Fractal Kitty

Drafted a 3rd inquiry activity for D3 and D4 play with flips and rotations.

I am hoping it's not to dry. I'll read through it again tomorrow.

#mtbos #dihedral #grouptheory #iteachmath

claude

searching for any structures / theory that involve a particular operation on non-empty lists of postitive integers like "the length of the list multiplied by the least common multiple of all the items in the list"

any ideas? references to any literature would be very appreciated if you know of any.

#askfedi #math #maths #mathematics #combinatorics #GroupTheory #DiscreteMath

2something

Me: If you're a student in my class you will do group presentations, meaning that you will organize into a subset of the class, stand in front of the room, and talk about math to the rest of the class.

Also me: Let's talk about group presentations, a completely abstract algebraic concept that have absolutely nothing to do with standing in front of the room talking about your work.

Also also me: Okay, time for group presentations
about group presentations.

#Algebra #ITeachMath #GroupTheory #GroupPresentation

HoldMyType

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existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way.
einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral.[3] Such anisohedral tiles were found by Karl Reinhardt in 1928, but these anisohedral tiles all tile space periodically.
Partition of a plane in closed set - tile
2022, hobbyist David Smith discovered a "hat"-shaped tile formed from eight copies of a 60°–90°–120°–90° kite (deltoidal trihexagonals), glued edge-to-edge, which seemed to only tile the plane aperiodically.[8] Smith recruited help from mathematicians Craig S. Kaplan, Joseph Samuel Myers, and Chaim Goodman-Strauss, and in March 2023 the group posted a preprint proving that the hat, when considered with its mirror image, forms an aperiodic prototile set.[
Wiki
#grouptheory

Feb 25, 2025, 06:58 · · · 0 · 0
Jon Awbrey

Riffs and Rotes • Happy New Year 2025
inquiryintoinquiry.com/2025/01

\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)

\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)

\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)

No information is lost by dropping the terminal 1s. Thus we may write the following form.

\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]

The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.

Riff 2025
inquiryintoinquiry.files.wordp

Rote 2025
inquiryintoinquiry.files.wordp

Reference —

Riffs and Rotes
oeis.org/wiki/Riffs_and_Rotes

#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes

Jan 03, 2025, 19:00 · · · 1 · 0
Alex Nelson

Want to formalize something in Mizar but don't know where to begin?

I'm starting a new series of posts with project ideas, starting with Loops! They're needed to formalize the sporadic groups (as found in, e.g., Aschbacher's book "Sporadic Groups").

#Mizar #proofassistant #mathematics #grouptheory

thmprover.wordpress.com/2024/1

Loops in Mizar

Project Summary: Formalize Section 12 of Aschbacher’s…

Ariadne's Thread
Joshua Grochow

A shop called "Wreath Products" that sells mathematical puzzle toys, mobiles (like for baby cribs or art)...and, sure, also decorative wreaths.

#math #GroupTheory #algebra

Joshua Grochow

Sat Dec 7, 2024 on zoom & in-person

Session in memory of Richard Parker at the annual Nikolaus conference at Aachen (on group & representation theory). Main speakers:

Gerhard Hiß
Gabriele Nebe
Colva Roney-Dougal

math.rwth-aachen.de/Nikolaus20

#Math #GroupTheory #RepresentationTheory #Algebra

Nikolaus Conference 2024

www.math.rwth-aachen.de
Fedor Indutny

This textbook has heavy “demon summoning” vibes to it 😅

#GroupTheory

183231bcb

A normal subpost is a subpost whose left and right coposts are equal.

#GroupTheory #SubPost

Alex Nelson

Formalizing groups in Mizar, a review and assessment. (Originally, I wanted to entitle this "Get in losers, we're formalizing groups", but my accordion player vetoed it.)

This is an experiment where we start with "book definitions" as found in, say, Bourbaki, and then work our way towards its corresponding formalization in #Mizar.

#ITP #GroupTheory #Math

thmprover.wordpress.com/2024/0

How Mizar Formalizes Groups

Whenever I learn a new proof assistant, I always look…

Ariadne's Thread
Joshua Grochow

@gilesgardam Adding hashtags like #GroupTheory #Algebra will help people find this post (more important on mastodon than some other microblogging platforms, b/c "there is no algorithm here", or rather, the timeline is only chronological based on follows & searches)

Mauro Artigiani🍉

What are good references for #GroupTheory beyond the introduction? @ProfKinyon?

#Math

Wannes Malfait

Awesome video showcasing how group theory appears in the mathematics of error detection: youtu.be/yaoSdFAL4UY
#Math #GroupTheory

Markus Redeker

@j2kun So addition and negation, when operating on the 𝑛-bit integers, generate the dihedral group (en.wikipedia.org/wiki/Dihedral) that expresses the symmetries of a \(2^{n-1}\)-gon.

I didn't think of this before. 😃

#GroupTheory

Dihedral group - Wikipedia

en.wikipedia.org
CMD
@jeffcliff >a #grouptheory textbook

Which one? Unless it goes beyond Galois Theory or goes into the details of how we have classified every finite group, I don't see why this shouldn't be Dummit and Foote or Lang or something standard...but I wouldn't call such textbooks "group theory," as they go into Rings and Fields as well.

>Turing's Pure Mathematics
>A slog
...I'm going to wager it's the Lie Groups that are throwing you?

The big hint is that Lie Groups are their own thing, and you should read a book devoted to them. It's one of those "every Math Ph.D. knows of them but not every Math Ph.D. knows how to work with them" things. I'd strongly recommend shelving the text until you get through the group theory text, and at that point you should skip over the Lie Group paper.
Oscar Cunningham

Here's a tip for beginners in #AbstractAlgebra, #GroupTheory or #NumberTheory.

Sometimes it's useful to think in terms of the divisibility ordering instead of the usual ordering where 0 ≤ 1 ≤ 2 ≤ …. Divisibility gives a partial ordering on the natural numbers where we say that n ≤ m if n divides m. Or we can extend this order to all integers, in which case n and -n become equivalent in this order because they divide each other.

The main difference is that this isn't a total order. For example 7 doesn't divide 8 and 8 doesn't divide 7. The other difference is that while 0 is the least natural number in the usual ordering, it's the *greatest* number in the divisibility ordering, because everything divides 0!

Thinking this way make several definitions make more sense:

1. The least common multiple of 12 and 10 is 60, even though 0 is also a common multiple of 12 and 10. This is because 0 is greater than 60.

2. If an element of a group never becomes the identity when you repeatedly multiply it by itself, then we say it has order 0. This is because the order of an element is the least power of the element that equals the identity, under the divisibility ordering.

3. Likewise we say that a ring has characteristic 0 if 1 + … + 1 is never 0. Again we can use the divisibility ordering to say that the characteristic is the least integer for which adding together that number of 1s gives 0.

#Math #Maths #Mathematics