STATUS: Draft


Solving Polynomial Equations (18)


Aim: Understand the formal object that is a sum of all polygons with all of their subdivisions into triangles


Let $F1$ for a function that implements $C$.

Let $P3$ be a matrix of values retlating the part of the formula that calculates unsigned coefficients: $ \frac{(2 m_{2} + 3 m_{3})!}{(1 + m_{2} + 2 m_{3})!m_2!m_3!}$

Observe: It is possible to create operations such as addition and multiplication on newly defined types

Let $P$ be an ongoing sum of the of polygons of increasing sizes arrangeed in different combinations of triangles, expressed as

$$ P = | + P \bar{\triangledown} P $$

Observe: Note the $|$ operator here represents a 1-gon and the $\bar{\triangledown}$ operator is yet to be defined

Ref: This is taken from Concrete Mathematics (Ex. 22, Ch7) as part of a question to devise a new operation to explain the relationship between the solutions to a polynomial equation and the subdivisions of a polygon.

Observe: This conjecture introduces both a new type, $P$, and a new operation, $\triangledown$

Observe: This conjecture deals with the sum of the number of ways of subdividing a 2-gon into triangles + the the number of ways to subdivide a 3 gon into triangles + the number of ways ot subdivide a 4-gon into triangles and so on.

Observe: A way to make sense of this is take a multiset point of view

Definition: A Multiset is a data structure, part of the different available data structures including lists, ordered lists, multisets and sets.

Definition: Consider the following example:

  1. Let F1 be a 5-gon with a distinguisd edge, subdivided in triangles
  2. Let F1 be a 6-gon with a distinguised edge, subdivied in triangles
  3. Join the distinguised top of the two polygons join to the free edges of the $\bar{\triangledown}$ operator.

Observe: This creates an 10-gon. Note the distinguised (top) edges get glued to the free edges of $\bar{\triangledown}$.

Observe $\bar{\triangledown}$ is a multiplication with a trianglular subdivision and distinguised top edges

Aim: Create a rooted planar tree version of this

Observe: The trees can be joined to the existing edges of the $\bar{\triangledown}$ operator which is represented as the top tree with root $R$

Observe: Recall the above conjecture:

$$ P = | + P \bar{\triangledown} P $$

Observe: For a polygon with that an be subdived by traingles with distinsuished top edge, there can be seen a distinguised interior triangle with that top edge (seen in the $\bar{\triangledown}$ operator.

Observe: Every polygon that can be subdivided into triangles with a distinguised top edge is the product of 2 unique such polygons, one on left, and one on the right, with the exception of the degenerate case (2 sided polygons)

Observe: Therefore $ P = | + P \bar{\triangledown} P $ yields all combinations + addition of $|$ to account for the 2 sided case.

Summary: The above has only defined an operation that is possible to undertaken in relation to polygonal subdivision. More work is required to be able to compute with this operation


Aim: Understand this in the context of multiset data structures

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Definition: a multiset is an unordered collection with repetitions allowed

Observe: The $\bar{\triangledown}$ has been defined as an operation on planar polygon subdivisons, or planar rooted complete binary trees

Observe: The $\bar{\triangledown}$ binary operation is is not commutative and is not associative but can be extended to multisets

Observe Tree like structures using this operator

Observe: This structure naturally has addition and natural number multiplication.

Observe: Trees can be combined in the manner above.

Observe: this can be written as $2T_1 + T_2 \bar{\triangledown} T_3$

Observe: This represents an algebra of MSets.

Todo: Find a way to do this computationally