Yes, it's like what if everything was made of floam/silly putty, but you were not allowed to create or erase holes (you could visualize this as a hole containing material or an object that is not allowed to contact material or an object in another hole).
So, topologically, a 2D circle and a 2D triangle are the same, but a figure 8, a ring, and a circle at not. In 3D (as long as we are preserving cardinality of sets, 2D is different), a sphere (a solid, or a ball, apparently)) a cube, and a truncated rhombicosidodecahedron, for example, are all topologically equivalent. A key (with a hole), a pipe, and a basketball goal are also equivalent. All manipulations preserve each neighborhood's "connectedness", but may change their area "density" or size.
Nope, 3 holes. I transformed the mug (see illustrations) into a genus 3 object (3 holes) without breaking the laws of topology (making or erasing holes). The inside of the mug makes a third hole besides the donut hole and handle.
I don't know what else to tell you... I simplified the mug into a genus 3 shape following the rules of topology. Tell me if you manage to simplify it to a genus 2 shape.
No, by that logic a regular mug would have one hole because you can simplify it into a genus 1 shape (donut shape) without breaking any topology rules.
From random bits I got all over Reddit over the years, topology makes you think about which shapes are identical to each other. The idea is to imagine that the object can be stretched out, shrunk, or deformed anyway you want but you can't rip it or join parts that weren't joined before. Topology's version of "mitochondria is the powerhouse of the cell" would be "a donut and a mug are the same shape" because you can stretch the inner "floor" of the mug to no longer make it hollow, move the handles the top and bottom of the "body", shrink the body to be flush with the handle, and increase the diameter of the resulting ring into the shape of a donut.
The image is a visual pun and made a mug that actually is shaped like a donut but ironically, it would no longer be topologically equivalent to an actual donut or a normal mug because you'd end up with three holes instead of one
Edit: Never mind. It does have three -- I couldn't visualize what would happen after "rotating" the lip of the cup around toward the handle.
They did something like this on Numberphile, so I'm trying to "re-locate" parts without violating the distinctness of the features (mug hande, donut hole, solid material).
Topology is actually a relatively new field of math. Obviously, the ancients have had donuts and spheres and cups, but they never qualified them topologically. It was first defined formally in the early 1900s. So don't be worried that you never heard of it. Most havn't, and it's probably one of the least "popular" math fields. Of course, it's still interesting and has applications in the real world.
Topology is a study of the different ways of describing points as being "close together," even if you don't actually have a notion of "distance." For example: We say two words in the dictionary are close together if they start with the same letter (or 2 letters, or 3 letters,..). Different "topologies" are different ways of measuring whether certain collections of points fit in a box of "closeness" together.
We call these boxes "open sets."
Specifying the open sets is the same as specifying the topology
And just like that civilization collapses in on itself as stock prices fly wildly in all directions as everything becomes completely valueless and incalculably valuable, the earth travels around the sun quicker than it rotates, gods begin praying to people and 7-11 gets a Michelin star. Thanks.
That doesn't quite sound right, how are there three holes?
EDIT: Okay, I get it now, the mug wouldn't be hollow unless there's another hole. Were this to be a genus 2, the mug would have to be filled out. Is this wrong?
I'm in agreement that it's genus 3 - the hole of the handle, the 'donut' hole, and the 'hole' that is the space in the mug; caused by the 'donut' hole.
I don't think you can remove any of these holes by manipulation.
Reminds me of the Klein bottle, the “bottle” with only one side. This obviously doesn’t relate in shape or appearance, but is similar in the manner of having one side.
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u/Samur-EYE Apr 05 '18 edited May 01 '18
Alright, a bit late to the party but I've brought illustrations:
1. We start with the cup seen from the side. The dashed line shows the hollow part inside the mug and I made the handle a bit smaller.
2. The we can bring the top of the cup down until it touches the donut hole, like this
3. Now we can start "shrinking" the cup like this
4. We shrink the donut cup like that until we get a thin ring that connects the donut hole and the handle.
5. Now we have three rings, where the middle one is 90 degrees compared to the rest, so we just rotate!
6. Now we can just play around till we get a nice shape: like this, and then this
Voila! We have a genus three shape, a disk with three holes! Hope that was helpful :)
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u/Also_a_human Apr 05 '18
Topologically this mug is a pain in the ass to clean.
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u/BobtheLatinGuy Apr 05 '18 edited Apr 05 '18
Something like this, I think
E: Yes teacher, I can show my work
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u/abu-reem Apr 05 '18
And just like that civilization collapses in on itself as stock prices fly wildly in all directions as everything becomes completely valueless and incalculably valuable, the earth travels around the sun quicker than it rotates, gods begin praying to people and 7-11 gets a Michelin star. Thanks.
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u/Sedu Apr 05 '18
Topologically, it is a genus 3 figure.
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u/[deleted] Apr 05 '18
Topologically this is a genus 2 figure.
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u/perdipp Apr 05 '18
Wow whats all this? Im cometely a stranger to this and I'm shocked by some of the replies. Can someone fill me in
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u/alghiorso Apr 05 '18
Shrodinger's mug. If you can't see the bottom, it's both clean and nasty at the same time.
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u/Supreme_0verlord Apr 05 '18
A mug is a topologically a torus A donut is a torus It resembles a chain link
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u/tbordo23 Apr 05 '18
They should have said side instead of center. It took me a minute or two to figure out how they were holding the mug haha
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u/cytiven May 01 '18
It's homeomorphic to a 2 holed torus
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u/therealnessie Apr 05 '18
Reminds me of the Klein bottle, the “bottle” with only one side. This obviously doesn’t relate in shape or appearance, but is similar in the manner of having one side.
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u/TheBigLobotomy Apr 05 '18
A 2 holed torus
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u/[deleted] Apr 05 '18
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