Heesch Numbers and Tiling - Numberphile
video description
Specifically, if you locate the red piece with the highest vertical corner, follow it down and right toward the most opposite corner, select the only other red piece touching that corner, and follow it down and right to its lowest corner, you can locate an edge of contact between red and yellow pieces. Essentially the yellow region is not completely bounded by the green region in the current arragnement.
It looks like, if the second red piece I identified was changed to green, and the first red piece I identified was moved down and right by one fifth of the distance of its length in that direction, the diagram would be correct.
Alternatively, if the yellow piece I identified was switched to red, the diagram would be correct.
If I understand the rules correctly, only the second method I presented would actually resolve the system, but I could be mistaken if a piece in some layer n does not have to contact a piece in later n-1.
Date: 2022-04-09
Related videos
Comments and reviews: 9
Hanyang
I think you should search for something based on hexagon and makes the heesch number 2 before you go on any further. Your current answer to 2 doesnt make any sence except for it meets your rules. For Heesch 1, it has 5 60degrees positives and 2 negetives. 3 has 4 positives and 7 negetives. 4 has 2 180 degrees positives and 3 negetives( if counted by 60 degrees it should be -6 and +9. Means it's always 180 degrees positives or negetive.
And I looked the 5th up and found it has 11 negetives(180 degrees) and 10 positives.
Another point of view. Say if 0th circle has A perimeter and B negetives and C positives(say B>C, counted in 180 degrees. The 1st layer has 3A and 6B and 6C. 2nd -> 5A 12B 12C, etc.
When Heesch== 1, B is -2/3, and C is 5/3. For the 1st layer, emmm, OK, I quit. Whatever.
Any way I think the pattern is lengthened enough from heesch = 5, it would get to 6. And the formula should be found with the -another point of view-upon.
reply
I think you should search for something based on hexagon and makes the heesch number 2 before you go on any further. Your current answer to 2 doesnt make any sence except for it meets your rules. For Heesch 1, it has 5 60degrees positives and 2 negetives. 3 has 4 positives and 7 negetives. 4 has 2 180 degrees positives and 3 negetives( if counted by 60 degrees it should be -6 and +9. Means it's always 180 degrees positives or negetive.
And I looked the 5th up and found it has 11 negetives(180 degrees) and 10 positives.
Another point of view. Say if 0th circle has A perimeter and B negetives and C positives(say B>C, counted in 180 degrees. The 1st layer has 3A and 6B and 6C. 2nd -> 5A 12B 12C, etc.
When Heesch== 1, B is -2/3, and C is 5/3. For the 1st layer, emmm, OK, I quit. Whatever.
Any way I think the pattern is lengthened enough from heesch = 5, it would get to 6. And the formula should be found with the -another point of view-upon.
reply
marcoscolga24
7: 31 I love this idea that even cheats can be useful information and progress the field of mathematics. It sort of acts as a reminder of the superficiality of mathematics for me, that it's a merely a human creation based on the real world, built purely on axioms that we can choose to disregard for the sake of progression.
On a more contemplative note, could we say that as of 2021, C, where C is the set of all countries in the world, has a Heesch number of 2?
reply
7: 31 I love this idea that even cheats can be useful information and progress the field of mathematics. It sort of acts as a reminder of the superficiality of mathematics for me, that it's a merely a human creation based on the real world, built purely on axioms that we can choose to disregard for the sake of progression.
On a more contemplative note, could we say that as of 2021, C, where C is the set of all countries in the world, has a Heesch number of 2?
reply
Melinda
I have no problem with tiles with discontiguous regions. Sure, a contiguous region is neater, but mathematically seems no more valid than anything else. It's sort of like star polygons. A pentagram is a polygon that intersects itself, but so what? Well star polygons with even numbers of vertices are discontiguous too, but it's pretty hard to claim they're not equally valid polygons as those that are contiguous.
reply
I have no problem with tiles with discontiguous regions. Sure, a contiguous region is neater, but mathematically seems no more valid than anything else. It's sort of like star polygons. A pentagram is a polygon that intersects itself, but so what? Well star polygons with even numbers of vertices are discontiguous too, but it's pretty hard to claim they're not equally valid polygons as those that are contiguous.
reply
La
Things I'm curious about:
1. That bit about it being an -unbalanced tile- was dropped fairly quickly, how does that relate to it having a finite Heesch number? The fractions of a circle seem relevant.
2. That tile with Heesch number 5 looks like five identical tiles stacked next to each other. What's the Heesch Number of just one of those five?
reply
Things I'm curious about:
1. That bit about it being an -unbalanced tile- was dropped fairly quickly, how does that relate to it having a finite Heesch number? The fractions of a circle seem relevant.
2. That tile with Heesch number 5 looks like five identical tiles stacked next to each other. What's the Heesch Number of just one of those five?
reply
Sadrho
What if we think of the plane in 3d and instead of thinking of Socolar-Taylor's Tiling as unconnected; we instead think of it as connected by an unseen x axis above or below the level of the plane to be filled? Sorry if this is nonsense. I only know about as much as this video on the subject.
reply
What if we think of the plane in 3d and instead of thinking of Socolar-Taylor's Tiling as unconnected; we instead think of it as connected by an unseen x axis above or below the level of the plane to be filled? Sorry if this is nonsense. I only know about as much as this video on the subject.
reply
James
My intuition tells me that the cheat tile must be some kind of 4D+ shape that its projection(s) can cover 2D space. That may also be a generalization of fractal filling curves of fractology. Mathematics can be both dishearteningly comolex and fundamentally beautiful at the same time.
reply
My intuition tells me that the cheat tile must be some kind of 4D+ shape that its projection(s) can cover 2D space. That may also be a generalization of fractal filling curves of fractology. Mathematics can be both dishearteningly comolex and fundamentally beautiful at the same time.
reply
Lotrfan2001
This reminds me of the one where theres an infinite chessboard and a knight moving around on it, but it can only go to the lowest number (which are laid out in a spiral) it goes on for thousands of turns but eventually runs out of moves
reply
This reminds me of the one where theres an infinite chessboard and a knight moving around on it, but it can only go to the lowest number (which are laid out in a spiral) it goes on for thousands of turns but eventually runs out of moves
reply
Anthony
Great video to follow. My 4 year-old grandson chose to watch this with me as he liked the colours. When the question came up about the Heesch number of a circle he called it. (He might have just called the shape zero: D )
reply
Great video to follow. My 4 year-old grandson chose to watch this with me as he liked the colours. When the question came up about the Heesch number of a circle he called it. (He might have just called the shape zero: D )
reply
Sarainia
Get a super Computer or even an A. I. or a Quantum Computer to try to get to like 6 or 7 or so on kinda thing I'm certain an A. I. at very least might solve a scaling pattern as it goes up to each Heesch number!
reply
Get a super Computer or even an A. I. or a Quantum Computer to try to get to like 6 or 7 or so on kinda thing I'm certain an A. I. at very least might solve a scaling pattern as it goes up to each Heesch number!
reply
Add a review, comment
Other channel videos
