User:Garfungled/sandbox

In geometry, the Value of a Cube is a numerical representation of a cube using Values of its vertices, edges, or faces. It can be used to distinguish cubes given values, arbitrary or not, for its features. It's also a mathematical problem given in math competitions such as the AMC.

Definitions
When evaluating a cube, it is important to first understand how the evaluation works. The specific definitions can be found below, however, broadly, the Value of a Cube is the sum of the Values of its faces. Then, the value of the faces is the sum of the value of its defining edges. Finally, the value of the edges is the sum of the two vertex values that make up said edge. Therefore, if $$S$$ is defined as the Value of a Cube then,


 * $$S = f_1 + f_2 + f_3 + f_4 + f_5 + f_6$$
 * $$\text{s.t.} \; f_n = e_k + e_i + e_j$$
 * $$e_n = v_i + v_j$$,
 * $$i, j \in \mathbb{Z}$$,

where the set of face values is $$f$$, set of edge values is $$e$$, set of vertex values is $$v$$, and $$i, j$$ represent some arbitrary position chosen for each feature in the cube (see visual in figure 1.1).

Values
Values can be extended to the lower parts of a cube, that being its vertices, edges, and faces. A Value is a real number used to represent any given object or element of a list. For any Value, there should be a corresponding element of the list, functionally mapping the value to the element. Therefore, a new set can be created using the values. Such a set is named the Value Set, which is formally defined as,


 * $$\forall X = \{x_1, x_2, ..., x_n\},$$
 * $$\exists V_{s} \; \text{s.t.} \; V_{s} = \{g| x \mapsto g\}$$,

where $$X$$ is some set with $$n$$ elements, and $$g$$ is the mapped values for each $$x$$. Value Sets can be mapped via elementary operations, functions, or through definition.

Value of Vertices
The Value of a Vertex can be defined in any predetermined way. This is because the Values will further determine other values, therefore, it can be thought of as the independent value for the Values in a given shape. Therefore, for the set of vertices in any given shape $$v$$ of length $$n$$, the Value Set is defined as,


 * $$v = \{v_1, v_2, ..., v_n\}$$
 * $$V_{v} = \{g|v \mapsto g\}$$.

Thus, the Value set for the vertices in a shape have no distinct rule, only that, each value, $$g$$, must map to their corresponding vertex, $$v$$.

Value of an Edge
The Value of an Edge is defined to be the sum of the Values of the vertices that make up said edge. By the definition of an edge, there are exactly two vertices that make up an edge since an edge is two vertices conjoined by the edge, or line segment. Therefore, if $$v_k$$ and $$v_j$$ are the values of the vertices that make up edge $$e_n$$, then the value of the edge, $$g_n$$, is said to be,


 * $$g_n = v_k + v_j$$

From here, the Value set of edges is defined as so


 * $$e = \{e_1, e_2, ..., e_n\}$$
 * $$V_{e} = \{g|e \mapsto g\, g = v_k + v_j\}$$.

Value of a Face
The Value of a Face is defined to be the sum of the Values of the edges that make up said face. For a face, $$f_n$$, with $$n$$ edges, the Value of the face, $$g_n$$ is defined to be,


 * $$g_n = e_1 + e_2 + ... + e_n$$,

where the set of Values for the edges that make up face $$f_n$$ is $$e$$.