* With the horn torus model we leave 'classical mathematics'. Though we pragmatically make use of well established skills in three dimensions
to illustrate the dynamic, the model is not dimensional - neither Euclidean nor non-Euclidean, neither related to any vector space
nor even connected to Hilbert spaces and the respective formalism - at least not in an obviously predominant way. Horn tori, in particular
the 'small' ones, induce an extremely dynamic and complex geometry and simultaneously - the 'big' ones - are capable to form a nearly static
'flat space' with Euclidean rules when approaching 'infinite size'. Conventional methods are insufficient or too labourious to describe such
a hybrid of dynamic and static space properly and we maybe have to await further progress in quantum computing for an appropriate approach
and adequate treatment. Then mathematics surely will experience changes, and surprises in physics cannot be ruled out. But note: here and
now, on these pages, we only play a preliminary game!|
( following quotations from the game are W.D. 's very personal views and don't claim to lead to any enhancement in comprehensible science :-)
" ... despite I am an ardent apologist of the mathematical universe idea, I am deeply convinced that the known and widely accepted classical mathematics does not reveal the secrets of physical reality. Our mathematics is incomplete, doesn't provide the unique code that rules our world, and on the same time it contains a lot of ballast that is adverse to physical interpretation. A physically relevant complete mathematics has to comprise the mentioned code and has to be self-consistent, without contradictions, regardless of Gödel. To obtain such a mathematics, the classical formalism has to be extended significantly, but for self-consistency simultaneously knocked into proper shape in many respects. Physical laws have to be 'true' and 'real' and a priori valid, not invented by men but discovered, while mathematical rules often don't comply with these conditions in a consistent way. Too many axioms have been established intuitionally during the long history of mathematics. Physical reality is not based on axioms and also does exist in absence of human consciousness, then admittedly without all the properties we perceive, associate and attribute to 'nature'. Physical reality, even when not observed, at least exists as a kind of mathematically representable pattern. These patterns and/or 'structures' don't contain such familiar 'self-evident' constructs like three-dimensional space, round spheres, straight lines, massive objects, ..., they don't distinguish hard and soft or free of mass, don't know heat, colours, locomotion, velocity, ..., they are neither dark nor bright, neither loud nor silent, ..., they don't discern cause and effect, don't feature any order or rule, not even 'physical variables', ..., but all works! These patterns and structures are inevitable synergetic effect of fundamental entities, which exist from the 'earliest beginning' of our universe and which interact with one another in a simple, deterministic, fantastically intertwining manner ... "
" ... there still are physicists who believe in - e.g. - the 'simple' Bohr atom model, only a bit alterated by quantum mechanical interpretations. I don't. I never and nowhere can recognize small spherical bodies that are orbited by tinier ones on paths within different shells with certain velocities. Even the introduction of probabilities of presence at certain locations for any 'particle' within an orbital isn't really an abstraction from the old image. I know, the concept is enormously effective, but yet I prefer to see - as an image too - series of trajectories, 'unrolling' at one another, and thereby to identify 'uncoiling' complex numbers, forming complex superpositions as computable regular patterns. This unrolling process - as a visualisation of complex numbers - is describable analogously by nested or interlaced horn tori, performing revolutions around the bulge, which represent the imaginary part, and rotations around the main symmetry axis, the real part. Place of 'interaction', where a horn torus unrolls on a trajectory, always remains the center, the 'spatial point', to which any 'particle' in the universe is connected as one portion of its individual 'entity'. It is a simple, descriptive, reliable mechanism ! ... "
( here a very condensed quintessence of the texts which you can find on this website :)
" ... in the horn torus model all infinite many 'particles' of the universe are represented in every 'spatial point'. Every horn torus shares the common tangent with all other horn tori, when they are nested into one another at their centers. Size of a horn torus symbolises the 'distance' to the location, where the associated entity (the dynamic coordinate) converges to size zero. Different 'spatial points' differ in the combination of horn torus sizes. All combinations with natural numbers of sizes are possible, forming an infinite-dimensional regular pattern. All paths through this discrete pattern (space) are equally possible, and every horn torus can unroll at a trajectory formed by any other horn torus as interaction at any 'spatial point'. This space is by its definition a multiverse (due to infinite many paths) with non-local correlations between 'particles'. A limited neighbourhood of every 'spatial point' contains the complete information about one selected universe. With this property one immediately recognizes the possibility that big bang has not started out of one tiny spot but rather took place all-over a preexisting just one-dimensional infinity ... " (curious? continue!)