Hovering between dimensions, occupying space without truly filling it
November 27, 2024 11:33 AM Subscribe
The Menger problem would be their first time moving beyond school workbooks with answer keys. “It was a little bit nerve-racking, because it was the first time I was doing something where truly nobody has the answer, not even Malors,” said Nazareth. Maybe there was no answer at all. from Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal [Quanta]
Menger sponge, previouslies
Menger sponge, previouslies
You will be shocked to learn I came here with the intent to post this. Since you did the work, I can skip to my petty complaint: the article (not the paper, the kids are alright) goes seemingly out of it's way to not ever namecheck Sierpinski, referring to the Sierpinski tetrahedron/pyramid only indirectly as "another related fractal" or "a tetrahedral version of the Menger sponge". They namecheck Cantor, and yet. Did Waclaw ruin Gregory Barber's marriage or something?
Anyway, this is delightful and I will try to see what I can make of the paper itself, though the article gets at a decent gist of it it feels like. Mostly I'm curious about things like (a) how the number of crossing in a given knot corresponds to needed iterations of the Menger sponge its embedded in and (b) what exactly they ginned up for the mapping into the tetrahedron.
posted by cortex at 12:43 PM on November 27 [8 favorites]
Anyway, this is delightful and I will try to see what I can make of the paper itself, though the article gets at a decent gist of it it feels like. Mostly I'm curious about things like (a) how the number of crossing in a given knot corresponds to needed iterations of the Menger sponge its embedded in and (b) what exactly they ginned up for the mapping into the tetrahedron.
posted by cortex at 12:43 PM on November 27 [8 favorites]
how have i posted literally none of the previous threads about Menger sponges
this was my big chance
posted by cortex at 12:45 PM on November 27 [6 favorites]
this was my big chance
posted by cortex at 12:45 PM on November 27 [6 favorites]
At a previous place of employment, everyone was handed a huge box of 1000 business cards. We where all software developers, and had next to no need for them.
So we build a business card Menger sponge.
The smallest unit was a cube (without holes) made out of 6 business cards folded to "self close", and was about 2 inches on each edge. Tier 1 required 20 such cubes (so 120 cards) and was about 3 base cubes high, so 6 inches on each edge (plus a bit of fudging), and Tier 2 required 20 of those (so 2400 cards) at 18 inches on each edge. Reaching Tier 3 would require 48,000 cards and was 54 inches on each edge (almost human sized).
(For some reason, I remember the cube being larger than 18 inches, but smaller than 54).
Before we got too far along, the company name changed ... and for some reason, they didn't issue us new business cards.
posted by NotAYakk at 1:51 PM on November 27 [4 favorites]
So we build a business card Menger sponge.
The smallest unit was a cube (without holes) made out of 6 business cards folded to "self close", and was about 2 inches on each edge. Tier 1 required 20 such cubes (so 120 cards) and was about 3 base cubes high, so 6 inches on each edge (plus a bit of fudging), and Tier 2 required 20 of those (so 2400 cards) at 18 inches on each edge. Reaching Tier 3 would require 48,000 cards and was 54 inches on each edge (almost human sized).
(For some reason, I remember the cube being larger than 18 inches, but smaller than 54).
Before we got too far along, the company name changed ... and for some reason, they didn't issue us new business cards.
posted by NotAYakk at 1:51 PM on November 27 [4 favorites]
Okay, stumbled through the paper without falling into the sort of terror and paralysis that usually hits me when the formulae start droppin', if mostly just because I'm a sicko for the underlying concepts. I couldn't quite get the combinatoric projection of the tetrahedron to click for me but the idea felt okay, and the fact that they essentially said "let's build a (weirdly distorted) Sierpinski tetrahedron inside a Menger sponge" to establish a relationship between the two was delightful.
One of the core bits of their process here, of using only Cantor dust coordinates for all the vertices of any given knot's arc presentation, makes me think a lot about the process of constructing physical (and virtual "physical") Menger sponges, something I've done way too much of over the years. Those Cantor dust coordinates where from all the way from the front face to the back face of a Menger sponge there are no gaps? Are what I've ended up thinking of as the load-bearing columns of a physical Menger sponge construction: they are the framing 2x4s, so to speak. Lots of other material gets filled in to complete the form, but those are good keystone pieces to start with. And in these folks' formulation, creating the knot projection in the Menger sponge is kind of like doing wiring that runs up and down from the ceiling to the floor and back, along those framing columns.
In that sense, their system would work just as well with, rather than a whole Menger sponge, only a pair of Sierpinski carpets of the same iteration level, with a layer of Cantor dust of the same iteration level in between— they don't need the full depth of a sponge, they just need two distinct layers. And in fact if you slice off just the first three distinct layers of any iteration of a Menger sponge, you'd have exactly that! For a first iteration sponge, it'd be unchanged, still just a 3x3x3 object. For a second iteration sponge, you'd slice three layers off of a 9x9x9 to end up with a 9x9x3 object. And so on.
In that sense there's a clear huge growth in potential workspace for describing knots within a Menger sponge; at higher iterations it's possible you could use some more complex process of projecting/embedding a knot that uses the full 3D space of the sponge to more densely pack a given knot compared to the sort of 2.5D nature of the Arc projection method here. Whether or not it's actually *possible* to achieve that denser embedding I have no idea in practice, but it feels like an interesting possibility to explore!
posted by cortex at 2:20 PM on November 27 [1 favorite]
One of the core bits of their process here, of using only Cantor dust coordinates for all the vertices of any given knot's arc presentation, makes me think a lot about the process of constructing physical (and virtual "physical") Menger sponges, something I've done way too much of over the years. Those Cantor dust coordinates where from all the way from the front face to the back face of a Menger sponge there are no gaps? Are what I've ended up thinking of as the load-bearing columns of a physical Menger sponge construction: they are the framing 2x4s, so to speak. Lots of other material gets filled in to complete the form, but those are good keystone pieces to start with. And in these folks' formulation, creating the knot projection in the Menger sponge is kind of like doing wiring that runs up and down from the ceiling to the floor and back, along those framing columns.
In that sense, their system would work just as well with, rather than a whole Menger sponge, only a pair of Sierpinski carpets of the same iteration level, with a layer of Cantor dust of the same iteration level in between— they don't need the full depth of a sponge, they just need two distinct layers. And in fact if you slice off just the first three distinct layers of any iteration of a Menger sponge, you'd have exactly that! For a first iteration sponge, it'd be unchanged, still just a 3x3x3 object. For a second iteration sponge, you'd slice three layers off of a 9x9x9 to end up with a 9x9x3 object. And so on.
In that sense there's a clear huge growth in potential workspace for describing knots within a Menger sponge; at higher iterations it's possible you could use some more complex process of projecting/embedding a knot that uses the full 3D space of the sponge to more densely pack a given knot compared to the sort of 2.5D nature of the Arc projection method here. Whether or not it's actually *possible* to achieve that denser embedding I have no idea in practice, but it feels like an interesting possibility to explore!
posted by cortex at 2:20 PM on November 27 [1 favorite]
Another thought about the unused depth in all those unused layers of higher-iteration sponges: you could potentially use that space to embed additional knots, one or more in each XxXx3 "sandwich". So there's the interesting question of what iteration M of sponge would you need to accommodate a mapping of all distinct classes of knot with up to N crossings? How fast does the needed M grow with respect to larger N?
posted by cortex at 2:38 PM on November 27 [1 favorite]
posted by cortex at 2:38 PM on November 27 [1 favorite]
I'm only surprised Cortex didn't zoom here in time to be the first commenter.
:)
posted by Greg_Ace at 3:09 PM on November 27 [2 favorites]
:)
posted by Greg_Ace at 3:09 PM on November 27 [2 favorites]
And in fact if you slice off just the first three distinct layers of any iteration of a Menger sponge, you'd have exactly that
I misspoke here: the general idea is solid but it's not generally true that the middle layer of an XxXx3 slice off a Menger sponge would be strictly Cantor dust; in most cases (other than 3x3x3) it'd be some superset that includes, like every layer, Cantor dust, plus some other stuff. It *is* generally true that the exact middle layer of any whole Menger sponge iteration is a layer of Cantor dust, though. There's some neat patterns involved in there.
posted by cortex at 5:13 PM on November 27 [1 favorite]
I misspoke here: the general idea is solid but it's not generally true that the middle layer of an XxXx3 slice off a Menger sponge would be strictly Cantor dust; in most cases (other than 3x3x3) it'd be some superset that includes, like every layer, Cantor dust, plus some other stuff. It *is* generally true that the exact middle layer of any whole Menger sponge iteration is a layer of Cantor dust, though. There's some neat patterns involved in there.
posted by cortex at 5:13 PM on November 27 [1 favorite]
I discovered this totally different maths problem the other day: Moser's Worm Problem. Looks simple but currently unsolved.
posted by TheophileEscargot at 12:38 AM on November 28 [2 favorites]
posted by TheophileEscargot at 12:38 AM on November 28 [2 favorites]
NotAYakk - we did the same thing, with more-or-less the same result (we didn't change name, but engineers no longer got business cards by default). I wonder how many software companies stopped issuing their engineers with business cards for precisely this reason.
posted by parm at 4:51 AM on November 28 [2 favorites]
posted by parm at 4:51 AM on November 28 [2 favorites]
I discovered this totally different maths problem the other day: Moser's Worm Problem.
Oh, I didn't know about the worm problem! That's delightful, and I always wonder with bounds-chasing problems like these whether (a) we'll ever get to a concrete, proven minimum form and (b) whether it'll be at all aesthetically appealing as a specimen of geometry or if it'll turn out to be something like John Bidwell's 17 squares packed in a square monstrosity.
Also, reminded me immediately of the similar moving sofa problem, which is similarly appealingly approachable and similarly as yet unproven.
posted by cortex at 9:06 AM on November 28 [2 favorites]
Oh, I didn't know about the worm problem! That's delightful, and I always wonder with bounds-chasing problems like these whether (a) we'll ever get to a concrete, proven minimum form and (b) whether it'll be at all aesthetically appealing as a specimen of geometry or if it'll turn out to be something like John Bidwell's 17 squares packed in a square monstrosity.
Also, reminded me immediately of the similar moving sofa problem, which is similarly appealingly approachable and similarly as yet unproven.
posted by cortex at 9:06 AM on November 28 [2 favorites]
Speaking of which, here is a brand new and as yet not thoroughly vetted claim to prove in ~200 pages that the current best known solution to the moving sofa problem is in fact optimal. Will keep an eye on this and maybe turn it into a post if it seems to have its feet under it because Big If True etc.
posted by cortex at 9:41 PM on December 1
posted by cortex at 9:41 PM on December 1
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