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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.
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
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
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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
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)
\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)
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
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.png
Rote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.png
Reference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes
#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes
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").
Project Summary: Formalize Section 12 of Aschbacher’s…
Ariadne's ThreadA shop called "Wreath Products" that sells mathematical puzzle toys, mobiles (like for baby cribs or art)...and, sure, also decorative wreaths.
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
This textbook has heavy “demon summoning” vibes to it
A normal subpost is a subpost whose left and right coposts are equal.
#GroupTheory #SubPost
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.
https://thmprover.wordpress.com/2024/02/08/how-mizar-formalizes-groups/
Whenever I learn a new proof assistant, I always look…
Ariadne's ThreadLooking forward to tomorrow’s 2nd workshop on #symmetry, invariance and #NeuralRepresentations at the #BernsteinConference: #GroupTheory, #manifolds, and #Euclidean vs #nonEuclidean #geometry #perception … I’m pretty excited
@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)
What are good references for #GroupTheory beyond the introduction? @ProfKinyon?
Awesome video showcasing how group theory appears in the mathematics of error detection: https://youtu.be/yaoSdFAL4UY
#Math #GroupTheory
@j2kun So addition and negation, when operating on the 𝑛-bit integers, generate the dihedral group (https://en.wikipedia.org/wiki/Dihedral_group) that expresses the symmetries of a \(2^{n-1}\)-gon.
I didn't think of this before.
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.