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How many holes?

The image above shows two viewing angles of the same three dimensional object. I recently posted a problem with this image, and the question asked how many holes it had. Not a single person guessed that it could have \(3\) holes, which has led me to create this note.

Firstly, How do you define a hole, and how is one made?

My initial thought was that it is a piece of area in an object that has been completely "punched out", so that you can pass through the entire object without passing through any solid matter.

If this were the case, then a hole in the ground would mean that you have not made a hole in the Earth, as it does not pass completely through it. Rather, you have transformed the Earth so some area has been moved to a difference space.

Secondly, Does an object maintain the amount of holes it has even after undergoing a transformation?

For the answer of the original holes problem, I posted this gif which shows that the object would have 3 holes if it was transformed. Is this a valid proof to show that the unchanged object has 3 holes?

Take a copper pipe, for example. It is a hollow cylinder with \(2\) openings at either end. However if we flattened it into a disc, there would only be one area that is completely "punched out", which means one hole by this definition.

For the green 3D object in question, if you were to fill in the three holes you would see that both versions of the object, before and after the transformation, have no completely "punched out" areas. The initial version looks like a teacup, and the flattened version looks like a pancake. So does that means a teacup has no holes, despite the fact its curved shape would suggest that the object has a big hole in the middle for the liquid to sit?

Thirdly, Can holes be created or destroyed?

Do holes have to be conserved? If so, how many holes would the object have?

Are there any other questions to consider?

I would love to hear your thoughts on this topic.

- Credit to the original creator of the problem: Reddit user Philip_Pugeau
- Thanks to Calvin Lin for suggesting the idea of creating a note for this conundrum.

Note by Michael Fuller
1 year, 9 months ago

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How many holes does this have?

Answer: Just one.

In topology, the genus of a connected, orientable surface is the same as the number of holes it has. The genus of a sphere is 0. The number of simple, non-intersecting cuts it takes to render any such surface topologically equivalent to a sphere is the genus. Hence, it takes only one cut in the above figure to change it into a strand. It takes 3 cuts to render your figure similarly into a strand. Cutting it 4 times is overkill, and you end up with two pieces, not one.

For those not familiar with topology, a sphere and a ball are not the same thing. A sphere is a 2D surface, while a ball is a 3D solid. The topology of 2D surfaces is dominated by analysis of lines and loops on the surface, so that a "cut" is really about a continuous transformation of the 2D surface as one loop on it shrinks down to a point, much like how a clown makes animals with long balloons. In topology, such surfaces can pass through itself, as in the case of a non-orientable Klein bottle, the same way loops can pass through itself on a surface, but kinks are noted, i.e. treated as a discontinuity of an otherwise continuous transformation. Michael Mendrin · 1 year, 9 months ago

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My solution:

Since holes are preserved with stretching, stretch one of the holes until the entire shape becomes flat. Then clearly we have a surface with 3 holes.

Very interesting problem! Daniel Liu · 1 year, 9 months ago

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@Daniel Liu The idea of "stretch one of the holes until the entire shape becomes flat" isn't always applicable. See Michael's image as an example. Calvin Lin Staff · 1 year, 9 months ago

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@Calvin Lin A good point, Calvin. The first impression one gets from a layman's book about topology is that it seems too simple and obvious. But then it can quickly become far from obvious, which is why most of topology doesn't rely on pictures and diagrams. It should go without saying that it's a rich field of study. Michael Mendrin · 1 year, 9 months ago

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@Michael Mendrin I'm interested in knowing, how does topology look like without diagrams? What types of problems are considered "far from obvious"? Daniel Liu · 1 year, 9 months ago

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@Daniel Liu One "not obvious" problem that comes immediately to mind is, "what's the fewest mobius strips can a Klein bottle be dissected into?". But even a clever guy with a pencil should be able to work this out using very good drawings. A far more difficult problem is "can a sphere be turned inside out by means of continuous transformation?" As I noted above, lines and surfaces can pass through themselves, but kinks are to be avoided because they represent discontinuities. If you attempt to simply push a sphere through itself in the obvious way, you'll immediately end up with a kink around it. This time, it's extremely difficult to try to solve relying on pictures and diagrams. Go try it.

Here's a text on that sphere eversion problem On Eversion of Spheres

And the more general theorem which addresses this in higher dimensions Smale-Hirsh Theorem

I doubt if anybody can draw pictures for this one in any higher dimensions. Michael Mendrin · 1 year, 9 months ago

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@Michael Mendrin Ah I remember that problem from watching this video a long time ago. It's a great video, really easy to follow. Daniel Liu · 1 year, 9 months ago

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@Daniel Liu Nice video!

I first heard about topological sphere eversion described in a Scientific American article back in 1966. It was so cool I decided to look more into the subject. Wow, has it been almost 50 years now?

Most all of the discussion we've been having here in this note involves homotopy. Here's a typical math course paper (probably graduate level) on homotopy theory Homotopy to give you an idea of what it'd be like to "do topology". Also see Elementary Homotopy Theory Michael Mendrin · 1 year, 9 months ago

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@Daniel Liu The diameter of a set is the supremum (maximum) distance between any 2 points in the set.

Given any set in \( \mathbb{R} ^3 \), can we always split it up into \( 3 + 1 \) sets of strictly smaller diameter?
Given any set in \( \mathbb{R} ^4 \), can we always split it up into \( 4 + 1 \) sets of strictly smaller diameter?
Given any set in \( \mathbb{R} ^{2015} \), can we always split it up into \( 2015 + 1 \) sets of strictly smaller diameter?

I am offering a $1000 reward for a correct solution to the question about \( \mathbb{R}^4 \). Calvin Lin Staff · 1 year, 9 months ago

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@Calvin Lin If I could find that correct solution to get that $1000, I think I might end up winning some kind of a prize in mathematics. This sounds like Borsuk's Conjecture, which is false for dimensions higher than something like 64 but still unproven for 4?

What's the reward for the \( \mathbb{R}^{2015}\) case? Michael Mendrin · 1 year, 9 months ago

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@Michael Mendrin Right. It has been proved true for 2, 3, false for 64 +, and unknown for everything in between. I have spent a chunk of time trying to figure out the 4 case. Calvin Lin Staff · 1 year, 9 months ago

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@Calvin Lin I find that there is an interesting connection between Borsuk's conjecture and Poincare's conjecture, which, as we know, Perelman knocked off a while ago. I have no hopes of knocking off either. Michael Mendrin · 1 year, 9 months ago

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@Michael Mendrin Oh, what's the connection? It would be interesting to see how they are related.

Borsuk's interesting cases happens when we have a discrete set of points in \( \mathbb{R} ^ d \), while Poincare is only about 3-manifolds. The overlapping area seems to be combinatorial topology, but I don't understand how they would share a particular object to study. Calvin Lin Staff · 1 year, 9 months ago

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@Calvin Lin Ah, the Bing-Borsuk conjecture has already been proven for dimensions \(n\le 4\), so it's not the same thing. You're right. The Bing-Borsuk conjecture doesn't address any cases of discrete set of points, even though it's said to be a stronger form of Poincare's conjecture.

See my comment elsewhere about my arm going to sleep. Michael Mendrin · 1 year, 9 months ago

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@Michael Mendrin Delving initially into topology I found it just absolutely fascinating: counterintuitive and rich with examples of fun, interesting objects. In recent times my enthusiasm has waned, but I'm still interested in topology; do you know of any good books/online lectures that can expose me to some intermediate-level topology? Jake Lai · 1 year, 9 months ago

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@Jake Lai They're buried in my huge pile of books in my garage. Maybe I should go dig through it to find those books. It's been a long time now.

Edit: In looking around, I've found an online PDF textbook on introductory topology that you can download. It's very similar to one of the good books I have on the subject. And it's free.

Introductory to Topology Michael Mendrin · 1 year, 9 months ago

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@Michael Mendrin Very much appreciated! Thank you :) Jake Lai · 1 year, 9 months ago

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@Calvin Lin I see, tl;dr cutting loops until there are no more is a more general way to solve it. Daniel Liu · 1 year, 9 months ago

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