Unsustained awareness implied by gamma inversion

Psychosocial Implication in Gamma Animation (Part #11)

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Curiously, what is effectively an inverted gamma, is widely used in so-called "awareness ribbons" worn on the lapel -- possibly as a symbol of protest.

Awareness ribbons

These include, with variants in different countries and on different occasions:

• Blue ribbon (numerous causes)
• Red ribbon (notably solidarity with HIV/AIDS victims, prevention of drunk driving)
• White ribbon (notably violence against women)
• Pink ribbon (breast cancer awareness)
• Yellow ribbon (various causes)
• Green ribbon (notably various illnesses)
• Orange ribbon (notably self-harm)
• Purple ribbon (multiple health-related causes)
• Brown ribbon (notably for anti-tobacco, colon cancer)

Aside from its inversion as gamma, of particular relevance to the argument above is the extent to which the form of these ribbons implies a another dimension through the overlap. This might be considered as recognition of a topological form of the simplest kind -- possibly the beginning of a knot, or a half-knot. It is appropriate to note how the more aesthetic depictions of gamma as a lower case letter (see below) offer a similar sense of curving overlap, implying a degree of encirclement of an incipient hole.

The result can be fruitfully interpreted as an uncompleted Möbius strip -- with all that may imply in terms of the paradoxical engagement required by the psychosocial causes for which a ribbon may be used. These have been otherwise explored by Steven M. Rosen (Science, Paradox and the Moebius Principle: the evolution of the transcultural approach to wholeness, 1994). The awareness in question might be considered as implying potential completion or enablement by a "complementary focus" of awareness in the form of another ribbon. This is suggested by the image on the left. This could be understood as in progress of transformation into the paradoxical form on the right -- the completed Möbius strip.

Relating awareness ribbons to a Mobius strip

Further to the argument above regarding symbols, there is a case for recognizing the significance associated with their elaboration -- as the mind follows the lines and curves of the form, and their twists and inversions. This is most evident in the case of calligraphy considered as a spiritual discipline. Sanskrit calligraphy is considered to be a direct way of invoking energies and realities, as with those Arabic script or Chinese characters. The process is an invocation of a certain form, concept, idea or energy. Also of interest is the relevance of "gamma" as an accepted symbol in the disparate variety of contexts evident from the array of examples cited above. Consistent with the configuration of the awareness ribbons, curiously the calligraphic construction of gamma (below) suggests a degree of three-dimensional twist.

As concluded by Robert W. Gunn (Intimacy, Psyche and Spirit in the Experience of Chinese and Japanese Calligraphy, Journal of Religion and Health, 40, 2001), calligraphy is a mode of self-discovery and self-development that opens people to a substantial dialogue between cultures and the paths of inner conversation.

For Wong Ping Ho (The Chinese Approach to Learning: the paradigmatic case of Chinese calligraphy, 2005):

The case of Chinese calligraphy is representative, for "the tracing of characters is often described as if it were the definitive act of Chinese learning"... We would therefore take take a look at the learning of Chinese calligraphy, see how its seemingly mechanical learning methods are in fact necessary for the achievement of the states alluded to by qi and shen, and how it provides a paradigm for interpreting the point of learning in Chinese culture, at least as understood by the sages (p. 159)

Potential implications in forming letters with this perspective, most notably in the light of their significance in politics and religion, include movement to the "left", or to the "right", and movement "up", or "down" -- as well as their associations to "positive" or "negative"..

The question posed here is the cognitive implication of "gamma" as a traceable form, given the pattern of mnemonic associations above.

The argument may be taken further by considering a very concrete form of tracing and travelling a form, as is exemplified in complex road junctions. The elegance of the forms these may take follows from the mathematics governing so-called Euler spirals (also known as spiros, clothoids or Cornu spirals). These are curves whose curvature changes linearly with the curve length (the curvature of a circular curve is equal to the reciprocal of the radius).

The spiral is fundamental to transition curve design in railroad and highway engineering for connecting and transiting the geometry between a tangent and a circular curve, on which there is a very extensive literature. The governing equation is related to the so-called Gamma function, the Gauss error function, and is a special case of the confluent hypergeometric function (Raph Levien, The Euler Spiral: a mathematical history. 2008). The question is then how this might apply to any "transition curve" between strategic models based on "linear thinking" and those dependent on a sense of "curvature", as discussed separately (Clothoid as a psychosocial transition curve: from linear to circular, 2012). Is there a corresponding literature on "transition curve design" with respect to governance?

Euler's spiral or Clothoid (See also screen shots from an interactive representation The spiral converges to the centre of the holes in the image as the length of the spiral (measured from the origin) tends to positive or negative infinity.

It is of course the suggestive visual similarity of the above curve to a portion of the lower-case gamma depiction which is of interest here, together with the "transiting" curvature in the lower half of its aesthetic representations. As used in the Gamma function, it is however the upper-case variant of gamma that is used.

In the context of the above argument, the associated Euler equation offers a key to understanding the transformation between linear and circular cognitive modes. It has been suggested that Euler's formula describes two equivalent ways to move in a circle (Kalid Azad, Intuitive Understanding of E.uler's Formula, 2010). This is potententially vital to engaging with psychosocial dynamics constrained by different forms of "linear thinking" and requiring a degree of reconciling "curvature"

As a mathematical device for converting between polar coordinates and rectangular coordinates on the complex plane, its wider signifiance merits continuing exploration, as discussed separately (¿ Embodying a Way Round Pointlessness ? 2012). This included consideration of Enabling a reconciliation between one and nothing: p and the mysterious Euler identity and Spiralling around "nothingness" and "pointlessness": the implication of phi.

Appropriate to this argument, one early exploration of transition curve design argues that, from a driver-perspective, there is little to distinguish between one based on an Euler spiral and one based on a lemniscate (A. J. Tyson, Highway Transition Curve Design, Survey Review, 10, 1950). The lemniscate of Bernoulli is commonly used inroadwork where it is required to have thecurve transitionalthroughout, and without any intermediate circular curve. Lemniscate refers to a variety of figure-of-eight curves. As a three-dimensional dynamic system, the Lorenz attractor exhibits a lemniscate shape. It may of course be compared with the Möbius strip. Curiously (with respect to the awareness ribbons above) the etymology derives from a sense of being "decorated with ribbons", associated with the the ancient Greek island ofLemnnoswhere ribbons were worn as decorations.

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