Text: € Space is divided into at least 12 dimensions. There are the three conventional dimensions, and each of these is divided into four sub-dimensions. € The first and basic sub-dimension is the quaternion, the unit of which has a diameter of one unit, or a cube which is one unit on a side. This unit is so miniscule that we may consider it dimensionless. We take this unit to be on the order of 10^-14 millimeter. The quaternion is the equivalent of a point in space, in Quantum Arithmetic. € The second sub-dimension is the product of two quaternions, and is considered on the same basis as a line. It is used here as a root, and has length but its width and thickness is the unit quaternion. The root is the equivalent of a point in space in conventional mathematics. It is represented by b, e, d, or a. € The third sub-dimension is the product of two roots or the product of four quaternions. It is an area in Quantum Arithmetic, but considered as a line in conventional methematics. It is represented by any of the capitalized identities, A through K. This dimension has length and width but its thickness is the unit quaternion. € The fourth dimension in Quantum Arithmetic is the product of two of the third-dimensions; of four of the roots; or of eight of the quaternons. It will have measurable length, width and thickness. It is represented in the capitalized identities by L, (L = beda/6), or by the squares of any of the other capitalized identities, (C^2 + F^2 = G^2). In Quantum Arithmetic this dimension is a solid. In conventional methmatics it is undefined. (The area of an ellipse is the product of pi and three roots, d^3, and two quaternions, ˆa, and ˆb, in square units. For the same measurement in circular units it is : Area = d^3ˆab). € This last dimension has had little discussion in this text, partly to avoid going into the higher dimensions, and partly because this higher dimension and its relationship to conventional dimensions, has not been investigated. When these lower dimensions are more fully understood then the upper dimensions will fall into place. It is anticipated that a 5-way division is involved, and that the conventional 3-dimensions, the 4-dimensions of Quantum Arithmetic, and the unexplored 5-dimension system will, working through Synchronous Harmonics, make a 60-way division of space, hence the viability of the sexigesimal system of numerical notation. € The system of integers forms an associated symmetry around a square of an integer, D. Each pair of integers will have a complimentary system of characteristics for par-value, primeness/factorability, and shape. The arrangement goes into greater detail as more subsidiary integers are added to the framework. € General: Although the ellipse is derived through porisms, the aspects of the ellipse discussed here; the role of C in maintaining the orientation of the ellipse; the role of E as a mediator of the ellipse and the role of J, F, G, and K in the operational phases; are all in the field of Pseudaria. In retrospect, and particularly in the derivation of fractional equivalents, lead to a deeper meaning in that the ellipse may be only a frame of reference in which these are taken. This has all been done through constructive geometry because it would have been relatively impossible to approach and describe in an abstract sense. This four-way division of 'duties' between C, D, E, and the operators, J, F, G, and K can be extended outside the realm of methematics and into many other fields. Plato refers to these 'duties' in Republic, but in a philosophical sense. Quantum Arithmetic leads almost naturally into the humanities and far beyond the limits of conventional mathematics.

See Also: SPACE; DIMENSIONS; SCALAR; FOURTH DIMENSION; ONE

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