A tipi is a portable structure made of poles and covered in hides. Native American tipis varied in size and the number of poles used. A 12-pole tipi framework is shown in the following figure. The framework of a tipi lean toward the west so the apex was not centered above the floor. Name the polyhedron formed by the framework of poles and the ground and give the number of faces, edges, and vertices. Does Euler’s formula hold for Native American tipis?
it's just a pyramid.
And, of course Euler's formula holds.
To determine the polyhedron formed by the framework of poles and the ground in a Native American tipi, we can examine the structure of the 12-pole tipi framework shown in the figure.
In this case, the poles form a cone-like shape with the ground forming the base. The poles lean towards the west, causing the apex of the framework to be offset from the center above the ground. This cone-like structure closely resembles a frustum of a cone.
A frustum of a cone is a polyhedron with two parallel circular bases and a curved surface connecting them. In this case, the circular bases are formed by the contact points of the poles with the ground and the apex, while the curved surface is formed by the poles themselves.
Now let's determine the number of faces, edges, and vertices of this frustum-like polyhedron.
- Faces: The frustum has three faces - the two circular bases and the curved surface.
- Edges: The frustum has a total of 20 edges. Each pole connects one point on the base to a point on the curved surface, resulting in 12 edges. Additionally, there are four edges connecting points on the top base to the apex of the frustum, and four edges connecting points on the bottom base to the ground.
- Vertices: The frustum has a total of 14 vertices. Twelve vertices are formed by the contact points of the poles with the ground and the apex. The remaining two vertices are formed by the points where the poles meet at the top and bottom bases.
Now let's consider Euler's formula, which states that for any polyhedron with V vertices, E edges, and F faces, the equation V + F = E + 2 holds.
In the case of the frustum formed by the 12-pole tipi framework, we have:
V = 14
E = 20
F = 3
Substituting these values into Euler's formula, we have:
14 + 3 = 20 + 2
17 = 22
Since 17 does not equal 22, Euler's formula does not hold for this specific frustum formed by the Native American tipi.
However, it's important to note that Euler's formula is a general formula applicable to any polyhedron, and it may not hold for all specific configurations or structures.