#dimensional numbers

Mandalic geometry (MG) is a new kind of mathematical methodology based on a worldview having roots that predate written history.

It is a discrete geometry which currently consists of just a coordinate system but can be extended as Descartes did his to encompass an entire analytic geometry.

Mandalic geometry introduces and is based on a new number system, the probable number system (or probabilistic number system.)

Just as the complex number system combines real numbers and imaginary numbers and is more robust than either, the probable number system combines real numbers and probable numbers and is more robust than either.

The probable number system is also more robust than the complex number system. Complex numbers combine real numbers with imaginary numbers to form the single complex plane. Composite numbers combine real numbers with probable numbers to form six interdependent composite planes.

Axiomatic to the system is the contention that numbers can exist in different dimensions and therefore can be described as being of some particular dimension. Numbers are always viewed and treated within context of a stated dimension.

Probable numbers are an extension of the real numbers to higher dimensions and are independent of imaginary and complex numbers.

Mandalic geometry does not admit the existence of square root of -1 in the real world other than in mathematics invented by the human mind. In place of square root of negative numbers, MG introduces the new concept of contra-square root. In brief this involves substitution of a combination form of interactive two-dimensional analogues of +1 and -1 for -1 as currently used in imaginary number contexts. This is more fully explained elsewhere in the blog.

Put another way, in place of imaginary numbers MG posits the existence of probable numbers. These can be considered the result of what is essentially wavelike interactions of higher dimensional numbers to form the real numbers we know in the 3-dimensional world.

Higher dimensional numbers can interact with one another through wavelike constructive and destructive interference to generate ordinary 3-dimensional numbers. Numbers are not viewed as constants to be acted upon as Descartes so views them but rather as being themselves active and changeable. They participate in process. This feature alone enables composite numbers to mediate between mathematics and physics better than either real or complex numbers can.

The interactions of higher dimensional numbers in the process of dimensional compositing to yield 3-dimensional numbers is a function of time and therefore probabilistic from our limited ordinary point of view. From this perspective, certain probablity distributions are the result of dimensional compositing and the consequent mandalic form. MG considers the probabilistic nature of quantum mechanics likely to be based on such.

The probabilistic nature in three dimensions of what are here called probable numbers is what gives rise to the mandalic form which can in a sense be considered the 3-dimensional evolution of 6-dimensional numbers from protean representations through progressive differentiation of form to the stage of maximal differentiation and back again to the undifferentiated state of greatest probability.

The mandalic form has a geometric progression of its line structures in the three Euclidean/Cartesian dimensions such that series of numbers of the form 1:2:1, 2:4:2, and 4:8:4 occur throughout all of those dimensions when a hybrid 6D/3D coordinate system results from performing 2:1 compositing from six to three dimensions.

Mandalic geometry views points and lines in three dimensions as convenient fictions that exist only as evanescent probabilistic concurrences of analogous entities in higher dimensions.

The probabilistic nature of MG makes it ideal for investigations and descriptions of quantum mechanics.

The exclusion of imaginary and complex numbers and substitution of probable and composite numbers which are easily reducible to ordinary algebraic/arithmetic forms and can be worked with using the same methods as those mathematical disciplines makes MG more utilitarian and appropriate to application to quantum mechanics than are complex numbers. All operations performed are based on simple inversion (reflection through a point) and on real numbers, maintaining all the usual rules and properties of ordinary arithmetic, including commutativity (which quaternions fail to preserve.)

MG is currently based on discrete numbers and is concerned mainly with the positive and negative integers. Fractions and irrational numbers are not excluded from the system but do not currently play a significant role. Future incarnations of MG will extend it outward beyond the unit vector cube to tile the geometric universe and inward to encompass fractional entities and fractals.

It is a hybrid geometry resulting from superposition of 6-dimensional numbers and 3-dimensional numbers and is fully commensurate with 3-dimensional Cartesian geometry.

It describes a linear mapping of two dimensions to one dimension which forms a field of probable numbers over the field of real numbers, analogous to the field of complex numbers but constructed on a different principle and extending to the real numbers in all three Cartesian dimensions rather than just one. The two independent higher dimensions so mapped become dependent variables in the mandalic "line" that results from the compositing of the two. This is expressed, in a sense, as two sine waves 180 degrees out of phase that mutually intersect a common Cartesian axis (x,y or z) at Cartesian +1 and -1 and are maximally separated at Cartesian 0.

This phase difference produces wave interference of both constructive and destructive varieties. So-called "points" or "particles" they represent come into existence only discretely and intermittently at Cartesian -1, +1, and 0, the locations of intersection or confluence (-1 and +1) and maximum separation, the maxima/minima of the two entangled sine waves that occur at Cartesian 0.

As the unit vector cube corresponds to and describes only half of each of the two sine waves, two unit vector cubes are required for a full cycle. Mandalic geometry as currently formulated with a single unit cube then needs to be extended to at least two of these. Extension in both directions of all three Cartesian axes is easily accomplished by repeatedly inverting the current single unit vector cube.

This means that mandalic coordinates alternate positive and negative on both sides of Cartesian 0. The extensions can be continued to infinity in both directions, but not, properly speaking, positive and negative infinity since the manner of extension has created what is essentially a convention-free coordinate system which consists of repeated units of consecutively inverted unit vector cubes in which positive and negative alternate ad infinitum and every Cartesian even-numbered coordinate becomes a "zero equivalent" , or better, a neo-zero in this extended mandalic coordinate system.

The resulting geometry is a dynamic one with "points", "lines", and "planes" coming into and passing out of existence intermittently in a time-sharing of corresponding Cartesian entities. It "persists" in time and space by means of continuous creation, destruction and re-creation and is "held together" by "force fields" produced and maintained by means of tensegrity which is based ultimately on dimension and number, and by a process that.might best be described as a "weaving of reality" with warp and woof.

The 2:1 compositing of dimension involved creates a new number system the members of which are like the real integers in all ways except that they map differently to a Cartesian geometric space. Whereas Decartes assumes that one number maps to one point, MG does not make this assumption which is just an unproved axiom that Descartes makes implicit use of.

The method of dimensional compositing automatically results in a mandalic formation having a geometric progression through three Euclidean/Cartesian dimensions from periphery to center (origin).

Currently MG is limited to a description of unit vectors in a composite hybrid 6D/3D geometry but can be extended to include all scalar values and any even number of dimensions.

The notation system used is borrowed from Taoism and foreign to most Western mathematicians. It is, however, basically equivalent to Cartesian coordinate signs (yin=minus; yang=plus); ordered pairs (=bigrams); and ordered triads (=trigrams); and extends these concepts to include ordered quads (=tetragrams) and ordered sextuplets (=hexagrams).

This notation system is used rather than the usual Cartesian notation because it is much easier for the mind to manipulate dimensional numbers using it. It takes only a little practice to become accustomed to using it. Without its use, understanding of mandalic geometry becomes extremely difficult, if not impossible.

As MG views a point as a concurrence of various different dimensions, it interprets Cartesian ordered pairs and triads, and their extensions to higher dimensions, as tensors and treats them as such. This makes it possible to apply operations of addition and multiplication to these mathematical entities in a manner analogous to the way William Rowan Hamilton applied these operations to complex numbers by way of what he called "algebraic couples".

The probabilistic mandalic form that is the hallmark of MG conveys and necessitates a new interpretation of zero(0). In MG "zero" is not the empty null that it is in Cartesian geometry and Western mathematics generally, but rather a fount of being, so to speak, and a logic gate spanning dimensions. Wherever a zero occurs in Cartesian coordinates two Cartesian-equivalent forms are found in mandalic coordinates. So in the mandalic cube based on unit vectors the twelve edge centers, having a single Cartesian zero, have two Cartesian-equivalent forms (hexagrams); the six face centers, having two Cartesian zeros, have four Cartesian-equivalent forms; and the single cube center, the Cartesian origin point with three zeros, has eight Cartesian-equivalent forms.

This alternative zero and the mandalic structure it inhabits force the creation of four different amplitudes of dimension in the 6-dimensional unit vector cube. These are not independent but all mutually dependent and holo-interactive within the composite 6D/3D coordinate system. All of this occurs in a context reminiscent of the one inhabited by nuclear particles. The mapping proposed by MG may in fact model the elementary force fields, electromagnetism and quantum chromodynamics. It suggests a possible mechanism for formation of the state of matter known as a quark-gluon plasma. Hidden within it may even be the secret of quantum gravity.

© 2016 Martin Hauser

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