Cayley-Dickson Algebras and Finite Geometry

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a < b <c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG$(N-1,2)$, the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) is the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) coincides with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) is identical with a particular $(21_5,35_3)$-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. We also draw attention to a remarkable nesting pattern formed by these configurations, where ${\cal C}_{N-1}$ occurs as a geometric hyperplane of ${\cal C}_N$. Finally, a brief examination of the structure of generic ${\cal C}_N$ leads to a conjecture that ${\cal C}_N$ is isomorphic to a combinatorial Grassmannian of type $G_2(N+1)$.Comment: 26 pages, 20 figures; V2 - the basis made explicit, a footnote and a couple of references adde

Blaise Pascal

Blaise Pascal was a mathematician with a great impact. He began his mathematical journey from a young age, and, throughout his lifetime, made significant contributions in geometry, probability, philosophy and religion. Toward the end of his short life, Pascal became focused on his faith, which led to a change in his thoughts and has served as an example to Christian mathematicians ever since

Intimations of a Spiritual New Age: IV. Carl Jung\u27s Archetypal Imagination as Futural Planetary Neo-Shamanism

This series of papers on early anticipations of a spiritual New Age ends with Carl Jung’s version of a futural planetary-wide unus mundus rejoining person and cosmos, based on his psychoid linkage of quantum physics and consciousness, and especially on the neo-shamanic worldview emerging out of his spirit guided initiation in the more recently published Red Book. A cognitive-psychological re-evaluation of Jung’s archetypal imagination, the metaphoricity of his alchemical writings, and a comparison of Jung and Levi-Strauss on mythological thinking all support a contemporary view of Jung’s active imagination and mythic amplification as a spiritual intelligence based on a formal operations in affect, as also reflected in his use of the multi-perspectival synchronicities of the I-Ching. A reconsideration of Bourguignon on the larger relations between trance and social structure further supports the neo-shamanic nature of Jung’s Aquarian Age expectations