Views of a fourth degree 'chain' and its near-hyperbola .

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The figures in this post show x, (between ±4), with the roots of : ( (x-1) * (x-1/3) * (x+1/3) * (x+1) ) ^ (1/4) .

As a fourth root is taken, there will be four root curves .

The figures in the left three columns show these together .

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Because the value of the function will be zero whenever x equals the opposite of one of the numbers it's paired with,

its roots yield a chain with 'links' between the x values of : -1 & -1/3, -1/3 & 1/3 and 1/3 & 1 .

It is in the nature of these links that they alternate orientation in complex space ;

the center link holding an axial orientation, with curves at (0, π/2, π, 3π/2), (0°, 90°, 180°, 270°) ;

and the links to either side holding an inter-axial, with curves at (π/4, 3π/4, 5π/4, 7π/4), (45°, 135°, 225°, 315°) .

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The limbs extending from this chain are close in shape to being fourth degree hyperbolic, as i define a fourth degree hyperbola ;

please see views of fourth and second degree hyperbolas and circles, (post of 8 November) .

However, they take different bearings in complex space, (axial, rather than inter-axial) .

This continues the alternation of orientations expressed in the chain .

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The links are evenly spaced in this model .

An unshown variant mapping the roots of : ( (x-1) * (x-1/2) * (x+1/2) * (x+1) ) ^ (1/4),

would have unevenly spaced links where the center link was larger than those to either side ;

but would remain very similar in its hyperbolic limbs to the model shown .

This raises the possibility/imaginability that a given n-degree chain may have a number of 'isotope' forms ;

which differed in the size and spacing of their links,

but which could be, (to some extent), inter-applicable within whatever natural situations a chain might apply .

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The figures in the right-hand three columns show a single curve, (as i compose it), along with, (turquoise), its limbs' tending line .

The assembled curve is discontinuous at x=0, where it 'vaults the origin' .

This is done to provide, (from the four options available for each segment),

a form which is symmetrical with respect to its tending line, and whose hyperbolic limbs occupy opposite quadrants .

I feel that this is true to the nature of hyperbolic curves .

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An infinite-degree chain would have an infinite number of root curves, which, taken together, would describe solid surfaces .

If the chain expressed, (as the examples do), a series of unique first degree statements,

the chain would have infinite links, and resemble a string of (∞-1) beads .

If the links were evenly spaced, and the chain constrained in length to the (x) distance between -1 and 1,

(as these examples are), it would appear as a thread of 0 thickness .

If the links were not evenly spaced and sized, but such that the center link was largest,

and each progressively to either side were smaller unto the infinitesimal,

the peaks of the chain might describe a wave function .

A cyclical oscillation, between or including these states, might be imagined in which the tread appeared to 'vibrate' .

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My application of the term 'chain' to these mathematical objects is idiosyncratic,

(as is my use of the term 'hyperbola' ; please be aware that my development of hyperbolas is non-standard) .

I do not consider the limbs of these chains to be exactly hyperbolic, (as i understand them) …

{ except in such cases as the second degree form ( (1-x) * (1+x) ) ^ (1/2), and the fourth degree form ( (1-x^2) * (1+x^2) ) ^ (1/4), which are not shown here }

… but do consider them close relatives of, and analogs for, the hyperbola which may have a place in nature .