polytopes

Let me start by admitting that this blog is a social experiment to see how long people take to ask me “what the fuck?” (I am kidding)

(seriously though, I am always a bit of chaos. Nothing out of the ordinary, though admittedly a bit sick and depressed at the moment. Thank you and all the love for caring y’all. Love you all 😉 )

Now, on to the other stuff: At some point in my life, I was doing my PhD and being an all around useless student (I spent most of my office computer hours watching Game of Thrones and Gossip Girl; this is not to say I did not work, but I usually cannot sit in an office chair and work. I paced around outside. Honestly I feel most time was wasted because I was anxious about seeming to work and ending up watching series rather than going out and thinking my own way), my advisor PhD Günter Ziegler and I looked at a gherkin and said: this is going to make a fine math paper. Anyway, tonight I was visited by three ghosts, and they told me a tale.

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I am currently visiting my friend Paco Santos in Santander, Cantabria, and things are off to a rocky start. As I enter his office, he challenges me to a duel.

“I hold a theorem” he says. And after thinking for a few seconds, he adds: “I am also holding a corollary about symmetries”

I am stumped. It is early, and I lack the mental fortitude of morning coffee. I have to think…

The answer: the vertices of the dodecahedron can be partitioned into 5 regular tetrahedra (these are the green diagonals).

Now, you can use this fact to compute the group of symmetries of the dodecahedron! Well, clearly you can take the vertices of one of the tetrahedra to itself. That is the alternating group $A_4$ . But you can also take any tetrahedron to any other tetrahedron, leading to conjugate copies of the same group. Those are 5 copies. Hence, what you obtain finally is the alternating group $A_5$ .