Donut Has A Hole
Hold up a donut. Now hold up a coffee mug. They look nothing alike -- one is round and sweet, the other has a handle and holds hot chocolate! But a special kind of math person called a topologist would say they are actually the same shape. Why? Because both of them have exactly one hole. Topologists don't care about size or color or shape -- they only care about holes.
What You'll Learn
How to count the holes in everyday objects. Why a donut and a coffee mug count as the 'same shape' to a topologist. How squishing and stretching (without cutting or tearing) can turn one shape into another.
Counting Holes
A donut has 1 hole, right through the middle. A button usually has 2 or 4 holes for the thread. A pretzel, if it's the twisty kind, can have 3 holes. A basketball has 0 holes -- it's a solid ball shape with nothing going all the way through it, even though the surface has little dimples. The trick is asking: does a hole go all the way through the object? A bowl looks like it has a hole on top, but that's just a dent -- nothing pokes all the way through the bottom, so a bowl actually has 0 holes, just like the basketball.
To check if something really has a hole, imagine poking a pencil straight through it. If the pencil comes out the other side without breaking anything, that's a real hole! If the pencil just gets stuck, it's a dent, not a hole.
Stretchy Shapes
Topology is sometimes called 'stretchy geometry' or 'rubber sheet geometry,' because topologists imagine shapes are made of squishy clay or stretchy rubber. If you could smoosh a donut without tearing it or poking new holes, you could slowly reshape it into a coffee mug: the hole in the donut becomes the hole in the handle! But you could never turn a donut into a basketball this way, no matter how much you squish -- because a donut has 1 hole and a basketball has 0. To change the number of holes, you would have to tear the clay or pinch two parts together, which topology doesn't allow.
Match each object to how many holes it really has.
Terms
Definitions
Drag terms onto their definitions, or click a term then click a definition to match.
According to topology, why are a donut and a coffee mug considered the same shape?
Why does a bowl have 0 holes in topology, even though it looks like it has an opening on top?
Squish a Donut Into a Mug
Roll a ball of play-dough or clay into a donut shape with one hole poked through the middle. Without tearing it or closing the hole, slowly reshape it into a coffee mug with a handle, keeping that same hole as the handle's hole the whole time. Then go on a hole hunt around your house: find 3 objects with exactly 1 hole, 1 object with 0 holes, and (if you can) something with 2 or more holes. Draw or list what you found.
Topologists actually count the holes in the human body too -- and depending on how you count, a person is topologically similar to a donut, because of one long tube running through the middle!
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