Math Is Fun Forum

dolgopolov

Member

Registered: 2011-01-26

Posts: 16

teaching euler characteristic of a plane

hello guys and girls,

first of all forgive my english.

i am wanting to teach euler characteristic of a plane in a really fun way to get the kids to discover the identity for themselves.

i thought about getting the students to use paper to draw different shapes in 2D and note different things about them. or should i just say straight out to note down the vertices, the number of edges, etc?

for 3D, i can use those colorful polygon connector things and do the same, no?

what is the best way to teach this in a way that the kids will not find boring?

thank you!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: teaching euler characteristic of a plane

hi dolgopolov

What age are your kids?

I'd start with nets for tetrahedron, cube, octahedron, dodecahedron and icosahedron and get them to make them and discover their properties.

See http://en.wikipedia.org/wiki/Euler_characteristic

Then start an investigation:  (i) Draw a plane shape or network  (ii) count nodes, arcs and regions (iii) summarise results in a table (iv) ask "What do you notice?"

You might look up http://en.wikipedia.org/wiki/Schlegel_diagram as well.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

dolgopolov

Member

Registered: 2011-01-26

Posts: 16

Re: teaching euler characteristic of a plane

hello bob bundy, thank you so much for your help!

they are 15-16 years old.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: teaching euler characteristic of a plane

hi dolgopolov

You are very welcome. 

That's older than I had imagined but I stay with my suggestions.

Do you know the puzzle about 3 houses that need electricity, gas and water pipes.  How can you connect them across the surface without crossing pipes?

Picture below.

At the start of this topic you could set this challenge for your class.

Later, when they assure you it's impossible, you can use topology to show them a surprising answer.

I leave this as a puzzle.  Post back if you cannot find a solution.  He he !:lol:

Bob

Last edited by Bob (2011-01-26 05:24:25)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

pi_muncher

Guest

Re: teaching euler characteristic of a plane

I've never seen this puzzle before, what do you mean by 'across the surface'?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: teaching euler characteristic of a plane

You are not allowed to dig tunnels.

In the real world, of course, that's exactly what you would do; but this puzzle isn't from the real world.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

pi_muncher

Guest

Re: teaching euler characteristic of a plane

What is the way to solve it using algebraic topology?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: teaching euler characteristic of a plane

hi pi_muncher

You want the answer so soon?  I expected at least a few days of thinking ......

Ok. There was a hint in an earlier post so here it is again.

http://en.wikipedia.org/wiki/Euler_characteristic

Oh yes and here's a hint that is useful for many problems, mathematical and real life:

If you cannot find a solution to a problem;  question your assumptions.

Bob

Last edited by Bob (2011-01-26 07:40:51)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

pi_muncher

Guest

Re: teaching euler characteristic of a plane

I'm not the OP, and I'm pretty sure there's no solution by the topic of this thread, Euler's Characteristic of a Plane, that

.

However, I would like to know your solution...how is this possible in two dimensions with algebraic topology?