Shapes you may stretch and squash but not tear: a loop with one side, what is left when you cut it down the middle, and why a mug and a doughnut are the same shape.
Step 1 of 7 · One side
Step 1 of 7
One side
Topology asks which properties of a shape survive stretching, bending and squashing, without tearing or gluing. Lengths and angles change; the number of sides, edges and holes stays.
Cut a strip of paper and glue its ends into a Loop. The paper has two sides, orange and blue. Predict what will change if you turn one end half a turn before gluing, then look.
Step 2 of 7
Cutting down the middle
When you cut a plain loop along its middle line, two narrow loops are left, each as long as the original. The Half-twists are fixed at one now: a Möbius strip.
Step 3 of 7
Three half-twists
The number of half-twists decides whether the loop has one side or two: an odd number gives one side and one edge, and an even number gives two. What is left after the cut depends on that number too.
Try it: after checking, try two half-twists as well, and cut.
Step 4 of 7
Mug and doughnut
Drag Squeeze the mug from 0, a coffee mug, to 1, a doughnut: the mug squashes into a doughnut without tearing or gluing. The mug’s cup fills in, its body shrinks into the handle, and a doughnut is left. The mug’s only hole, through the handle, stays all the way. So to a topologist a mug and a doughnut are the same shape.
Step 5 of 7
One edge
A half-twist changes more than the sides. Follow the strip’s edge.
Step 6 of 7
A full twist
We have seen what a cut does to a loop with , and half-twists. That leaves .
Step 7 of 7
How long each is
Cut down the middle is on, and the two loops are linked.
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Loading the interactive version…
Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.
Practice · Predict, then look
The loop has two sides: the ant walking the middle line never reaches the other side without crossing the edge. What will happen if you add one half-twist before gluing?
Practice · Predict, then look
The loop has two sides: the ant walking the middle line never reaches the other side without crossing the edge. What will happen if you add one half-twist before gluing?
Practice · Predict, then look
You cut the Möbius strip along its middle line, all the way until you are back at the start. What will be left?
Practice · Predict, then look
You cut the Möbius strip along its middle line, all the way until you are back at the start. What will be left?
Practice · Predict, then look
Now a loop with three half-twists. What will you get when you cut it down the middle?
Practice · Predict, then look
Now a loop with three half-twists. What will you get when you cut it down the middle?
Practice · Choose
A coffee mug has one hole, through its handle, so it is the same shape as a doughnut. How many holes, in topology’s sense, does a glass with no handle have?
Practice · Choose
A coffee mug has one hole, through its handle, so it is the same shape as a doughnut. How many holes, in topology’s sense, does a glass with no handle have?
Practice · Calculate
An ordinary loop, with no twist, has two edges: one at the top and one at the bottom. How many edges does the Möbius strip, a loop with one half-twist, have?
Practice · Calculate
An ordinary loop, with no twist, has two edges: one at the top and one at the bottom. How many edges does the Möbius strip, a loop with one half-twist, have?
Practice · Predict, then look
A loop with Half-twists: , one full twist. It has two sides and two edges, like an ordinary loop. Predict: what will be left when you cut it down the middle?
Practice · Predict, then look
A loop with Half-twists: , one full twist. It has two sides and two edges, like an ordinary loop. Predict: what will be left when you cut it down the middle?
Practice · Calculate
The loop with Half-twists has been cut down the middle into two linked loops. How long is each of them, compared with the original loop?
Practice · Calculate
The loop with Half-twists has been cut down the middle into two linked loops. How long is each of them, compared with the original loop?
Lesson complete. Every practice question here is done.