"...excess points each inflict Damage, and the Armor value is reduced to zero for the remainder of the situation."
Where I'm really tossing around is the implications that Supplement 4 introduces. While its careers are basic, they add several new skills and I see some skill assignments that stretch my ideas that I'm getting from the original books. That makes me curious what other folks think.
So again, thanks for your thoughts. I look forward to continuing to read your posts (here, blog, and on COTI) and anyone else who is on a similar path.
ALL OF THEM.
In math you get to make up the rules of the game... but then you have to follow them with utmost precision. You can change the rules... but then you're playing a different game. You can play any game you want... but some games are more worthwhile than others.
If you play one of these games long enough, it doesn't feel like a game - it feels like "reality", especially if it matches up to the real world in some way. But that's how games are.
Unfortunately, most kids learn math by being taught the rules for a just a few games - and the teacher acts like the rules are "true". Where did the rules come from? That's not explained. The students are never encouraged to make up their own rules.
In fact, mathematicians spend a lot of time making up new rules. For example, my grad student Alissa Crans made up a thing called a shelf. It wasn't completely new: it was a lot like something mathematicians already studied, called a 'rack', but simpler - hence the name 'shelf'. (Mathematician need lots of names for things, so we sometimes run out of serious-sounding names and use silly names.)
What's a shelf?
It's a set where you can multiply two elements a and b and get a new element a · b. That's not new... but this multiplication obeys a funny rule:
a · (b · c) = (a · b) · (a · c)
That should remind you of this rule:
a · (b + c) = (a · b) + (a · c)
But in a shelf, we don't have addition, just multiplication... and the only rule it obeys is
a · (b · c) = (a · b) · (a · c)
There turn out to be lots of interesting examples, which come from knot theory, and group theory. I could talk about this stuff for hours. But never mind! A couple days ago I learned something surprising. Suppose you have a unital shelf, meaning one that has an element called 1 that obeys these rules:
a · 1 = a
1 · a = a
Then multiplication has to be associative! In other words, it obeys this familiar rule:
a · (b · c) = (a · b) · c
The proof is in the picture.
A guy who calls himself "Sam C" put this proof on a blog of mine. I was shocked when I saw it.
Why? First, I've studied shelves quite a lot, and they're hardly ever associative. I thought I understood this game, and many related games - about things called 'racks' and 'quandles' and 'involutory quandles' and so on. But adding this particular extra rule changed the game a lot.
Second, it's a very sneaky proof - I have no idea how Sam C came up with it.
Luckily, a mathematician named Andrew Hubery showed me how to break the proof down into smaller, more digestible pieces. And now I think I understand this game quite well. It's not a hugely important game, as far as I can tell, but it's cute.
It turns out that these gadgets - shelves with an element 1 obeying a · 1 = 1 · a = a - are the same as something the famous category theorist William Lawvere had invented under the name of graphic monoids. The rules for a monoid are that we have a set with a way to multiply elements and an element 1, obeying these familiar rules:
1 · a = 1 · a = a
a · (b · c) = (a · b) · c
Monoids are incredibly important because they show up all over. But a graphic monoid also obeys one extra rule:
a · (b · a) = a · b
This is a weird rule... but graphic monoids show up when you're studying bunches of dots connected by edges, which mathematicians call graphs... so it's not a silly rule: this game helps us understand the world.
Puzzle 1: take the rules of a graphic monoid and use them to derive the rules of a unital shelf.
Puzzle 2: take the rules of a unital shelf and use them to derive the rules of a graphic monoid.
So, they're really the same thing.
By the way, most math is a lot more involved than this. Usually we take rules we already like a lot, and keep developing the consequences further and further, and introducing new concepts, until we build enormous castles - which in the best cases help us understand the universe in amazing new ways. But this particular game is more like building a tiny dollhouse. At least so far. That's why it feels more like a "game", less like "serious work".
Even slight changes to the statistics from the human baseline creates some incredibly interesting cultural assumptions.
As an easy example, let's create an uplifted Ape's genetic profile, but keep it close to the human baseline, humans are a type of ape, after all. Let's adjust Str by +1D and Soc by -1D and change Dex to Agi to see how these things change what it means to be sapient
Ape: SAEIDS (compare: Humaniti-SDEIDS) with D of 322221 for each stat.
Clearly they're stronger than humans with a 6+2D roll for Str (a chimp, even only topping 4' can rip the arms off of human quite easily), the average ape uplift is as strong as the strongest humans (Ape str goes from 7-18, with the average at 13, or D).
Instead of Dexterity for Hand-Eye coordination, they have Agility, which gives them Whole Body coordination - their sense of propioception doesn't accumulate in the hands, but extends equally throughout the body. Careers that care about it call for C2 as a Controlling Characteristic and don't call out Dex (rendering checks on Agi as Agi/2), so clearly either there are adaptive control schemes available or apes really are as good with every inch of their bodies as humans are with their hands. The Merchant career's requirement of "To begin Spacehand: Dex" means you're not going to see many ape-uplifts on Merchantmen. Perhaps this has something to do with their Soc...
The Soc dice of 1 gives them a maximum Social stat of 6.
This could mean a couple of things - one, instead of going their own way, they wish to work within 'the system' (geneering, perhaps?), but are limited in their ability to do so. Maybe they have trouble with human social cues, which can be quite subtle, or are subject to discrimination outright. Or it could simply mean that they naturally form smaller, but stable (otherwise Charisma would be the stat), groupings than humaniti does: topping out at 25, say, as opposed to humaniti's 50 or so. Either one alone or both together are just as good and interesting.
Here's a more difficult example: The uplifted Octopode.
The genetic stats could easily be SGVIIC with dice of 122232. This is very different. Their stats are Str, Grace, Vigor, Int, Instinct, Charisma.
They're not nearly as strong as humans, averaging 3.5, while humans easily double that, but they have no bony structure to provide for leverage in assisting every-day tasks in a human dominated culture. They might easily be stronger than humans, but Str 1D shows how difficult it is to operate as a non-skeletal being in a skeletal world.
They have Grace, instead of Dex, meaning their control over their limbs is phenomenal. Considering that they have between 8 and 10 depending on the species, this is probably a good thing.
Vigor indicates that their stamina is just not as good as that of humans, the Octopodes excel at short term tasks, but longer-term planning is cut short due to the limited time per day that they are active.
Instead of Education, Octopodes have Instinct, and 3D at that, giving them 6+2D in it. Their lineage doesn't include much parenting, and they learn everything as they go and by genetic memory. However, they're GOOD at it, able to almost compete with humans at their institutions of higher learning. The average Instinct for an Octopode is 13, and that reduces down to a 4, quite below the human's Educational stat. Humans don't want to breed a superior being, and the troubles that Octopodes occasionally encounter with educational systems helps keep humaniti feeling better about themselves. Fortunately, most educational institutions allow the use of native Int to pass courses, which the Octopi are easily the equal of humaniti in.
Octopodes use Charisma instead of Social Class, they don't have a species history of congregating or self-organising (except for mating, but that's not quite the same), so the most eloquent or simply the loudest of a group usually gets the others to go along. Long term social creations are a new thing to these newly fully-sapient beings, and they're still having trouble adapting to the human's world.
Think of the real line as a mountainview silhouette. Pick two points on that jagged horizon line. Given the first's exact
position, can you imagine finding out the second point's (apparent) elevation just by adding the miniscule changes you notice while slowly sliding your gaze along the edge towards the target?
On the real line there's just one way to get from here to there. In the complex plane there are many, so the appropriate notion here is the path integral. Instead of an area below some function graph, it should compute the target point's position, solely from the starting point and the changes measured while traversing the path.
Coming from there, it seems quite natural to require the function to behave nice enough so it wouldn't matter which path we chose. If you try to make up a situation where the result differs depending on the path you took, you'll notice that there must be some kind of jump, a cliff in the landscape.
Functions without cliffs like these are called smooth. Many functions do have cliffs (and isolated singularities that look like needles), but are smooth almost everywhere else, or at least on a subset of C. Typically an open subset. Complex geometers love open subsets! You find the definition of path integrals in the first formula in the picture below.
Puzzle 1: Who came up with it? Euler??
The path integral is going to be your favorite tool for everything! The other two formulas compute winding number and path length of a path. Go and insert the definition into the formulas! Funny creatures, eh?
Puzzle 2: What if a path visits a point twice, crosses itself, or is partially on top of itself?
Let's get visual: You should know about manifolds! Here's a simple one: The inverse of z -> z² also known as the complex square root function. Like the real square root, it is a multi-valued function. To get -1, you can either square i, but -i squared also yields -1. There are always two possible solutions, except at the origin (a branch point), where the two roots coincide: -0 = +0
Put your finger on the point called 1, say our function returns +1 there. Now slowly move your finger counterclockwise in a half circle towards -1. The function returns a point halfway on the arc between your finger and the starting point 1. As your finger gets to -1, our instrument displays i, like in the example from the previous paragraph. Moving further on the circle, still counterclockwise, and cycling back towards 1 we finally get it to show... -1!
Contradiction!! Or is it? Since the two roots are always 180° apart from each other, we'd have noticed if a small movement of our finger would have sent the corresponding function result all the way over to the other root. How about this: Moving once around the origin brought us to the other sheet! Moving around twice will bring us back again! In pictures:
At least, that's how you're supposed to visualize it. If you let a computer draw a phase diagram for you, it'll show you only one sheet with a jump, a discontinuity along some curve. In our case that curve could be the negative real line, and the opposite colored (negative) of the portrait happens to look like the other sheet!
Puzzle 3: What's the value of a path integral along a closed path (a loop) for our square root function, if the origin is not inside?
Puzzle 4: What if the origin is inside?
Puzzle 5: What if we run the path backwards around the origin?
For more complicated functions you may get more complicated manifolds. The combinatorial diagram at the bottom depicts an example.
It seems as if the square root function could accomodate an argument angle range from 0 to 720°... It indeed does make sense to put it like that, because we can compute a square root using the exponential function and its inverse, the complex logarithm:
sqrt(z) = z^(1/2) = e^(log(z)·1/2)
Geometrically, before insanely stretching or squeezing things, meanwhile converting multiplication to addition(!), the exponential translates its input from cartesian to polar coordinates! And the logarithm does the reverse. Because the exponential function is periodic and repeats after 2·pi·i, its inverse, the logarithm, has an infinite number of sheets. Given one, you can get to the next by adding a full turn:
e^z = e^(z + 2·pi·i)
z_0 = log(n),
z_k = log(n) + k·2·pi·i
Notice that our square root, according to the recipe, will repeat after two full 360° turns (adding 4·pi·i)! Or just look at the picture here:
Okay, studying paths is a nice way to explore complex functions. It then often seems to be a good idea to only look at small pieces of our function at a time. But there is another way to quickly get at function values:
Cauchy's integral formula: If you know the behaviour of a function on a simple loop, you can compute it for all the points on the inside as well. Here's how ("Sy" is my ascii-art attempt to write an integral sign over a path gamma):
f(a) = 1/2·pi·i Sy f(z)/(z-a) dz
Puzzle 6: What's wrong with the purple image in the example section I just linked to?
Even better, differentiation can also be written as a path integral!
f'(a) = Sy f(a)/(z-a)² dz
Integrating and differentiating almost look alike! And that's a very good thing, wikipedia explains like this:
> complex differentiation, like integration, behaves well under uniform limits – a result denied in real analysis.
Discovering these things must have meant excitement even to great names like Felix Klein, Augustin-Louis Cauchy, Leonard Euler, Carl Friedrich Gauss, Bernhard Riemann, and many others. What disappointment when we learned that fields are not only of exceptional beauty, but also very rare...
The geometric emphasis is due to Carl Weierstrass, who, after he had cleaned up the material a bit, successfully popularized this approach. I should read his accounts, I heard they still look very modern!
To be continued...
part 1 in this series: Complex numbers explained in shocking facts
Green's theorem relates a path integral to a double integral over the enclosed area. If you're used to see integrals as surface areas you may want to have a look:
I'm indebted to Elias Wegert for writing things up nicely in his book, together with many cool pictures:
Elias Wegert: "Visual Complex Functions - An Introduction with Phase Portraits" (2012)
#complex #numbers #explained
I have a VM rack that's disconnected from the internet and need to put the new Fedora 22 on it, but the Live CD only comes w/ about 1.5GB of OS. Is there a full 7-8GB DVD lying around? My search-fu is failing me.
I'd really rather not have to manaually select RPM packages and import them into the environment one by one, I'd rather just grab the latest complete standard repository once a month or so and up date software that way.
Before my recent move to the fine state of Maryland:
I am late of the Board of NARAL Pro-Choice VA
I am a former Director of Communications of Progressive Democrats of America NoVA Chapter
I was the Co-President for the Presence Committee for the Loudoun County Democratic Committee
When I have a few seconds spare time, I like to play OSR style D&D and Munchkin.
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