@VisualAlgebra

Associate Professor (Clemson) | AIMS Lecturer (South Africa) | Author: "Visual Algebra" (forthcoming) | YouTuber | First Gen | Homesteader | Dad to Ida & Felix

 Clemson, SC
math.clemson.edu/~macaule/
Joined July 2009
  • Tweets6.2K
  • Following1.4K
  • Followers4.9K
Replying to @CaptainJoey63 @lateforapex
Nah, they’re in for a surprise attack there. Defense won’t hold.
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Replying to @ChrisCuomo
@ChrisCuomo
MS Word’s paper clip: “It looks like you’re trying to write a sentence. Would you like help?”

ALT MS Word’s paper clip: “It looks like you’re trying to write a sentence. Would you like help?”

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Replying to @bipolarguy86 @KalshiSports
That one was my #1 heartbreak for over a decade, until…
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ALCS Game 7, 2025

ALT ALCS Game 7, 2025

Super Bowl XLIX

ALT Super Bowl XLIX

Kalshi Sports
@KalshiSports9 Dec 2025
What's the biggest heartbreak you've ever experienced?
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Replying to @GretchenCarlson
This is perhaps the most surprising part of it…
“We are highly confident this text is entirely human”

ALT “We are highly confident this text is entirely human”

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Replying to @sorryforbrie
I bet they’re all from the east. I’m seeing the exact opposite sentiment. The PNW is just special. 🔱
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Replying to @raincityseries
Guilty as charged
Bobby Ayala

ALT Bobby Ayala

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Replying to @missmaree73 @joshtpm @brianbeutler @AJDelgado13
Actually he did! (I agree that most in his party would not have.)
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Replying to @chamath @NYSE
@chamath @NYSE
"Corporate needs you to find the differences..." Elizabeth Holme's Partner has a new start-up, vs. Chamath planning on launching another SPAC.

ALT "Corporate needs you to find the differences..." Elizabeth Holme's Partner has a new start-up, vs. Chamath planning on launching another SPAC.

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“Worst person you know just made a great point”

ALT “Worst person you know just made a great point”

Ted Cruz
@tedcruz18 Jun 2025
Tucker Carlson is obsessed with defending Russia and the KGB thug that runs it.
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Replying to @WhiteHouse
Hey Vatican, you know what would be the ultimate troll? 👀
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Replying to @RepJackKimble @LauraLoomer
He’s too young to be pope. But you know who isn’t? And is actually Catholic? Pope Brandon I
Pope Brandon I

ALT Pope Brandon I

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The Vatican has the chance to do the funniest thing...
Pope Joe Biden

ALT Pope Joe Biden

Lindsey Graham
@LindseyGrahamSC29 Apr 2025
I was excited to hear that President Trump is open to the idea of being the next Pope. This would truly be a dark horse candidate, but I would ask the papal conclave and Catholic faithful to keep an open mind about this possibility!   The first Pope-U.S. President combination has many upsides. Watching for white smoke…. Trump MMXXVIII!
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Replying to @cperruna
And then there are those whose agenda prevents them from even disclosing the point or percentage drops… 🤦‍♂️
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Replying to @omarshehab
My only insight here is on the concept of a hidden subgroup, not applications or algorithms. The condition "f(a)=f(b) iff aH=aH" can be viewed w/ cosets partitions. Lecture 3.3: Cosets youtu.be/kFDzrzhgrbk Lecture 3.4: Normalizers & normal subgroups youtu.be/SkmcLkKfo6A

Visual Algebra, Lecture 3.3: Cosets

Due to the structural regularly of Cayley graphs, if a subgroup con...

youtube.com
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Here's a summary of our 4 examples actions and the fundamental features. Please share, I think this will be helpful to algebra students and instructors alike! Lecture 5.4: Examples of actions youtu.be/2mMClKyFHNw 16/16
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**Don't let someone tell you that you can only take the quotient by a subgroup if it's normal!** You can ALWAYS quotient by the right cosets and get a G-set. If it's normal, that G-set happens to be a group. [In Ch 3, we constructed G/H by collapsing by *left* cosets.] 15/16
Collapsing the Cayley graph of D₆ by the left cosets of <f> fails b/c it's not normal, so the quotient isn't a group. However, if we collapse by the right cosets, we get a G-set!

ALT Collapsing the Cayley graph of D₆ by the left cosets of <f> fails b/c it's not normal, so the quotient isn't a group. However, if we collapse by the right cosets, we get a G-set!

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Our last action is new, but arguably the most important: G acting on cosets of some subgroup H≤G. Also, *every* transitive G-set is isomorphic to such a G-set! This is constructed by collapsing the *right* cosets in a Cayley graph. 14/16
Five fundamental features of a group action of G on the cosets of some subgroup H.

ALT Five fundamental features of a group action of G on the cosets of some subgroup H.

Action graph of D₆ acting on a set of binary hexagons, and an isomorphic action graph of D₆ acting on the cosets of <f>.

ALT Action graph of D₆ acting on a set of binary hexagons, and an isomorphic action graph of D₆ acting on the cosets of <f>.

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One of my favorite examples is A₄, and it's something we've seen since Chapter 3. These last 2 pictures are from an old lecture, way back when we introduced normal subgroups: Lecture 3.4: Normalizers and normal subgroups youtu.be/SkmcLkKfo6A 13/16
Action graph of A₄ acting on its subgroup by conjugation, superimposed on the subgroup lattice.

ALT Action graph of A₄ acting on its subgroup by conjugation, superimposed on the subgroup lattice.

Fixed point table of the action of A₄ on its subgroups by conjugation.

ALT Fixed point table of the action of A₄ on its subgroups by conjugation.

Partition of cosets of three subgroups of A₄: cosets in the normalizer are blue, and others are red.

ALT Partition of cosets of three subgroups of A₄: cosets in the normalizer are blue, and others are red.

Subgroup lattice of A₄ with conjugacy classes circled.

ALT Subgroup lattice of A₄ with conjugacy classes circled.

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