# A new way to visualise spread polynomials

(**s**_{a} + **s**_{b} + **s**_{c})^{2} – 2 (**s**_{a}^{2} + **s**_{b}^{2} + **s**_{c}^{2}) – 4 **s**_{a} **s**_{b} **s**_{c} = 0.

By fixing two of the spreads, parametrized in the variable **s**, the formula reduces to a quadratic equation in the third spread. Being quadratic there are two such solutions in the spread variable corresponding, geometrically, to cases where the angle measured by the first spread is oriented positively or negatively with respect to the second. For the simplest case, using **s**_{a} = **s**_{b} = **s**, the TSF becomes:

(2**s** + *S*_{c}(**s**))^{2} – 2 (2**s**^{2} + *S*_{c}(**s**)^{2}) – 4**s**^{2} *S*_{c}(**s**) = 0

=> *S*_{c}(**s**) (*S*_{c}(**s**) – 4**s**(1 – **s**)) = 0,

giving *S*_{c}(**s**) = 0 or 4**s**(1 – **s**).

*S*(**s**) = 0 is indeed the spread when a positive angle corresponding to **s** combines with a negative angle of equal measure, while 4**s**(1 – **s**) (the ‘double angle formula’) is the spread produced when equal, like oriented angles are combined.

Using **s**_{a} = **s ,**

**s**

_{b}= 4

**s**(1 –

**s**), the TSF then becomes:

(**s** + 4**s**(1 – **s**) + *S*_{c}(**s**))^{2} – 2 (**s**^{2 }+ 16**s**^{2}(1 – **s**)^{2} + *S*_{c}(**s**)^{2}) – 64**s**^{2}(1 – **s**) *S*_{c}(**s**) = 0

=> (*S*_{c}(**s**) – **s**)(*S*_{c}(**s**) – **s**(3 – 4**s**)^{2}) = 0,

giving *S*_{c}(**s**) = **s** or **s**(3 – 4**s**)^{2}.

The TSF also gives the ‘solution-set’ of every spread triple where *three* relations, rather than a single spread relation, are considered at once.

If we take the spread **s** plotted as (**s**, 0, 0) and its double angled spread relation 4**s**(1 – **s**) plotted as (**s**, 4**s**(1 – **s**), 0) and allow **s** to run through all values in the interval [0,1] the familiar Logistic Curve is drawn in the unit square in the first quadrant of the *x-y* plane.

The *pair* of spread polynomials (first and second) then combine in the solution of the TSF to form an appropriate *triple*:

1^{st}, 2^{nd} and 3^{rd} { {**s**, 4**s**(1 – **s**)} , **s**(3 – 4**s**)^{2}} or 1^{st}, 2^{nd} and 1^{st} { {**s**, 4**s**(1 – **s**)} , **s**}

each entry a polynomial in **s** which, plotting all three entries at once, describe *co-ordinates*:

(**s**, 4**s**(1 – **s**), **s**(3 – 4**s**)^{2}) or (**s**, 4s(1 – **s**), **s**)

lying on separate curves in three dimensions. The ambient space is referred to as a *phase-space* (sometimes, in 2-d, as *phase portrait*) when the variables plotted represent the effects of changes occurring in some actual space (rotations of angles in 2-d planar space, for example). In the phase-space either solution forms the locus of a curve whose projection in *x-y* and *x-z* planes are respectively the spread polynomials

(**s**, *S*_{n}(**s**), 0) and (**s**, 0, *S*_{n-1}(**s**))

and whose projection in the *y-z* plane is then an implicit curve of the form

(0, *S*_{n}(**s**), *S*_{n-1}(**s**)).

A recursive formula for spread polynomials (obtainable from the TSF) gives, for each new polynomial:

*S*_{n+1}(**s**) = 2(1 – 2**s**)*S*_{n}(**s**) – *S*_{n-1}(**s**) + 2**s**

This provides a means of successively generating the spread triples that make up these separate spread triple ‘functions’ *S*(**s**,n):

(**s**, **s**, *S*_{2}(**s**)).. (**s**, *S*_{2}(**s**), *S*_{3}(**s**)).. (**s**, *S*_{3}(**s**), *S*_{4}(**s**)).. (**s**, *S*_{4}(**s**), *S*_{5}(**s**)).. (**s**, *S*_{5}(**s**), *S*_{6}(**s**))

These functions can either be varied in the spread variable or the indexing parameter ‘n’. When the latter is carried out the successive values

(**s**, *S*_{2}(**s**)).. (*S*_{2}(**s**), *S*_{3}(**s**)).. (*S*_{3}(**s**), *S*_{4}(**s**))…

will all lie on an *ellipse* in the plane *x* = **s**. Developing these ellipses along the *x* direction traces out the surface resembling an inflated regular tetrahedron and called the Ellipson.

Alternatively, by varying the spread parameter, fixing on particular spread triples and indexing on these, the same surface is developed .

### An Animated Ellipson:

The Ellipson, f(a,b,c) = (a + b +c)² - 2(a² + b² + c²) - 4abc = 0