WEIRD SHAPES! J.P. McCarthy, CIT & UCC.

Sheldon’s Joke: Why did the chicken cross the Mobius Strip? …to get to the same side…

Making a Mobius Strip! What do I Need? Each pupil needs a few sheets of paper. Each pupil needs a scissors. Each student needs a pencil. I have loads of strips of Sellotape here --- Muinteoir Rebecca has more.

Start with a Simple Loop Cut an A4 sheet lengthways into four strips. Be precise! Now take one strip and make a loop. Put the two ends together and tape the paper together this way, being sure to tape across the entire strip.

Activities with the Simple Loop Draw a line down the middle of a strip, continue drawing until you meet up with your starting point. How many sides does a loop have? What do you think will happen if you cut along that line?

Spice up the Simple Loop Now we will make a Mobius Strip. Use one strip of paper. Put the two ends together, and give one end a half-twist. Tape the paper together this way, being sure to tape across the entire strip.

Ants on the Mobius Strip! Draw a line down the middle of the strip, continue drawing until you meet up with your starting point. Did you draw this line without lifting your pencil from the paper? Could an ant walking along your line walk until he met his starting point, without walking over an edge of the paper?

Cut the Mobius Strip in Two… How many sides does a Mobius strip have? What do you think will happen if you cut along that line? Cut along the line. What happens? What would happen if you cut in half again? Try it and see.

Mobius Handcuffs Make another Mobius Strip. Cut the new Mobius strip one-quarter of the way in from one edge. The cut will meet itself eventually.

Uses of Mobius Strips

Enter Euler! Nearly 300 years ago there was this Swiss mathematician called Euler Euler was a serious workaholic: when he lost his right eye he said “Great! Now I will have less distractions!!”

Euler’s Shapes Euler studied many things including shapes and discovered that all obeyed a formula: P - E + F = 2 P is the number of ‘points’, E the number of edges and F the number of ‘faces’

Example 1: Pyramid How many Points? How many Edges? How many Faces? So P – E + F = 5 – 8 + 5 = 2!

Example 2: Mystery Box How many Points? How many Edges? How many Faces? So P – E + F = 8 – 12 + 6 = 2!

Exit Euler! Poor Euler spent a lot of time showing why his formula was always true but 100 years later the Mobius Strip was discovered. How many Points? How any Edges? How many Faces? So P – E + F = 0 – 1 + 1 =0… NOT 2!

A New Way? Later on a mathematician came up with a correction to Euler’s Formula: P – E + F = 2 – (2xH) Can anyone guess what H stands for?? So for a Mobius Strip P – E + F = 0 = 2-(2x1) = 2 – 2H

Block Example What is P? What is E? What is F? So P – E + F = 16 – 32 + 16 = 0! …equal to 2 - 2H = 2-2 = 0

Cavities rather than Holes… What is P? What is E? What is F? So we have P – E + F = 16 – 24 + 12 = 4 How many holes though? Does this equal 2-2H?

OK Holes AND Cavities ? So another correction to Euler’s Formula: P – E + F = 2 – (2xH) + (2xC) (write this formula down!) This 2-2H+2C is called the Euler Characteristic Can anyone guess what C stands for?? ?

…NOPE Basically Mathematicians can make things as complicated as they want and some of the shapes they talk about we can’t even draw a picture of… OK a competition over the next few pages… raise your hand if you know the answer.

Raise your Hand if you Know If the Euler Characteristic is equal to P – E + F and is also equal to 2- 2H + 2C, what is P – E + F equal to for this shape:

Raise your Hand if you Know What is the Euler Characteristic of

Raise your Hand if you Know What is the Euler Characteristic of

Raise your Hand if you Know What is the Euler Characteristic of

WEIRD SHAPES! J.P. McCarthy, CIT & UCC.