Views of a fourth degree hyperbola & its aperture circle, compared with a second degree hyperbola and circle .

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Mathematically, the circle and hyperbola can be modeled as expressions of the same formula,

(which may vary in the order and number of parameters used) .

For the figures in this post, i plotted x against the roots of :

( 1 - x ^ degree ) ^ ( 1 / degree ) .

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When degree equals 2, the formula's two roots produce a classic, round-nosed hyperbola, with a classic, round circle filling its aperture .

These are shown in the bottom two rows along with, (in the left-hand and center figures), the hyperbolic limbs' tending lines .

When degree equals 4, the formula's four roots produce a squarish-nosed hyperbola of four limbs,

with a similarly squarish circle of four 'sections' filling its aperture .

These are shown in the columns [above the text] : all four root curves between the X limits of ±6 in the left-hand three ;

and a single curve, (as i compose it), with its limbs' tending line, in the right-hand .

In all figures where more than one root curve is shown, the color-coding indicates how i believe they may run as individuals .

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I assemble the individual root points into curves discontinuous at 0, where each executes a leap across the origin .

The reasons are as follows :

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First, it provides a way, consistent for both second and fourth degrees, that the curves will be symmetrical with respect to their tending lines ;

something i believe is fundamental to the hyperbola's understanding .

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Looking at the second degree alone, (shown below with root curves which vault the origin),

one could imagine a more reasonable symmetry with respect to their, (turquoise & magenta), tending lines without such a jump …

a symmetry where the positive and negative circular arcs were described by uninterrupted curves, rather than exchanged ones.

But, the second degree is a special, to some extent 'tolerant', case in which the hyperbolic limbs are flat on the XI plane ;

the aperture circle thus on the XR plane ; and where the circular and hyperbolic arcs meet at a π/2, (90°), angle .

Either the uninterrupted-arc symmetry or the exchanged-arc symmetry could be applied here,

(the latter provided that curves which vault the origin are a valid composition) …

but the more powerful method will be that which also applies to the fourth and higher, (even integer), degrees .

Study of the fourth degree curves, above, shows that only by vaulting the origin can they become symmetrical with respect to their tending lines ;

so i apply this composition to the second degree also .

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An alternative symmetry, based upon using a plane parallel to an axis at the origin as a figurative mirror, would work for second and fourth degree hyperbolas ;

(reflecting both the curves and their tending line).

However, because it would not work for the third, (previous post), and fifth degrees … as these are not symmetrical from zero, but from their centers of curvature,

and as their lines of symmetry, (the limbs' tending line), are not parallel to an axis … i consider it a weaker approach .

And, characteristic of the third, (and other odd-integer degree hyperbolas), is that the limbs cross,

through the zone of aperture, to their respective opposite quadrants .

By composing even-integer degree hyperbolas as symmetrical to their, (unreflected), tending lines, that crossing can be preserved as a trait common to all integer degrees .

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Further, if the model of the second degree hyperbola and circle apply to the photon, (as i hope they do, please see post of 14 December 2010),

a mathematical vaulting of the origin, and peculiar properties which might be associated with it, could help account for some of the photon's behavior .

If this has validity, the reasoning might be extended to model the electron as based, (in part), upon an even-integer-near-infinite degree hyperbola and circle .

Between these models, the 2:1 ratio of the photon's and electron's 'spin' property might be understood as follows :

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Looking at the bottom row, (a second degree system, from left to right), one sees that the limbs of the hyperbola make a π - π/degree, (90°),

corner with the arcs of the circle at the origin of the, (Re, Im), complex plane .

If π, (180°, the angle between a hyperbolic limb and its coplanar, opposite quadrant sister-limb within that root curve), is divided by this, the result is 2 .

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Looking at the bottom figures of the first, third and fourth columns, (a fourth degree system),

one sees that the limbs of the hyperbola also make a π - π/degree, (here 135°), corner with their 'sections' of the circle, (as i have composed them) ;

an angle more oblique than that for the second degree .

If one were to imagine an even-integer-infinite degree hyperbola, the angle between a hyperbolic limb and its 'circle' arc, (again, by this method), would be π - π/∞ .

If π is divided by that angle, the result will be infinitesimally greater than 1 .

Comparing this with the quotient from the second degree hyperbola, (exactly 2), one gets a ratio which is very close to the 2:1 sought for these spins .

If this model is correct, and if this aspect is applicable to the spins of the photon and electron, their ratio should be very, very slightly, (perhaps infinitesimally), less than 2:1 .

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Please be aware that my development of n-degree hyperbolas is non-standard ;

thus my use of the terms, "{second, third, fourth, fifth} degree hyperbola", and/or "circle", to describe these systems

may conflict with other uses of the same terms by other writers describing different mathematical objects .

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♣   postscript, 10 November 2011   ♣

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An aspect of the composition, the choreography of the root curves which i didn't stress, but which i feel is important ……

For even degree hyperbolas and their circles,

the complex plane angle between a hyperbolic limb and its circle half-section should be that from among the available options which is closest to π, (180°, straight) .

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For the second degree, the difference between the options { ±π/2 }, (±90°), is moot, as they are equally close .

For the fourth degree, from the options { ±π/4, ±3π/4 }, (±45°, ±135°), i imagine the symmetry being based upon the 3π/4 option ;

though in the display figures, i opted for the, (clockwise rather than counterclockwise), -3π/4 angle .

For an eighth degree, from the options { ±π/8, ±3π/8, ±5π/8, ±7π/8 }, i imagine the symmetry using the 7π/8 angle … etc.

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Such an arrangement is not essential to the curves' symmetry from their center lines, (so long as they 'vault the origin', and so long as the same option is applied to all curves) …

but it is essential to the, (i hope accurate), representation of spin .

It also makes sense to me that the hyperbolic limb and circular section will be joined which, between them, have the flattest complex plane angle .