Further along my claim that what separates mathematicians from everyone else is:
and that learning 20th-century geometry might expand your imagination beyond the usual impoverished shapes of taxonomies.
Here are some calisthenics you can do with a pen and paper that I hope give you a feel for what a (mathematical) group is. (It’s a shame that “group”, “set”, “class”, “category”, “bundle” all have distinct meanings within mathematics. Another part of the language barrier.)
Think of “a group” this way. A group catalogues the relationships between “verbs”.
That is: think of a function as a “verb” and the thing it operates on as a “noun”. One of the tricks of abstraction is that these can be interchanged. Maybe what that might mean will already come clear from this example.
Starting with a pentagon, which I’ll just represent with five numbers for the points. (So: whatever works here might work on other “circles of five”—or "decks of 52"—or … something else you come up with!) That will be the one “thing” or “noun” and in the group exploration you’ll see that the “structure of the verbs” is more interesting than whatever they’re acting on. (This is why in group theory the name of the object is usually omitted and people just list the operations/verbs.)
In John Baez’s week62 you can read about reflection groups. I picked two “axes” in my pentagon ⬟ arbitrarily. If you’re writing along you can draw a different -gon or different axes. Reflection is going to mean interchanging numbers across the axis (“mirror”).
It’s the same as reflecting the Mona Lisa except you don’t have to re paint the portrait every time. The same 2-dimensional plane can be indexed by the numbers more easily than by the whole image. (Unless you’re following along with computer tools and you’ve chosen “the square” as your shape. Then transforming
Mona is probably more interesting.)
Without my saying so it’s probably obvious that reflecting twice would bring you back to the start. Flip Mona upside-down, flip the pentagon ⬟ along
a, then repeat.
If you wanted to give “starting point of
noun" a verb-name you could just say
What that establishes, formulaically, is that
⬟). Where ƒ is “flip”. We’ve also established that
ƒ=ƒ⁻¹. Trivial observation, maybe-not-trivial in formula form! After all, suppose you had some science problem and it included a long sequence of
ƒ(g(ƒ(ƒ(ƒ(h(g(X))))))) type stuff. You could make it shorter (and maybe the resulting formula or computation easier) if you could cancel ƒƒ’s like that.
That works for either of my pentagon ⬟ reflections
What group theory is going to talk about is how the two verbs interact. What happens when I do
a(a(b(a(b(⬟))))) ? Well I can already simplify it by reducing any trains of
Above are the first few results of
a∘b∘a[⬟]. (NB: “The first” operation is on the right since the thing it’s acting on only appears all the way to the right. So in group theory we have to read right-to-left ←.) I’ll write a bit more text for those who want to continue the chain on their own to give you time to look away. You could also try doing
b∘a∘b[⬟] where I did
a∘b∘a[⬟] (read right to left! ←).
Just like it’s “sort of amazing” in some sense that
- •••—•••—•••—••• (four groups of three…um, regular meaning of “group”!) is the same as ••••—••••—•••• (three groups of four)…and that not only in this specific case but we could make a “law” out of it
So is it also a bit amazing that maybe these reflection laws will be order-invariant in some sense as well.
That may seem like less big of a deal if you think “Everything in maths is commutative and symmetrical”—but it’s not! And most things in life are not commutative or symmetrical. Try to drink your milk and then pour it into the glass or don your underwear after your pants.
It’s also not so obvious (if nobody had told you the answer first and you just had to figure it out yourself) that
b∘a∘b∘a∘b∘a∘b∘a∘b∘a[⬟] = ⬟.
Another fairly easy shape to explore its groups is the square. (And what goes for the square, goes for the plane 𝔸²—or for 1-dimensional complex numbers ℂ.)
That’s the end of what a group is. Next: looking ahead to put them in context.
All of these activities amount to exploring the building blocks of a particular group.
But someone (Arthur Cayley) has also come up with a good way to look at the entire structure of the verbs.
Which is ultimately where this theory wants to go: to help us compare & contrast verb-structures. (Look up “group homomorphism”.) Or to notice that two natural phenomena exhibit the same verb-structure.
You can download a free program called Group Explorer to look at various Cayley diagrams.
In an upcoming post called The Shape of Logic, the Logic of Shape I’ll talk about the relationship between groups and manifolds.
Where I’ll ultimately want to go with this is to call groups a “periodic table of elements” for logic. That may not be exact but it’s a gist. Given that semigroups, groups, Lie groups, and other assumption-swapped variations on the group concept usually turn out to be “Factorable” into simple components (Jordan-Hölder, Krohn-Rhodes, etc.)—and assuming that the Universe somehow builds itself out of primitives sufficiently determined or governed by mathematics—or at the least, that normal people can learn this periodic table and expand their imagination with powers of 20th century geometry.