Likewise, the Periodic Table of Elements is a pictorial representation of otherwise fungible Elementargedanke energy into Volkergedanke forms, and synthetics by analogy are Volkergedanke local forms specific to their manifestation time, place, events, and persons. Understanding the remarkable nature of synthetics is of practical utility; it helps us regulate their not inconsiderable impact on our health and the environment. 
For more information on them, please see my article BASIC CARE INSTRUCTIONS  #15: Synthetic Fabrics
https://suityourself.international/appanage/index.php?_a=newsletter&newsletter_id=63

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SYMMETRY

The etymology of the word "symmetry" is "relation of parts, proportion," from Middle French symmétrie (16th century) and directly from Latin symmetria, from Greek symmetria "agreement in dimensions, due proportion, arrangement," from symmetros "having a common measure, even, proportionate," from an assimilated form of syn- "together" (see syn-) + metron "measure" (from root me - "to measure"). The meaning is thus "harmonic arrangement of parts", first recorded in the 1590's. (http://www.etymonline.com/index.php?term=Symmetry)

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       A musical spectrum: the major scales on the circle of fifths.

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A mathematical group is a very different thing from a group in everyday language. An everyday group is a set with some distinguishing characteristic that unites its members. A mathematical group is a set of transformations, not objects in the usual sense. A transformation acts on some object and alters it to a different object. For example, a "transformation" may simply move an object from one position to another. What it does is quite general and arbitrary; it is only necessary that there is an initial state and a final state with some fixed and definite relation to one another. All the transformations in a group are supposed to act on the same objects, and be capable of being applied successively.

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        Baseball pitcher; giving a new meaning to the phrase "curve ball"!

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Mathematics symmetry is a mathematical approach to elucidating grouped changes in orientation.  Symmetry can be of different kinds and types.  There are different ways to move things, and in mathematics, these are called "transformations". Things can be moved, for example, by rotation, reflection, glide reflection, and inversion. Combinations of these moves can define a group's "sequence" identity, and reveal the member's interrelations, even when the group members might otherwise seem unrelated.  

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      A spectrum of continuous surface Gyroids, groups of similar patterns are sorted by their symmetries. Bradley constructions.

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Mathematical groups express symmetry. In the macroscopic world, we observe coarse bodies and represent them by idealized points and curves that approximate reality. In quantum mechanics, on the other hand, symmetries are exact, not approximate, and the idealization is often the reality. Any two electrons are precisely alike, not merely close resemblances as would be two peas in a pod.  A circle is perfectly symmetrical, and just goes on looking the same no matter how you move it about, rotate it, reflect it as in a mirror, flip it upside down, right side up, rotate it in either direction, turn it inside out. The theory of angular momentum is the rotation group; the conserved quantity is the component of the angular momentum along the axis of the rotation.

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        Conservation of angular momentum; Make a Gif.

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Mathematical symmetry groups are used to indicate that some object or group of objects are invariant, meaning they don't fundamentally change, or are immune to, being moved about, being repositioned, in various ways, i.e. "transformations". Their overall attributes, such as their shape or volume, aren't altered by repositioning them or moving them about. They have some inherent fixed characteristic that relates them as a group, a family.

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        Poincare Invariance, from General Quantum Field Theory, cylindrical.

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If you happened to read my earlier discussion on algebra, you will have a clearer idea of what I mean here. Otherwise, check that out here:
 #36 Archetype - The "T" Intersection, in the section "Algebra & Mathematics: The Symplegades"
at either of these links:
https://suityourself.international/appanage/index.php?_a=newsletter&newsletter_id=84
https://www.suityourself.international/appanage/magazine-36.html


The ammonia molecule, a nitrogen atom with three hydrogen atoms forming a kind of triangular tent, is a symmetrical pyramid and it's precisely symmetrical, not merely closely symmetrical, as would be a molecular model.  The ammonia molecule belongs to a symmetry group called a point group (specifically C3v); a point group is a group in which one point remains fixed under all symmetry operations. The ammonia molecule can turn itself inside-out; that is, the nitrogen can pass through the plane of the hydrogens. This isn't easy, but the nitrogen can tunnel through, and there is a doubling in the emission band spectra as the result. The states divide into those symmetrical with respect to this inversion, and those that are antisymmetrical (change sign). 

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     Nitrogen inversion in  ammonia (Wikipedia).

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The inversion doubling is a very interesting phenomenon because it turns out to be possible to separate molecules in even and odd inversion states, and this led to the ammonia maser, the first of its' kind. Ammonia exhibits a quantum tunnelling due to a narrow tunneling barrier, and not due to thermal excitation; superposition of two states leads to energy level splitting, which is used in ammonia masers. 

For more information on this, see 
https://en.wikipedia.org/wiki/Nitrogen_inversion

An isolated atom is precisely spherically symmetric. 

 

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       Ammonia masar transitions in a time-dependent field (Wikipedia).

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In mathematics, various math operation sequences are sorted into groups of sequences, of patterns, called "symmetry groups". A  mathematical symmetry group is a group of spatial transformations that leaves an object unchanged; a symmetrical object, or symmetry group of objects, is considered symmetric if its' properties stay the same after applying some mathematical operation, or series of operations, to it.  The set of operations that "preserve a given property" of the object define a "symmetry group".  

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        Two litter mate puppies swap their food dishes.

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Mathematical symmetry operations, or transformations, allow rotations, reflection in a plane, and any combination of these. 

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       Trompe l'oeil elephant and giraffe holiday lights: the elephant appears to be partly attached to a wall and the other half to a board, however as the board pulls out, the elephant disappears, becoming a panorama of two giraffes; all are wired as lights.

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Combinations include inversion in the origin, rotation about an axis plus a reflection in a normal plane (the rotation and reflection do not necessarily have to be symmetry operations themselves), and so forth. 

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        Alexandra Soldatova catches; Olympic Rhythm Gymnastics.

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This is loosely analogous to the moves involved in dancing a waltz, a rhumba, or a square dance.  A group is some sequence of moves that the parts make, such that they all end up alike. In mathematical symmetry, group members start out the same size and shape, but have different orientations, and by moving each of them using the group's particular sequence, they all end up exactly like each other. 

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      Coordinating disparate symmetries over time; projecting the body into a moving hoop. Olympics Rhythm Gymnastics, Korea.

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Symmetry can be discussed as applying to one, two, three, four, or more dimensions, to one or many groups, as active or passive transformations, 
right-chiral or left-chiral, clockwise or counterclockwise, etc.

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     Anthony Howe Kinetic Sculpture "Di-Octo"  $225,000.00.

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Translation is similar to making one move in Chess, moving along the number line. On a triangle, this would be moving from a mid-point of a side, which is a point closest to the center, and along the side towards the closest angle point.  

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       "Remember, when Daylight Saving Time begins, we have to give everything a slight turn to the left." Cartoon by Sidney Harris, www.sciencecartoonsplus.com.

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Rotation is rotation about a fixed point, analogous to moving around the circle clockwise or counterclockwise. It is turning around an axis, and an object can rotate around the axis more than once (the number of rotations is called the "order of rotation"). 

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        A rotation perpetual motion bulldog.

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Reflection reflects an object across a line, as when looking in a mirror. 

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       Pi Pie mirror reflection. Does this = pie in the face?

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However, it's similar to changing axis, or direction, as in moving from latitude to longitude. Thus it can be a 'flip' as in a simple mirror reflection, and it can be combined with a translation to become a glide. 

 

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      Nautilus Gears.

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 A glide reflection is a composite of more than one move:  first, it is a reflection across a line, and then, it is a translation parallel to the line of reflection. A glide reflection will map a set of left and right footprints into each other.  

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        Example of a glide reflection; a composite of a reflection across a line and a translation parallel to the line of reflection (Wikipedia)

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Imagine someone has turned you upside down, and holds you while you walk across the ceiling, leaving your footprints traced there. The impressions left by your left and right feet will alternate, indicating movement in one direction as you walked along the ceiling.  

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        Fred Astaire, upside down.

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Inversion is turning something inside out, for example, taking a glove off your left hand and then turning it inside out so it will fit on your right hand, inside out.  The inversion returns the cycle to another octave, another level, another range of manifestation. 

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        Inversion: Ordinariat für Geometrie - Universität für Angewandte Kunst. 

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One mathmatical way to represent symmetry is via matrices. 

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        Matrix math.

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With matrices, symmetry can be represented as a series of moves, a sequence of translations, which is then arranged as a unit and used as mathmatical interactions. This can reveal when flow changes.

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      "An interesting look at configurations beyond the Standard Model of physics. The particles and forces of the Standard Model can be embedded in larger and larger symmetry groups, corresponding to grand unified theories (GUTs) of increasing complexity. Two-dimensional projections of some of these GUTs are shown here, with particles plotted along the gray axes according to their charges. Bosons (force-carrying particles) are represented as circles and squares; fermions (particles that constitute matter) are shown as other polygons." (Toptechnicalsolutions.com)

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Symmetry can tell us our hands are symmetrical, and that two symmetrical hands are clapping.


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In a comment on mathflow.net, contributor Dustin G. Mixon had this to say:
"When I introduce matrix multiplication in linear algebra, everyone has seen it before, and so I inject some "fun" by claiming that multiplication is commutative. The more outspoken students read my smile and speak up with an emphatic "No, it isn't!" I then proceed to make my case by multiplying 1×1 matrices and 2×2 matrices that happen to commute. Eventually, a student suggests matrices."
(https://mathoverflow.net/questions/281447/mathematical-games-interesting-to-both-you-and-a-5-year-old-child)

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TO CONCLUDE

The center-point is the infinite perfection of the archetype and the circle is the archetype manifested  as mastership. It represents the truth, goodness, beauty, the inspirational power, of the fulfilled work of art. It is the refined and sensitized consciousness of the artist as manifestor--interpreter, with "interpreter" meaning "teacher" as well as "performer". It is the integrated "Golden Fleece".   It is nothing less than the fulfillment of the sacred endowment of the artist as a spiritual instrument, to perceive archetypes and to manifest them, to thus illuminate the consciousness of others so the archetypes are able to be integrated into our lives.   The circle's center-point represents the source, of manifestation, on all planes, octaves, and cycles.

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       Astrolabe: stereographic projection on tympan (Wikipedia).

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The circle is the result of connecting two points to create a line, the center point and the second point, thus creating the one radii joining the points. This is the Elementargedanke manifesting as the Volkergedanke. The result in time is the manifestation of two diameters, polar Volkergedanke, and the circle itself is their perfected ensemble. 

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     Animated polarizer in front of a computer flat screen monitor. LCD monitors emit polarized light, typically at 45° to the vertical, so when the polarizer axis is perpendicular to the polarization of the light from the screen, no light passes through (the polarizer appears black). When parallel to the screen polarization, the polarizer allows the light to pass and we see the white of the screen.
https://en.wikipedia.org/wiki/Polarizing_filter_(photography)

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We can only effect systematic and productive change to the degree that we understand these facts and the "forms" by which energy manifests, and by "forms" I mean the patterning of the relative strength and weakness of the various forces operating in our lives, unfolding as rhythm, patterns, sequence, seasons, timing.  When hands are clapped together, they are heard, and they are clapping in response to what they have perceived.

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       Harry Whittier Frees photograph, c1935; Blink and Daddy pups boxing in the ring.

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This knowledge enables us to become aware of when circumstances are advantageous, and when they are less so, of beginnings and endings, and should we want to, we can begin to unfold our undeveloped qualities, correct faulty traits, and eliminate destructive propensities. 

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      He doesn't realize he's won the lottery.

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We can jump up, and down, clap our hands, and then, jump rope.

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        Harry Whittier Frees photograph, c1935; kitties are jumping rope with their doll.

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To everything, there is a season and a time for every purpose. To the degree that we possess this knowledge, and to the degree that we have the freedom of choice to learn how to use it and control it in ourselves, to develop discernment and wisdom, can we begin living our lives in the world as it is, in ways that make this easier and less of an uphill battle, for ourselves and others.  

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       Harry Whittier Frees photograph, c.1935; kitty is bowling with ten pins.

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This is what the ancient stories meant by revealing to us that the passage of the Argonauts through the Polarity of the Symplegades was one way. The myths tell us that their successful passage with Triton's help silenced these rocks forever, and that they could thus return by a different route. Their successful passage through the Symplegades, unbeknownst to them, symbolized the end of one journey, and the beginning of another. After all, at this point in their journey, they still had not got the Golden Fleece.   

 

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        "When you feel how depressingly slowly you climb, it's well to remember that things take time."- Piet Hein

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However, once the Golden Fleece is in hand, a way must be found to integrate it into the culture. Some way must be found to introduce it into the culture, so the culture can use it to heal itself.  These ancient stories also tell us our fate should we fail to integrate what we've brought back from this voyage: Jason and Media's story spell out the consequences.  

This is what I discuss next.

 

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        "Mr. Osborne, may I be excused? My brain is full." Cartoon by Gary Larson.

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ADDITIONAL RESOURCES


The sources for the material in this Archetypes series are listed in one place,  #40 Archetypes Additional Resources:
 https://suityourself.international/appanage/index.php?_a=newsletter&newsletter_id=88
or
https://www.suityourself.international/appanage/magazine-40.html

If you're interested in exploring these topics further, then I recommend the resources at the above links. 
 

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I sign our magazine articles "See Into The Invisible". Thanks for reading.

Best Wishes, 
Debra Spencer

All Content is © Debra Spencer, Suit Yourself™ International. Technical Library FAQ Index ISSN 2474-820X. All Rights Reserved. Please do not reproduce in part or in whole without express written consent. Thank you.
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All Content is ©2019 Debra Spencer, Appanage™at www.suityourself.international Suit Yourself ™ International, 120 Pendleton Point, Islesboro Island, Maine, 04848, USA 44n31 68w91 Technical Library FAQ Index ISSN 2474-820X. All Rights Reserved. Please do not reproduce in part or in whole without express written consent. Thank you.

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All Content is ©2019 Debra Spencer, Appanage™at www.suityourself.international Suit Yourself ™ International, 120 Pendleton Point, Islesboro Island, Maine, 04848, USA 44n31 68w91 Technical Library FAQ Index ISSN 2474-820X. All Rights Reserved. Please do not reproduce in part or in whole without express written consent. Thank you.
Success consists of going from failure to failure without loss of enthusiasm. ~ Winston Churchill