g-conjecture - Numberphile
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However, what intrigued me was that some h-vectors were identical to the lines of Pascal triangle. Coincidence? No. I conjectured early this morning that it happens with the octahedron family (called orthoplex or cross polytope. Thinking about it, I couldn't go back to sleep so I got up and spent 2 hours proving it, haha! Time to have breakfast now --hungry--
This also makes more wonder, does the h-vector follow some general pattern, but more specific than just a palindrome that goes steadily up as you go to the middle. Of course, with 1D sphere (circle) triangulations' h-vectors will be (1, n, 1) for all n and I found triangulations of 2D sphere with h-vectors (1, n, n, 1) for all n. Not sure what happens in higher dimensions (apart from the two families above, I'll look into it after breakfast.
Date: 2022-04-09
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Comments and reviews: 9
Twentydragon
4: 40 - _-A one-dimensional sphere is a circle. -_
No sir. That's a _two_-dimensional sphere. You're -triangulating- a two-dimensional shape with one-dimensional shapes.
5: 40 - _-A zero-dimensional sphere is two points. -_
No, that's one-dimensional. A zero-dimensional anything is a single point.
In one dimension (line-space, you're using points (0D) to approximate the surface of the sphere.
6: 20 - _-Although it is hard to imagine three-dimensional spheres. -_
We call those -spheres-. I think you mean four-dimensional spheres, which would be hard to imagine.
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4: 40 - _-A one-dimensional sphere is a circle. -_
No sir. That's a _two_-dimensional sphere. You're -triangulating- a two-dimensional shape with one-dimensional shapes.
5: 40 - _-A zero-dimensional sphere is two points. -_
No, that's one-dimensional. A zero-dimensional anything is a single point.
In one dimension (line-space, you're using points (0D) to approximate the surface of the sphere.
6: 20 - _-Although it is hard to imagine three-dimensional spheres. -_
We call those -spheres-. I think you mean four-dimensional spheres, which would be hard to imagine.
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heimdall1973
Euler formula works whatever polygons you divide the sphere into (or polyhedra instead of simplices in higher dimensions. But the h-vector won't be a palindrome anymore: if you cut up a sphere into a 2+n sided pyramid, the h-vector will be (1, n, 1, 1, which, apart from tetrahedron (n=1) is not a palindrome. The cube is even worse as the h-vector (1, 5, -1, 1) contains a negative number. But the h-vector will always start with 1 (which you write as the start of the series of 1s down the left) and end with 1 (because of Euler formula.
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Euler formula works whatever polygons you divide the sphere into (or polyhedra instead of simplices in higher dimensions. But the h-vector won't be a palindrome anymore: if you cut up a sphere into a 2+n sided pyramid, the h-vector will be (1, n, 1, 1, which, apart from tetrahedron (n=1) is not a palindrome. The cube is even worse as the h-vector (1, 5, -1, 1) contains a negative number. But the h-vector will always start with 1 (which you write as the start of the series of 1s down the left) and end with 1 (because of Euler formula.
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Anand
I am familiar to 2d sphere as circle 3d sphere as a ball 1d spere as 2 points on a line.
Somehow this guys one dimension less naming is confusing is it how it should be named and i have been wrong all this time. and why is it so called such way.
Only way i can think of is by angles circle ie. 1d sphere bu this naming maybe bcz point moves a single dimensional angle to get circle. while 3d sphere is formed by rotating in 2d angles.
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I am familiar to 2d sphere as circle 3d sphere as a ball 1d spere as 2 points on a line.
Somehow this guys one dimension less naming is confusing is it how it should be named and i have been wrong all this time. and why is it so called such way.
Only way i can think of is by angles circle ie. 1d sphere bu this naming maybe bcz point moves a single dimensional angle to get circle. while 3d sphere is formed by rotating in 2d angles.
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Sliyaroh
The 1 in the problems comes from the count of the highest level object which when you have only one is 1. e. g. A tetrahedron would have f = -4, 6, 4, 1- where the 1 in this set is the tetrahedron itself. When you extend this idea to multiple objects you get higher symmetries, such as two tetrahedrons connected at a point would be different from two separate tetrahedrons and would have distinct h vectors as a result.
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The 1 in the problems comes from the count of the highest level object which when you have only one is 1. e. g. A tetrahedron would have f = -4, 6, 4, 1- where the 1 in this set is the tetrahedron itself. When you extend this idea to multiple objects you get higher symmetries, such as two tetrahedrons connected at a point would be different from two separate tetrahedrons and would have distinct h vectors as a result.
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Sliyaroh
The g conjecture is merely a restatement of a truism: The most common elements of a simplex or n-polyhedra are those near the midpoint of f0-fn. Thus an subtractive pascal's triangle will always produce ascending values up to the midpoint(s) and descending afterwards. It can be proven to be fact simply on the basis of this realization but it is not any more fundamental than the h-vector itself.
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The g conjecture is merely a restatement of a truism: The most common elements of a simplex or n-polyhedra are those near the midpoint of f0-fn. Thus an subtractive pascal's triangle will always produce ascending values up to the midpoint(s) and descending afterwards. It can be proven to be fact simply on the basis of this realization but it is not any more fundamental than the h-vector itself.
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Michael
I'm not a mathematician so my terminology is bad. That said:
The palindrome effect is because of how there is a bell curve anytime you count the number of permutations of a given size in a total number of possibilities. Like 'how many combinations of 2/3/4/5 are there with a sample size of 6'. This directly correlates to the number of dimensions used in these tessellated spheres.
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I'm not a mathematician so my terminology is bad. That said:
The palindrome effect is because of how there is a bell curve anytime you count the number of permutations of a given size in a total number of possibilities. Like 'how many combinations of 2/3/4/5 are there with a sample size of 6'. This directly correlates to the number of dimensions used in these tessellated spheres.
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Maarten
Anyone else notice you can draw smaller triangles of the groups of three numbers that form a sum? So for instance inside the 16-cell one you can make triangles of (1-7-8, (1-6-7, (6-11-17, (17-15-32) and so on. The sum is always starting on the left, then points downward, then goes back up again on the right. A nice visual coincidence, perhaps.
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Anyone else notice you can draw smaller triangles of the groups of three numbers that form a sum? So for instance inside the 16-cell one you can make triangles of (1-7-8, (1-6-7, (6-11-17, (17-15-32) and so on. The sum is always starting on the left, then points downward, then goes back up again on the right. A nice visual coincidence, perhaps.
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Mike
Math people, please help. He keeps saying that everything is one dimension lower than it is. He refers to a normal sphere/tetrahedron/etc. as 2-dimensional, a circle as 1-dimensional, and a straight line as 0-dimensional. Am I wrong, or wouldn't 0-dimensional space be a point, 1-dimensional space on a line, and 2-dimensional space on a plane?
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Math people, please help. He keeps saying that everything is one dimension lower than it is. He refers to a normal sphere/tetrahedron/etc. as 2-dimensional, a circle as 1-dimensional, and a straight line as 0-dimensional. Am I wrong, or wouldn't 0-dimensional space be a point, 1-dimensional space on a line, and 2-dimensional space on a plane?
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Bret
The simplex in any dimension will always give you a string of 1's at the bottom because you are calculating Pascal's triangle sideways. Pascal's triangle encodes p choose n. f(n) of a simplex in p dimensions is p choose n. If you start with p choose n and subtract to get 1's it's the same as starting with 1's and adding to get p choose n.
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The simplex in any dimension will always give you a string of 1's at the bottom because you are calculating Pascal's triangle sideways. Pascal's triangle encodes p choose n. f(n) of a simplex in p dimensions is p choose n. If you start with p choose n and subtract to get 1's it's the same as starting with 1's and adding to get p choose n.
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