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 Post Posted: Mon Jun 17, 2013 12:56 am 
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Magic has 3 elements and 3 axes as explained by Sizemore in Book 1, Page 38a. Each of the 24 disciplines of magic is "aligned" with either Erf, Fate, or Numbers. Life, Motion, and Matter are called "elements", but then in Book 0, Episode 56 Wanda says that Life, Motion, and Matter are also axes of Magic.

That makes me wonder why they are called axes and it suggests to me that there must be some sort of 6-Dimensional space of Magic in which the various disciplines live, because that is the only reason I can think of. Using Cartesian coordinates to represent points in the space, we can give each axis a unit vector:
  • Life: (1,0,0,0,0,0)
  • Motion: (0,1,0,0,0,0)
  • Matter: (0,0,1,0,0,0)
  • Erf: (0,0,0,1,0,0)
  • Fate: (0,0,0,0,1,0)
  • Numbers: (0,0,0,0,0,1)
Then each discipline could be represented as the linear subspace whose basis is the unit vectors of its axes. All of Clevermancy would be lines, with Luckamancy being exactly the Erf axis, Healomancy being exactly the Fate axis, and Moneymancy being exactly the Numbers axis. Hocus Pocus, Spookism, and Stuffamancy would each be planes defined by two axes, while Eyemancy, Hippiemancy, and Naughtymancy would be 3-dimensional spaces.

Unfortunately the subspace idea has a serious problem because some disciplines would contain other disciplines. Healomancy would be completely contained by every Fate aligned discipline, and Carnymancy would contain every Fate aligned discipline. So instead of making each discipline into a linear subspace, the disciplines could be represented by non-overlapping shapes in 6-dimensional space. There's no way of telling what those shapes would be, but surely they must be contained in the subspace for each discipline, since doing otherwise would be giving a discipline size parallel to an axis that it doesn't have. We also know that each shape is aligned to one of Erf, Fate, or Numbers, which means that it has the same distance from all points on the axis, but the shape of a discipline isn't necessarily aligned to its other axes.

Wikipedia has an article about Six-dimensional space which may offer some clues about why Magic is defined on 6 axes in Erfworld. Looking at magic this way suggests that Erfworld in total is a 9-dimensional space (at least) which strongly reminds me of String Theory, just like all the talk of Grandiocosmic Strings. Should a true fan of Erfworld understand String Theory?

I'm just guessing that all 6 dimensions of Magic are spacial dimensions because that's the sort of dimension that people usually think about, but maybe the Fate axis runs along the temporal dimension so that Erfworld actually only has 8 spacial dimensions, the 3 obvious ones, plus 5 of the magical dimensions.

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     Post Posted: Tue Jun 18, 2013 11:47 am 
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    Lilwik wrote:
    Unfortunately the subspace idea has a serious problem because some disciplines would contain other disciplines. Healomancy would be completely contained by every Fate aligned discipline, and Carnymancy would contain every Fate aligned discipline.


    You just said they're linear subspaces. Do you mean, 'subspaces of dimension 1' or something else? Because the only way to get one line to contain another is for them to be the same line.

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     Post Posted: Tue Jun 18, 2013 5:11 pm 
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    I was sort of imagining Erf spells as branes, like the surface of a drum. You know, to continue the music sub theme magic spells seem to have.

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     Post Posted: Tue Jun 18, 2013 11:44 pm 
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    drachefly wrote:
    You just said they're linear subspaces. Do you mean, 'subspaces of dimension 1' or something else?
    A linear subspace is a special kind of subspace that could be a line but not always. It's a subspace that contains all linear combinations of the vectors of the subspace. So in other words, take all of the axes of some discipline of magic and that gives you a set of unit vectors that point in the direction of each axis, like {A, B, C}, then you can choose any scalars (a,b,c) and construct a vector aA + bB + cC and the subspace is all the vectors you can get that way. Wikipedia has a page about linear subspaces.

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     Post Posted: Thu Jun 20, 2013 10:14 am 
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    Okay, so you mean just a 'subspace in linear algebra', not 'a space with a single vector as its basis'

    I'm not sure I like this, for the reasons you gave - some fields would have only one dimension, others would have four.

    Also, I'm not sure you could ever generate a prediction with this. You have a representation, but it doesn't seem to do any work.

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     Post Posted: Thu Jun 20, 2013 4:05 pm 
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    drachefly wrote:
    Also, I'm not sure you could ever generate a prediction with this. You have a representation, but it doesn't seem to do any work.
    It cannot generate a prediction right now, at least that I can see, but it has potential for the future. Erfworlders have a theory of magic that we know exists but never get a complete explanation for. Casters have a far deeper understanding of magic than we will ever get by watching them work, but it should be possible to steal some of their insight by studying how they talk about magic. It could contain clues to the things which haven't been explicitly revealed, such as the nature of Weirdomancy.

    They must use the word "axis" for a reason. "Axis" doesn't have very many potential meanings. If this isn't what they are implying by the use of that word, then what else could they mean by it?

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     Post Posted: Thu Jun 20, 2013 10:48 pm 
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    They could mean something other than axis, and just suck at English... Also, it already sort of made a prediction that healamancy is included in all other fate magics. Or did I misunderstand? (Totes likely)

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     Post Posted: Thu Jun 20, 2013 11:08 pm 
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    Shai_hulud wrote:
    They could mean something other than axis, and just suck at English.
    I'm not aware of any evidence that suggests they suck at English.

    Shai_hulud wrote:
    Also, it already sort of made a prediction that healamancy is included in all other fate magics. Or did I misunderstand? (Totes likely)
    I presume that Healomancy is not contained within all other Fate disciplines and therefore the linear subspace idea is too simplistic and magic theory that casters understand in Erfworld is more complicated, but that's obvious for other reasons as well. I only suggest subspaces because if we are forced to imagine disciplines as regions in 6-dimensional space then subspaces are an easy way to do that.

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     Post Posted: Fri Jun 21, 2013 9:33 am 
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    Lilwik wrote:
    I'm not aware of any evidence that suggests they suck at English.
    Would improper use of the word "Axes" qualify?

    Lilwik wrote:
    I presume that Healomancy is not contained within all other Fate disciplines and therefore the linear subspace idea is too simplistic and magic theory that casters understand in Erfworld is more complicated, but that's obvious for other reasons as well. I only suggest subspaces because if we are forced to imagine disciplines as regions in 6-dimensional space then subspaces are an easy way to do that.
    Actually, for a long time I suspected that clevermancy really was intended by Rob to be the core magic that others were build up out of. I don't think that is necessarily true now (partly because of healamancy, partly because of the matter powers of moneymancy) but I thought so for a long time. So it's not that outlandish an idea. We would need more info on how healamancy works first.

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     Post Posted: Fri Jun 21, 2013 10:17 am 
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    Hmm. Now I'm thinking of function spaces. In particular, a single orthogonal basis for each class on the table.

    (life?x:1)(matter?y:1)(motion?z:1)e^-(xx+yy+zz)
    It's all normalized, and every class ends up orthogonal to the others.

    However, to fully distinguish the disciplines, we need to incorporate Erf, Fate, or Numbers. We could just take the direct product with C3, but we could treat it really like an axis - so that they're spin orientations. Each would have +/- x^, +/- y^, or +/- z^ in some linear combination thereof to make them distinct from the first set of directions.

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     Post Posted: Sat Jun 22, 2013 3:19 pm 
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    Isn't it odd that Clevermancy can do things that other magics can't? For example, we've very recently seen a Predictamancer needing the services of a Healomancer, even though Healomancy is Fate with no elements, and Predictamancy is Fate and Life. The only conclusion seems to be that the presence of the Life axis actually blocks Predictamancers from doing Healomancy somehow and it takes training to overcome that.

    Perhaps instead of thinking of representations for disciplines in an 6 dimensional space of magic it would be better to think of Erfworld as a 3 dimensional brane floating in 9 dimensional space. Then when magic is cast it can cause the a small portion of the brane to shift along one of the extra 6 dimensions.

    So for example shifting the brane along the Matter axis causes stuff to appear or disappear. Think of the brane as a landscape and the Matter axis as elevation, and some point on the Matter axis is the sea level; if the land is above sea level there is no water, and if it is below sea level there is water. So with the Matter axis there would be a stuff level and by lowering the brane below the stuff level a Stuffamancer can put stuff into Erfworld. Perhaps Life and Motion work similarly, though it's not as easy to picture.

    The meanings of Erf, Fate, and Numbers is not as clear, but surely the key to understanding them is in Book 0, Episode 1. "Zero is a flat country" means that the Numbers axis is numbered and zero is a point on that axis (like the stuff level) where the brane would be perfectly flat (in that dimension) if all points were at zero.

    "Its numbers were being shaped and guided by the firmament, by what magic theorists called the Erf Axis." So the Erf Axis is the firmament and according to Wikipedia: "The firmament is the sky, conceived as a solid dome. According to Genesis, God created the firmament to separate the 'waters above' the earth (the source of rain) from those below (in the underworld)." So in other words the Erf Axis is the sky, but that can't be right. That would be taking it too literally; it must mean the sky and everything beneath it, the whole of Erfworld. In that case, what separates the Erf Axis from Life, Motion, and Matter which seem to be included in Erf?

    Perhaps what I'm imagining as 6-dimensional isn't actually 6 independent dimensions but is actually only 3 independent dimensions and our two sets of axes: Life-Motion-Matter and Erf-Fate-Numbers are actually two ways to look at the same three fundamental degrees of freedom that magic has in reshaping Erfworld, much like Cartesian coordinates and spherical coordinates. So shifting the brane along the Numbers axis alone can end up shifting the brane along Life, Motion, and Matter, except that certain disciplines are locked on certain axes, so when Dittomancy shifts the brane along Numbers it can only do so in ways that don't change its position on Life and Motion. So creating the clone Slately was not a change in the Life axis and the clone shared the same soul as the original.

    Again I am troubled by Clevermancy since it is hard to imagine how the brane could be shifted along Erf, Fate, or Numbers without affecting Life, Motion, or Matter. Perhaps that means the brane has some degree of freedom beyond the three elemental axes, or there is a way to use juice that has nothing to do with moving the brane. And then there is Predictamancy which doesn't seem to move the brane at all, even though it should be free to move it along the Life axis.

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