The nice thing that happened in class today:
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
@futurebird Actually I have a (very undeveloped) concept of a lesson with respect to the symmetry group of the square.
Basically, after the class has been introduced at least to the intuitive approach to the symmetry group of the square, you give them a problem where they have to "solve" a substitution cipher from {a,b,c,d,e,f,g,h} or whatever to the symmetry group of the square, given the multiplication table of that substitution cipher.
The lesson here is that this problem doesn't have a single unambiguous answer: rather you can solve the substitution cipher for a few elements like the identity element and the "rotate by 180 degrees" element, but you can only classify the rest of the substitution cipher up to the symmetry group of the symmetry group of the square, more technically known as the automorphisms of D_4.
I was thinking maybe there's an angle to develop as like an alien linguist as part of a Star Trek science team, and perhaps even make it a trick question by making it seem like they are expected to find the one "true" solution.
It turns out that the automorphisms of D_4 is isomorphic to D_4, which is definitely a very yo dawg moment, but it turns out this is very much accidental. Groups G that are isomorphic to their own automorphism group include all complete groups, but this is one of a handful of sporadic exceptions of a group that is not complete but also isomorphic to its automorphism group. This includes D_4, D_6, D_∞, and may include a few more unknown examples.
It turns out that all the symmetric groups (i.e. groups of permutations of n elements) are complete except for n=2 and n=6. The n=6 exception actually pretty interesting, and @johncarlosbaez likes to talk about it.
constructive-symmetry/D002_Book_of_Algebra at master · constructive-symmetry/constructive-symmetry
A Philosophy of Math Education. Contribute to constructive-symmetry/constructive-symmetry development by creating an account on GitHub.
GitHub (github.com)
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@futurebird Actually I have a (very undeveloped) concept of a lesson with respect to the symmetry group of the square.
Basically, after the class has been introduced at least to the intuitive approach to the symmetry group of the square, you give them a problem where they have to "solve" a substitution cipher from {a,b,c,d,e,f,g,h} or whatever to the symmetry group of the square, given the multiplication table of that substitution cipher.
The lesson here is that this problem doesn't have a single unambiguous answer: rather you can solve the substitution cipher for a few elements like the identity element and the "rotate by 180 degrees" element, but you can only classify the rest of the substitution cipher up to the symmetry group of the symmetry group of the square, more technically known as the automorphisms of D_4.
I was thinking maybe there's an angle to develop as like an alien linguist as part of a Star Trek science team, and perhaps even make it a trick question by making it seem like they are expected to find the one "true" solution.
It turns out that the automorphisms of D_4 is isomorphic to D_4, which is definitely a very yo dawg moment, but it turns out this is very much accidental. Groups G that are isomorphic to their own automorphism group include all complete groups, but this is one of a handful of sporadic exceptions of a group that is not complete but also isomorphic to its automorphism group. This includes D_4, D_6, D_∞, and may include a few more unknown examples.
It turns out that all the symmetric groups (i.e. groups of permutations of n elements) are complete except for n=2 and n=6. The n=6 exception actually pretty interesting, and @johncarlosbaez likes to talk about it.
constructive-symmetry/D002_Book_of_Algebra at master · constructive-symmetry/constructive-symmetry
A Philosophy of Math Education. Contribute to constructive-symmetry/constructive-symmetry development by creating an account on GitHub.
GitHub (github.com)
I wonder if putting it in an addition table format might make it easier?
I've been wanting to do some symmetry group stuff. Bookmarking this for summer. I'd need to play around a lot to see if I can find a simple angle.
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The nice thing that happened in class today:
Grade 5 students solve a puzzle where they put cuneiform numbers in order (there is no guidance, just work with the symbols, how do you order them?)
I told them they are like archeologists cracking a code. They did it!
"But where is zero?"
"It wasn't invented yet." I said this seriously. I mean ... it's true.Later that day the same student asked if it was a joke. I got to tell them no! Zero had to be invented. Everything had to be invented!
@futurebird So very Cool!
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I wonder if putting it in an addition table format might make it easier?
I've been wanting to do some symmetry group stuff. Bookmarking this for summer. I'd need to play around a lot to see if I can find a simple angle.
@futurebird @johncarlosbaez addition table, multiplication table, it doesn't matter, its an abstract operation. But yeah, I do call it "addition", not multiplication, at least when introducing this stuff.
I think I have a reasonably simple angle for introducing the symmetry group of the square, and that's (imperfectly) represented in the repo as it currently exists. You should print out the calculator front-to-back and play with it for a bit.
I have somewhat developed ideas about how to move from the intuitive approach of my mechanical number line for D_4 to implementing the arithmetic of D_4 using pencil-and-paper calculations. Namely, I think the semidirect product, the 2x2 integer matrix approach, and the permutation-based (i.e. subgroup of S_4) approach are particularly notable.
I don't know where I'd place the lesson on automorphisms, as honestly it need not depend on anything other than the intuitive approach. On the other hand, I'd probably want to prioritize at least one or two of the pencil-and-paper approaches to performing addition in D_4.
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
Introduce them to 10-adic numbers, where there's more than one zero.
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@futurebird More... Zero? They do know what zero plus more zero is, right? 🤭
@faithisleaping @futurebird I mean, nonstandard analysis and infinitesimals are a thing, so maybe they're just really forward thinking.
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
@futurebird If there can be multiple infinities...just sayin'.
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
@futurebird Dedekind showed that any two models of Peano arithmetic are isomorphic. In laymen's terms, if there is something that works like we expect arithmetic to, it will have just the one zero.
This is not obvious, and your student is to be commended for trying things out!
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
@futurebird different from undef, null, positive zero, and negative zero? ... okey...
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@futurebird this story has saved my day from being bleh.
@Unixbigot @futurebird damn warp core runs on taking the local warp constant and dividing by the number of seconds since midnight. It crashes at midnight with an F_DIV_ZERO error.
Kit Bashir (@Unixbigot@aus.social)
“We’re out of warp, what’s wrong?” “Nothing, it happens every morning at this time. Just reset it. You haven’t been getting that on B-shift?” “No, and how long—holy crap!” “What?” “Warp degradation has added three days to our ETA so far. TELL ME if stuff breaks; if we miss the book sale on Rigel Four everybody’s getting Curium ash for christmas.” #Tootfic #MicroFiction #PowerOnStoryToot
Aus.Social (aus.social)
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@futurebird that's awesome. I don't share my work very often, but especially because you mention cuneiform, I actually have "invented" a new zero, called zo, in a modern base-60 number system, inspired by the Babylonian system and Wu Xing
hyxos_numerals/GRAMMAR.md at main · hyxos/hyxos_numerals
A rust library for working with the Hyxos Numerals - hyxos_numerals/GRAMMAR.md at main · hyxos/hyxos_numerals
GitHub (github.com)
There is a very poorly written and not maintained api to generate the glyphs at https://hyxos.io/docs
I'm plodding away in my spare time trying to turn it into something more usable to make it more accessible for everyone... up to this point it's mostly been used by my wife and I to build card game prototypes.
I'm hoping to release a much more polished glyph builder this year, I really want to make a typeface, and oh boy, that is a deep, deep rabbithole
@macbraughton I love Mastodon.
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