The key feature of these structures is that they involve a relationship between paradigms which are qualitatively different. One attempts to relate the (formal) linear paradigm which is horizontal to the (formless) circular paradigm which is vertical.

The most famous transcendental number p - which involves a quantitative relationship between the line (i.e. diameter) and the circle (circumference) - fully symbolises this situation.
 
 

Transcendental structures (psychologically) are diagonal where different phenomena are experienced in varying dimensions. This involves the relationship between both the horizontal and vertical levels of experience and cannot be expressed satisfactorily in terms of either. As the horizontal (linear) is inherently discrete and finite, and the vertical (circular) continuous and infinite, the relationship between both inevitably involves both finite and infinite aspects. In psychological terms therefore, such structures involve the explicit interaction of reason and intuition.

Transcendental numbers parallel this situation. A transcendental number is really a diagonal number which cannot be expressed solely in either horizontal or vertical terms.

This is why it inevitably involves both finite and infinite aspects requiring both reason and intuition for its (unreduced) interpretation.

At the most fundamental level of physical reality this highly subtle mathematical structure is replicated. The most minute particles involve the dynamic relationship as between explicit and formal (phenomenal) manifestations (linear) and implicit and formless emptiness (circular). Such particles are inseparable from the dimensions they inhabit and have mathematically - expressed in real terms - a transcendental existence.
 
 

There are fascinating psychological complements to many famous mathematical hypotheses e.g. Gelfond’s Theorem and Fermat’s Last Theorem.
 
 

Important psychological structures develop at the point level which can be precisely modelled in psycho-mathematical terms.

The transition from circular to point level involving both (transcendent) spiritual union and a somewhat confused form of projected phenomenal experience can be intimately related to the three roots of unity.

The (advanced) superstructures of the point level - which involve the attempt the "high level" cognitive attempt to relate the rational (linear) and holistic (circular) paradigms represent "real" transcendental structures. There is both a positive form (global external reality) and negative form (specific internal reality).

The substructures of the point level - which involve the complementary "low level" affective attempt to relate projected phenomena (linear) and holistic archetypal understanding represent "imaginary" transcendental structures. Again these have a positive direction (the external world as archetype) and negative direction (internal self as archetype).