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horn torus
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and/or mathematical universe texts

by Wolfgang W. Daeumler, Perouse

preliminary exercise (yet again):

a horn torus shall roll (perform a revolution of its bulge) along the main symmetry axis t, and thereby continuously change its size.

the envelope of all tori is a truncated cone and when size zero is included it is a cone (when projecting into the space of imagination!)

now the linear size of the horn torus shall represent its 'distance' from the 'origin, from size zero, i.e. with this condition

the circumference of the torus bulge has to increase exactly to the unrolled distance on the t-axis when rolling: 2 x Pi x dr = dt

the opening angle of the enveloping cone is then 4 x arctan (1 / (2 x Pi)) = 36,172244316152° (all in the space of imagination only,

within our new space there are no cones and such angles!) - the horn torus now additionally shall rotate around the main symmetry axis:

a point on its surface then describes a certain line (unrolling line, trajectory, cycloid) during rolling and simultaneous rotation.

this trajectory represents an

the most significant of these self-contained Lissajous figures represent elementary particles, fermions, arranged as a 'cascade' within the entity,

whereas lines that are non-closed within one revolution or within a few revolutions of the bulge represent exchange particles, bosons

a bit more complicated:

the condition that the linear size of the horn torus corresponds to the unrolled distance on the main symmetry axis shall now be altered:

the size shall correspond to the length of the total 'unwound' unrolling line, just the trajectory.

the opening angle of the envelope then strictly is dependent on the location of the horn torus within the entity,

in very small tori with a very large ratio of bulge revolution to rotation the angle has a maximum (starts with 360°!!)

this means: close to the 'origin' of the entity the horn tori increase size abruptly (extremely fast revolution → extremely fast increase → 'inflation')

for bigger tori the angle becomes more and more acute, the envelope asymptotically approaches a cylinder, space gets 'flat' and nearly linear

surprising (but already known) interim finding:

all entities show exactly identical patterns and identical dynamic properties when the revolution speed is assumed as constant

what happens when this speed (circumferential speed of the horn tori) is changed globally, i.e. for all (infinitely many) entities?

nothing at all!! the patterns and dynamic properties remain exactly the same, only - viewed from the outside - everything is enlarged or scaled-down.

within the entities and their 'interactions' no change can be noticed because the 'intrinsic scales' are also increased or decreased with the patterns

- perfect self-similarity! - a change in the circumferential speed has no effect, it appears constant

the circumferential speed of the unrolling horn tori we already have identified with speed of light - or if not, we do it hereby.

→

but note: for very small horn tori one revolution is completed within a fractional part of Planck time (= one revolution of unit torus),

so here, below the level of hadrons, either light speed is not defined or bigger than in the 'real world', e.g. during phase of inflation.

Furthermore, don't forget: the horn torus model is a geometrical aid to visualise abstract mathematical entities, complex manifolds -

horn tori do not occur 'in our nature', but numbers and manifolds do:

the universe is mathematical

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