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of Polyhedral Solids with a Cavity |
The regular polyhedra were known at least since the time of the ancient Greeks. The names of the more complex ones are purely Greek. But despite their being known for close to two millenia no one apparently noticed the fact that the sum of the number of faces F and the number of vertices V less the number of edges E is equal to two for all of them; i.e.,
The value of (V-E+F) is usually denoted by the Greek letter Chi (Χ). Thus Χ(cube)=2.
It was the Swiss mathematician Leonhard Euler who recognized and published this fact. The value of two is said to be the Euler characteristic of each of the polyhedra. This value is not changed by stretching or shrinking any side or face or even shrinking a side or face to zero. This means that the Euler characteric is a topological invariant because it is not altered by any continuous mapping.
AnotherSwiss mathematician, Simon Lhuilier (L'Huillier) (1750-1840),of French Hugenot background, found a slight generalization of Euler's formula to take into account polyhedra having holes. Lhuilier's formula is
where G is the number of holes in the polyhedron. Thus the Euler characteristic is 2 for a regular polyhedron but 0 for a torus-like polyhedron.
This is elegantly simple result. The following material is an extension of the Euler and Lhuilier formulas to polyhedral solids. The language is ambiguous in this matter. The term cube in ordinary discourse refers to a cubic solid whereas for the Euler formula it refers to the surface hull of the cubic solid. For the case of sphere there is an alternate term, ball to use for the solid leaving sphere to apply for the surface. For the other figures there are no convenient alternate terms and therefore terms like cubical solid and cubical surface must be used in stead.
The Euler Poincare characteristic for a polyhedral solid will be denoted as Χ_{3} and defined as
where C is the number of solid cell components in the figure.
It is most convient to use a two-dimensional polygon to illustrate the difference between a hole and a cavity. Consider a solid square. The Euler characteristic for a solid two dimensional is denoted as Χ_{2}.
For a solid square:
In contrast a square with a square cavity is as shown below.
Thus a hole connects with the exterior of the figure within the space of the figure. For a cavity there is no connection with the exterior within the space of the figure.
The Euler characterist for the solid square is V-E+F=4-4+1=1. For the solid square with a hole the Euler Characteristic is V-E+F=8-8+2=2. For the square with a cavity a decomposition gives 8 vertices, (4+4+4)=12 edges and 4 faces so its Euler characteristic is 8-12+4=0. Thus the effect of one hole is to increase the Euler characteristic by 1. The effect of a cavity is decrease the characteristic by 1.
Two holes result in 12 vertices, 12 edges and 3 faces so the characteristic is 3. If G is the number of holes then the two dimensional euler characteristic Χ_{2} is 1+G.
For two cavities the decomposition is a bit more complex but the end result is 12 vertices, 12+4+2=18 edges and 5 faces for a characteristic of Χ_{2}=12-18+5=-1. Thus one more cavity reduced the characteritic by 1. For a solid square with H cavities the Euler characteristic would be 1-H. Presumably a solid square with G holes and H cavities would have a characteristic of 1+G-H.
The testing of this formula for G=1 and H=1 is shown below.
For the decomposition shown V=8+4=12, E=4+4+4+4=16, F=4+1=5. Therefore Χ_{2}=12-16+5=1, which is the same as 1+G-H=1.
It is convenient to work with prisms; i.e., polyhedra which have top and bottom faces which are the same polygon. Let N be the order of the polygon. This means that the number of edges and the number of vertices of the top and bottom faces are both N. Each edge of the polygon corresponds to a rectangular face on the side of the prism. Thus the number of faces on the sides of the prism is N. These together with the top and bottom polygons means that the number of faces is N+2. There is a side edge for each vertex of the top polygon and 2N edges for the top and bottom polygon. Therefore the number of edges is 3N. The vertices of the prism are the same as the vertices of the top and bottom polygon. Therefore the number of vertices of the prism is 2N. There is only one cell for the prism.
When these values are substituted into the formula Χ_{3} the result is
The reason for choosing to work with prisms is that it is simple to deal with holes for them. Consider what happens if a prismatic cavity is created in the interior of the prism. The subdivision of a cube is shown below. (This figure looks like what is usually shown for a tesseract, a four dimensional cube. This figure just depicts a cubic hole in a three dimensional solid cube.)
The count for the decomposition shown is V=8+8=16, E=12+12+8=32, F=6+6+12=24, C=6. Thus
This value is in contrast to the value of 1 for a solid cube and the value of 0 for a solid cube with one hole. Thus a hole decrements the characteristic by 1 and a cavity increments it by 1. The characteristic of a solid cube or any other solid polyhedron with G holes and H cavities is most likely to be Χ_{3}(P_{3}(G,H))=1-G+H. (The notation P_{n}(G,H) is for an n dimensional polytope having G holes and H cavities of n dimension.) The signs of the coefficients are reversed compared with the expression for Χ_{2}(P_{2}(G,H))=1-G+H.
Consider first the change in the Euler-Poincare characteristic of a prismatic surface with one hole. For a polygon with N sides the top of the subdivision creates N faces where only one was before. Each one of these subdivisions corresponds to a cell component below it. Thus there are N cells where there was only one before. This means the net increase in the number of cells is (N-1) at the top and (N-1) at the bottom.. The N subdivision faces along with N rectangular sides of the hole and the N subdivision faces at the bottom of the prism plus the N faces separating the N subdivision solids means that the number of faces is increased by ((N-1) + N + (N-1))=3N-2. The number of edges is increased by N around the top of the hole plus N from each vertex of the hole to the corresponging vertex around the top face of the prism. There are N edges down the sides of the hole and N edges of the hole at the bottom face plus the N edges from the vertices of the hole to the vertices of the bottom face. Thus the increase in the number of edges created for the prism by the hole is (2N + N + 2N)=5N. The number of vertices is increased by N around the top of the hole and N around the bottom of the hole for a total increase in the number of vertices of 2N.
The change in the Euler Poincare characteristic for a prismatic surface with a hole is therefore
This means that
For a prismatic solid there are N solids created around the hole where there was only one solid before, for a net increase of N-1. Additionally there are the N faces of these solids. This means that
This means that since the Euler-Poincare characteristic for a prismatic solid is +1,
For more on the Euler-Poincaré Characteristic of geometric figures see Euler.
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