e+7=f
What is your question?
e = f - 7
A convex polyhedron is defined as a solid with flat faces and straight edges so configured as to have every edge joining two vertices and being common to two faces.
There are many convex polyhedra, only five of which are considered regular polyhedra. Regular polyhedra satisfy three criteria. All the faces are congruent. All faces have the same number of edges equal to, or greater than, three. Each vertex joins the same number of edges. The five regular polyhedra are the tetrahedron, cube, octahedron, dodecahedron and the icosahedron.
Euler's famous equation, v - e + f = 2, applies to all convex polyhedra where v equals the number of vertices, e equals the number of edges and f equals the number of faces.
Given the number of edges of a regular polyhedron, is it possible to determine the numbers of faces and vertices knowing that v - e + f = 2. It is possible given that the given number of edges does, in fact, represent a real convex polyhedron in the first place. Given just any number, there will not always be a polyhedron with that number of edges.
The smallest number of edges possible is 6 for the 4 sided tetrahedron consisting of 4 equilateral triangles joined along their edges to form a 3 sided pyramid. Not knowing the form of the polyhedra that contains the 6 edges, it is possible to derive the specific polyhedron having 6 edges. Given that v + f = e + 2 = 8, the only possible pairs of v and f are 3-5, 4-4, 5-3. If each face has m edges, then mf = 2e = 12. Similarly, if each vertex joins n edges, nv = 2e = 12. "m" must be 3 or greater leaving us with possible m's of 3, 4, or 6 and faces of 4, 3, or 2. Having already shown that "f" must be 5, 4, or 3, we are left with 4 or 3 as the only possible values for "f". Clearly 3 faces cannot form a closed convex polyhedron, leaving us with m = 3 and 4 faces. Knowing that we have 4 faces, "v" is also 4 and our polyhedron with 6 edges has 4 faces and 4 vertices, the tetrahedron.
The equation e + 7 = f represents a mathematical relationship between three variables: e, 7, and f. To solve this equation, we need to isolate the variable f, which means we want to express f on one side of the equation by itself.
To do this, we will subtract 7 from both sides of the equation:
e + 7 - 7 = f - 7
The 7 on the left side cancels out, so we are left with:
e = f - 7
Now, we have expressed f in terms of e. If you have a specific value for e, you can substitute it into the equation to find the corresponding value of f.
For example, if e = 5, we can substitute it into the equation:
5 = f - 7
To isolate f, we can add 7 to both sides:
5 + 7 = f - 7 + 7
12 = f
Therefore, when e = 5, f = 12.