I’m waiting at Terminal 4 S in Madrid, for my flight to Santiago de Chile. And I have here some pictures of a simple and nice excercise that I would like to share with all readers of this blog. It is about classifying a special surface with just two strips of paper, in two different ways.
STEP 1: Take two strips and glue them as follows: a Moebius strip (the pink one) penpendiculary with a normal strip (the yelow one).
This is a model of a Klein bottle with one hole! Yes!! If you follow the boundary, it has just one component, the hole (if you fill it with a disk, you would obtain a Klein bottle). You can prove this in two ways.
Let’s first do it with a standar method oftenly used. We compute its Euler characteristic, for example taking 3 faces: the square with its four edges, and two more rectangles glued to the edges of the square. Hence:
X = vertices-edges+faces= 4 – 8 + 3 = -1
Equating X to 2-g-1 (the Euler charactacteristic of a non oriantable surface of genus g, with 1 boundary component) you get g= 2. This corresponds, as we know by the classification of surfaces, to a Klein bottle with one hole.
Ok, but is there a way to see that you have in your hands a Klein bottle with one hole?
STEP 3: Then turn completely the yelow strip, until the pink ribbon goes through the yellow one (this is the reason why the yellow strip is crumpled, sorry).
Let’us see that this is really a Klein bottle with one hole. Let me replace these crumpled strips by new ones of rubber.
Put one of them “through” the tube of a Klein bottle, like in the next picture.
It would be great to have a movie showing how the little hole, where the bottle intersects to itsself, deformates until the two strips.
This idea came up by correcting an excercise of one of my students, Juanjo. I liked its way of gluing the two strips and how this allows us to visualize easily a Klein bottle with a hole.
Added on October 28th: At the end of STEP 2, one can also see the two strips as part of a Klein bottle (Thanks to Antonio Viruel to figure it out to me). Here is the picture.