Number of edges of a convex polytope with n vertices

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Homework Help Overview

The discussion revolves around determining the number of edges in a convex polytope with n vertices, specifically when all faces are triangular. Participants are exploring the relationship between vertices, edges, and faces using Euler's formula.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to apply Euler's formula and are questioning the logic behind their calculations. There is discussion about the relationship between edges and faces, with some participants suggesting different interpretations of the formulas.

Discussion Status

Some participants have provided guidance on the correct application of the formula, while others are exploring specific examples to clarify their understanding. Multiple interpretations of the relationships involved are being discussed, indicating an active exploration of the topic.

Contextual Notes

Participants are working under the assumption that all faces of the polytope are triangles, and there is a focus on ensuring the correct application of Euler's formula in this context.

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Homework Statement



What is the number of edges of a convex polytope with n vertices all of whose faces are triangles.


Homework Equations



# of faces + #of vertecies = # of edges + 2

The Attempt at a Solution



My reasoning is as follows:

n/3 + n = # of edges + 2

4n/3 - 2 = #of edges

This doesn't seem right though. Am I using the correct formula?
 
Last edited:
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Right formula. Wrong logic. Each edge meets two faces. Each face meets three edges. What's the relation between edges and faces?
 
3/2 * faces = edges?

Is the result

3/2*n - 2 = # of edges by any chance?

I just did

3/2*1/3*n + n = e + 2
 
Last edited:
Yes. Three times the number of faces equals twice the number of edges. If you have to do this again pick some example like tetrahedron E=6, V=4, F=4. Octohedron E=12, V=6 and F=8. It's a good sanity check.
 
That's actually how I found it. I drew a little tetrahedron with 4 vertecies, 6 edges and 4 faces. I don't know why but i just couldn't wrap my mind around it without a drawing. Thanks again for your time.
 

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