[5] **viXra:1308.0116 [pdf]**
*submitted on 2013-08-21 18:51:33*

**Authors:** John Frederick Sweeney

**Comments:** 35 Pages.

Polytope (3,3,5) plays an extremely crucial role in the transformation of visible matter, as well as in the structure of Time. Polytope (3,3,5) helps to determine whether matter follows the 8 x 8 Satva path or the 9 x 9 Raja path of development. Polytope (3,3,5) on a micro scale determines the development path of matter, while Polytope (3,3,5) on a macro scale determines the geography of Time, given its relationship to Base 60 math and to the icosahedron. Yet the Hopf Fibration is needed to form Poytope (3,3,5). This paper outlines the series of interchanges between root lattices and the three types of Hopf Fibrations in the formation of quasi – crystals.

**Category:** Topology

[4] **viXra:1308.0099 [pdf]**
*submitted on 2013-08-19 12:04:41*

**Authors:** John Frederick Sweeney

**Comments:** 64 Pages.

Mathematicians and physicists have long wondered why the Octionic Projective Plane (OP2), the Freudenthal – Tits Magic Square, or Magic Triangle and certain functions of the Octonions and Sedenions abruptly end. This paper lays out the various elements included in these conundra, with the assumption that irregularities and undiscovered relationships between these structures account for the anomalies. In addition to the above, this paper investigates the G2 to B3 to D4 to B4 to F4 Magic Triangle, the twisted product of S7 x S7 x G2, which leads to the Sedenions, the exceptional singularities, Kleinian singularities, Coxeter Groups H3 and H4, Polytope (3,3,5) , the 600 – cell and the binary icosahedral group.

**Category:** Topology

[3] **viXra:1308.0061 [pdf]**
*submitted on 2013-08-11 10:33:40*

**Authors:** John Frederick Sweeney

**Comments:** 82 Pages.

The final element of the Qi Men Dun Jia Model is the Boerdijk-Coxeter Helix, or the Tetrahelix of R. Buckminster Fuller, since this brings matter up to the level of DNA strings or lattices. Composed of Sextonions,Octonions, Twisted Octonions and Sedenions, the author examines the Boerdijk-Coxeter Helix from various perspectives to illustrate how BC – Helices play an important role in the formation of matter. The paper closely examines the eccentricities of the BC – Helix to determine whether these relate to diminished Octionic and Sedenion function, associativity and divisibility.

**Category:** Topology

[2] **viXra:1308.0051 [pdf]**
*replaced on 2013-10-11 17:31:47*

**Authors:** Nathan O. Schmidt, Reza Katebi

**Comments:** 34 pages, 7 figures, accepted in the Hadronic Journal

In a preliminary assessment, we begin to apply Santilli's iso-mathematics to triplex numbers, Euclidean triplex space, triplex fractals, and Inopin's 2-sphere holographic ring (HR) topology. In doing so, we successfully identify and define iso-triplex numbers for iso-fractal geometry in a Euclidean iso-triplex space that is iso-metrically equipped with an iso-2-sphere HR topology. As a result, we state a series of lemmas that aim to characterize these emerging iso-mathematical structures. These initial outcomes indicate that it may be feasible to engage this encoding framework to systematically attack a broad range of problems in the disciplines of science and mathematics, but a thorough, rigorous, and collaborative investigation should be in order to challenge, refine, upgrade, and implement these ideas.

**Category:** Topology

[1] **viXra:1308.0015 [pdf]**
*submitted on 2013-08-03 11:51:04*

**Authors:** John Frederick Sweeney

**Comments:** 31 Pages.

In Vedic Nuclear Physics, the number 28 plays a key role, and this will be discussed in a future paper. The 28 aspects must be extruded or dispersed along structures which contain factors of 12 or 6. The Poincare Dodecahedral Space contains these factors and relates to the 120-element binary icosahedral group, which double covers the simple 60-element icosahedral group. This, in turn, enjoys isometric relationships to the 60 Jia Zi and the 60 Na Yin of Chinese metaphysics, function to add Five Element and temporal qualities to matter as it becomes visible, in the Qi Men Dun Jia Model.

**Category:** Topology