Topology 101: The Hole Truth
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Topology 101: The Hole Truth
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Re: Topology 101: The Hole Truth
#2The author knows how to hook you right in the first paragraph:
>> If you’re looking to pick a fight, simply ask your friends, “Is Pluto a planet?” Or “Is a hotdog a sandwich?” Or “How many holes does a straw have?” The first two questions will have them arguing yay or nay, while the third yields claims of two, one and even zero.
Two, one, or zero? Continues reading...
Re: Topology 101: The Hole Truth
#3> A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes.
Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
Re: Topology 101: The Hole Truth
#4> A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes. Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
> This idea of an inner hole that is hidden and connected is quite unintuitive to me.
What do you mean by the hole in the torus being "hidden and connected"?
Re: Topology 101: The Hole Truth
#5> A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes. Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
> This idea of an inner hole that is hidden and connected is quite unintuitive to me. What do you mean by the hole in the torus being "hidden and connected"?
I suspect what was meant was that it's only apparent if you can view the torus from "a bird's eye view". If you were an ant walking on the surface at night, you would never run into a hole in the usual sense. You would be hard pressed to find a way to distinguish the surface from that of a sphere.
As an ant you could easily see that you were not on the surface of an infinite plane simply because you keep coming back to the same spot over and over again. In order to figure out that there was a hole (or more appropriately named as 'handle'), you'd need to leave some thread behind and notice that there was no way to contract various loops to a point.
Re: Topology 101: The Hole Truth
#6> A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes. Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
> This idea of an inner hole that is hidden and connected is quite unintuitive to me. What do you mean by the hole in the torus being "hidden and connected"?
This is how I understood the comment -
In a torus, there are effectively two holes -
1. the center hole around which there is a cylindrical ring
2. the whole inside the cylindrical ring, which is hidden and connected.
When we cut along the length of the cylindrical ring, we are effectively creating two edges, just like if we were to cut a circular ring of wire, we would end up with two points at the end.
The two edges then open up, effectively forming another cylinder, which has to be cut again.
Any other object which has only a single hole, would require only a single cut to flatten out.
Re: Topology 101: The Hole Truth
#7> A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes. Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
> This idea of an inner hole that is hidden and connected is quite unintuitive to me. What do you mean by the hole in the torus being "hidden and connected"?
There’s two holes: places you can draw a non-contractile circle, which can’t be turned into each other — “circumference” and “around”.
I think they mean the “around” holes have a center “inside” the donut.
Re: Topology 101: The Hole Truth
#8The author knows how to hook you right in the first paragraph: >> If you’re looking to pick a fight, simply ask your friends, “Is Pluto a planet?” Or “Is a hotdog a sandwich?” Or “How many holes does a straw have?” The first two questions will have them arguing yay or nay, while the third yields claims of two, one and even zero. Two, one, or zero? Continues reading...
Actually if you want to start a fight, ask whether pineapple belongs on pizza.
Re: Topology 101: The Hole Truth
#9Algebraic topology is an amazing part of mathematics. (If you like pictures, you might enjoy Topology Illustrated by Saveliev.)
Re: Topology 101: The Hole Truth
#10The author knows how to hook you right in the first paragraph: >> If you’re looking to pick a fight, simply ask your friends, “Is Pluto a planet?” Or “Is a hotdog a sandwich?” Or “How many holes does a straw have?” The first two questions will have them arguing yay or nay, while the third yields claims of two, one and even zero. Two, one, or zero? Continues reading...
Actually if you want to start a fight, ask whether pineapple belongs on pizza.
Nah, that's easy, if you think pineapple should be anywhere around pizza you cannot be my friend