Stretching Gifted Students with an Extension of Euler’s Polyhedra Formula

Anyone who knows anything about Mathematics knows Euler’s Formula connecting the number of faces (F), edges (E) and vertices (V) of polyhedra.

F – E + V = 2

 (A cube for example has 6 faces, 12 edges and 8 vertices so that 6 – 12 + 8 = 2).

But what if the shape has a hole in it? (Click the picture to go to the file)

Image1

Does the formula change? If so how does it change? And the most important question, why does it change?

Further to this, what if the the shape has 2 holes, 3 holes, …, n holes? How does this affect the formula?

Advertisements
This entry was posted in Algebra, Investigations, Shape and Measures and tagged , . Bookmark the permalink.

2 Responses to Stretching Gifted Students with an Extension of Euler’s Polyhedra Formula

  1. The Dude says:

    Hello good sir,

    the way you drew the shape, doesn’t the formula hold as is ? Wouldn’t you need to put edges linking the vertices of the “hole” to the vertices of the “original prism”, much as in

    for the formula to become
    F – E + V = 0
    or
    F – E + V = 2 – 2n
    where n is the number of holes.

    I’m maybe missing something.

    Good day to you!

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s