30:T6aa,md`# Geometric Algebra — a guided tour > *I'm making it public before finishing, and will keep updating it.* I first heard of **geometric algebra** back in college, completely unaware of its use in computer graphics, and it seemed all very fascinating; but I was a little disappointed to learn that it didn't make its way into mainstream mathematics—except for the most basic case of the so-called **exterior algebra**, and for a minor role in representation theory, better known under the name of **Clifford algebra**. Recently on Observable, I was pleasantly amazed at the JavaScript library \`ganja.js\` by [@enkimute](https://beta.observablehq.com/@enkimute) for making the full use of geometric algebra (for various "metrics") to create many mind-blowing mathematical illustrations and animations. In the process of learning it (and JavaScript) myself I'm putting together a little (idiosyncratic) introduction on the mathematical side, hoping that it may be of some help to others. Wikipedia does have a long and detailed description. Here [@kelleyvanevert](https://beta.observablehq.com/@kelleyvanevert/conformal-3d-geometric-algebra) gathered a number of references which may be more useful, and there are many more on the Web. One thing I find confusing about many of them is the plethora of new terminologies: *blades*, *rotors*, and even *bivectors* and *multivectors* I found not so illuminating; and such operations as *reversion*, *contraction*, etc. somewhat off-putting. (Full confession: I will be throwing around math jargons too.) One excellent resource that I recommend is http://geometry.mrao.cam.ac.uk/, especially the slides from two series of lectures from 2015 and 2016. `31:T47f,md`## Exterior algebra Obscure-sounding as it is, exterior algebra is of fundamental importance in modern mathematics; in some sense it is the "evil twin" of the polynomial algebra. The name refers to the exterior (or outer) product of two vectors in a vector space, in contrast with the more familiar inner product. One convenient way to *see* them is that the inner product of two ${tex`n`}-tuples gives a number: ${tex.block` \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} = \sum_i a_i b_i `} whereas their "outer product" ${tex.block` \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} \begin{bmatrix} b_1 & b_2 & \cdots & b_n \end{bmatrix} = \begin{bmatrix} a_1 b_1 & a_1 b_2 & \cdots & a_1 b_n \\ a_2 b_1 & & \\ \vdots & & \ddots \\ a_n b_1 & & & a_n b_n \end{bmatrix} `} is a more complicated object, *outside* the vector space. But this is not the "right" way to look at it. For one thing, these ${tex`n^2`} numbers are highly redundant, as the rows (and columns) are all linearly dependent. There is really just two vectors' worth of information, of course. `32:T519,md`Exterior algebra is only accepted more widely because of *differential forms*. Whenever you compute a multiple intergral, whether over a surface or a volume, you are actually working with differential forms. It is one of those things that are easy to work with, but hard to define rigorously. Assuming we are all familiar with ${tex`dx`} and ${tex`dy`}, viewed simply as *infinitesimals* in the ${tex`x`} and ${tex`y`} directions, then what is ${tex`dx\,dy`}? When we are just integrating it over a region of the plane, ${tex`\int dx\,dy`} gives the total area, so it's natural to regard ${tex`dx\,dy`} as the infinitesimal area "spanned by" the infinitesimals ${tex`dx`} and ${tex`dy`}. As often happens in calculus, it may best be regarded as *signed* area, so we *declare* ${tex`dy\,dx`} to be the negative of ${tex`dx\,dy`}. If you are in three dimensions, you could multiply all three infinitesimals in any order, but the result would be the same volume (up to the sign), to wit ${tex.block`dx\,dy\,dz=-dx\,dz\,dy=dz\,dx\,dy=\cdots`} Here you have it, an exterior algebra. The word *algebra* refers more specifically to the product, which is customarily denoted by a (scary-looking) ${tex`\wedge`} symbol between the infinitesimals, hence the product is often referred to as the *wedge product*. `33:T41d,md`Formally, then, the exterior algebra of ${tex`\mathbb R^3`} is a vector space "spanned" by the symbols ${tex.block` (1) \\ dx \quad dy \quad dz \\ dx\wedge dy \quad dx\wedge dz \quad dy\wedge dz \\ dx\wedge dy\wedge dz `} a total of ${tex`8=2^3`} symbols, which are naturally grouped into four types (being of *grade* or *degree* 0, 1, 2, and 3). If you only allow constants in front of each of the symbols, you get an honest 8-dimensional vector space, sometimes denoted ${tex`\bigwedge \mathbb R^3`}. To be an *algebra*, it comes equipped with the wedge product which is associative, but anti-symmetric (so, for instance, ${tex`dx\wedge(dx\wedge dy)=0`} because ${tex`dx\wedge dx=0`}). By analogy with the polynomial algebra, we may describe the exterior algebra as the algebra of "polynomials in anti-commuting variables"—weird, but it does have the advantage of being finite-dimensional. The concept is credited to Hermann Grassmann, who was way ahead of his time and developed much of (abstract) linear algebra as we know it today.`34:T6a5,md`However, when doing calculus, you would like to put any (say smooth or ${tex`C^\infty`}) function of ${tex`x,y,z`} you want in front of the "symbols", so it is actually an *algebra over (the ring)* ${tex`C^\infty(\mathbb R^3)`} as we say, as opposed to an algebra over the real numbers. This is denoted ${tex`\Omega(\mathbb R^3)`}. The advantage of having a (varying) function in front is that we can differentiate. Differentiating a 1-form we get a 2-form, and differentiating a 2-form we get a 3-form. This operation, simply denoted by ${tex`d`}, is known as the *exterior derivative*, an unassuming name for such an important concept in mathematics (in fact, the good old ${tex`dx`} of Leibniz may be regarded as a 1-form, the exterior derivative of the coordinate function ${tex`x`} as a 0-form). Though it had many precursors (namely in Green's and Stokes' theorems), it was only first brought out in its full glory by Elie Cartan in the early 20th century and gradually gained acceptance. The precise definition for ${tex`d`} is less important; the amazing thing is that, when we write the chain of maps as ${tex.block` 0\longrightarrow \Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \Omega^2 \xrightarrow{d} \Omega^3 \longrightarrow 0 `} the composition of two consecutive arrows is always zero: ${tex`d^2=0`}. (We include two 0's—the zero vector space, consisting of a single element—at the ends, since the only choice of ${tex`d`} there also follows the rule.) We call such a sequence of maps a *co-chain complex*, another obscure term for a central object in mathematics (more specifically, in a *cohomology* theory; the one described above pertains to *de Rham cohomology*). `35:T58b,md` A metric is characterized by its *signature*: ${tex`(p, q, r)`}, the number of positives, the number of negatives, and the number of zeros. More precisely, a metric, as a symmetric bilinear form ${tex`V\times V\to\mathbb R`} and denoted by ${tex`\cdot`} just like a dot product, can be "diagonalized" (thanks to the "baby" Spectral Theorem), i.e. there exists a basis ${tex`\mathbf{e}_1, \ldots, \mathbf{e}_n`} that are mutually "orthogonal" (${tex`\mathbf{e}_i\cdot\mathbf{e}_j = 0`} for ${tex`i\neq j`}, such that, after relabelling and scaling, ${tex.block` \mathbf{e}_i\cdot\mathbf{e}_i = \begin{cases} 1 & i=1,\ldots, p \\ -1 & i=p+1, \ldots, p+q \\ 0 & i=p+q+1, \ldots, p+q+r(=n) \end{cases} `} For example, our Eulidean 3-space has signature ${tex`(3, 0, 0)`}, and our Minkowski space-time has signature ${tex`(1, 3, 0)`}. (The case with ${tex`r>0`} is often regarded as uninteresting in mathematics, but it turns out to be very useful for some applicatinos of geometric algebra.) Such a basis is called a *standard basis*, and any vector ${tex`\mathbf{a} \in V`} can then be expressed as ${tex`\mathbf{a} = \sum_i a_i \mathbf{e}_i`}, or simply ${tex`\mathbf{a}=(a_1,\ldots, a_n)`}, and the inner product of two vectors is given by ${tex.block` \mathbf{a}\cdot\mathbf{b} = \sum_{i=1}^p a_i b_i - \sum_{i=p+1}^{p+q} a_i b_i`} (One may take this to be the definition of our metric.) `36:T4d8,md` Now, we can define the **geometric product** on the (full) exterior algebra ${tex`\bigwedge V`}, the dimension of which is ${tex`2^{\dim V}=2^n`}. The **geometric algebra** will then be the same vector space ${tex`\bigwedge V`} but with this new geometric product replacing the wedge product. (There is apparently no agreed-on notation for it; one might denote it by ${tex`G(V, \cdot)`} or ${tex`G(\mathbb R^{p,q,r})`}). Firstly, for ${tex`\mathbf a`} and ${tex`\mathbf b`} in ${tex`V`} (i.e., "degree-1" elements of ${tex`\bigwedge V`}), we define their geometric product as ${tex.block` \mathbf a \mathbf b := \mathbf a \cdot\mathbf b + \mathbf a \wedge \mathbf b`} mixing up the symmetric degree-0 part with the anti-symmetric degree-2 part. It is easy to recover the inner and outer products from the geometric product: ${tex.block` \mathbf{a} \cdot \mathbf{b} = \frac{1}{2}(\mathbf{a} \mathbf{b}+\mathbf{b} \mathbf{a})`} ${tex.block` \mathbf a \wedge \mathbf b = \frac{1}{2}(\mathbf{a} \mathbf{b} -\mathbf{b} \mathbf{a}) `} So, on a superficial level, the geometric product does contain all the metric information one would want in geometry. But there's much more than that, which I hope to demonstrate in the next section. `37:T58c,md` I suppose one of the conceptual hurdles for people who encounter all this the first time is "What is this ${tex`\mathbf{e}_{12}= \mathbf{e}_1 \wedge \mathbf{e}_2`} after all?" I'd go back to the "outer product" I described at the very beginning, and to clarify a bit. That outer product, simply by multiplying a ${tex`1\times n`} matrix with an ${tex`n\times 1`} matrix, can't be our wedge product since it is not anti-symmetric, but we can turn it into one. Take $n=3$ for simplicity, so that ${tex.block` \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \qquad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} `} and we can regard ${tex.block` \mathbf{e}_{12} = \mathbf{e}_1 \mathbf{e}_2^T - \mathbf{e}_2 \mathbf{e}_1^T = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} `} (Here juxposition denotes ordinary matrix multiplication, and ${tex`^T`} is transpose.) However, to write down ${tex`\mathbf{e}_{123}`} we'd need a ${tex`3\times 3\times 3`} "tensor"; and more importantly, it's rather bizarre to have expressions that add different types of objects together (scalars, vectors, matrices, tensors), even just formally. This discussion is only meant to discourage you from thinking in terms of matrices; instead you should embrace a vector space with basis vectors labelled by subsets of ${tex`\{1, 2, \ldots, n\}`}, and not to worry about what ${tex`\mathbf{e}_{12}`} *is*. `38:T598,tex.block` \begin{matrix} & 1 & \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 & \mathbf{e}_{12} & \mathbf{e}_{13} & \mathbf{e}_{23} & \mathbf{e}_{123} \\ 1 & 1 & \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 & \mathbf{e}_{12} & \mathbf{e}_{13} & \mathbf{e}_{23} & \mathbf{e}_{123} \\ \mathbf{e}_1 & \mathbf{e}_1 & 1 & \mathbf{e}_{12} & \mathbf{e}_{13} & \mathbf{e}_2 & \mathbf{e}_3 & \color{red}\mathbf{e}_{123} & \color{red}\mathbf{e}_{23} \\ \mathbf{e}_2 & \mathbf{e}_2 & -\mathbf{e}_{12} & 1 & \mathbf{e}_{23} & -\mathbf{e}_1 & \color{red}-\mathbf{e}_{123} & \mathbf{e}_3 & \color{red}-\mathbf{e}_{13} \\ \mathbf{e}_3 & \mathbf{e}_3 & -\mathbf{e}_{13} & -\mathbf{e}_{23} & 0 & \color{red}\mathbf{e}_{123} & 0 & 0 & \color{red}0 \\ \mathbf{e}_{12} & \mathbf{e}_{12} & -\mathbf{e}_2 & \mathbf{e}_1 & \color{red}\mathbf{e}_{123} & -1 & -\mathbf{e}_{23} & \mathbf{e}_{13} & \color{red}\mathbf{e}_{3} \\ \mathbf{e}_{13} & \mathbf{e}_{13} & -\mathbf{e}_3 & \color{red}-\mathbf{e}_{123} & 0 & \mathbf{e}_{23} & 0 & 0 & \color{red}0 \\ \mathbf{e}_{23} & \mathbf{e}_{23} & \color{red}\mathbf{e}_{123} & -\mathbf{e}_3 & 0 & -\mathbf{e}_{13} & 0 & 0 & \color{red} 0 \\ \mathbf{e}_{123} & \mathbf{e}_{123} & \color{red}\mathbf{e}_{23} & \color{red}-\mathbf{e}_{13} & \color{red} 0 & \color{red}-\mathbf{e}_{3} & \color{red} 0 & \color{red} 0 & \color{red} 0 \\ \end{matrix} `39:T77a,md`## Projective geometry It's not easy to give a quick one-liner description of what projective geometry is about. For those who remember high school plane geometry and conic sections, I can say that projective (plane) geometry is concerned with the "incidence relations" of points and lines (in particular, the *duality* between them), and those properties of the circle that hold true for a general conic section (ellipses, parabolas, and hyperbolas). In particular, lengths and angles don't feature in this geometry. The essence of projective geometry is the inclusion of new points *at infinity*, one for each direction, to form the *projective plane*. The standard way to realize it concretely is to add an auxiliary coordinate to the Cartesian coordinates ${tex`(x,y)`} of a point, and declare two triples are the same if one is a (nonzero) multiple of the other. Such a triple is called the *homogeneous coordinates* of the point. In other words, this gives the "construction" of the (real) projective plane: ${tex`\mathbb{RP}^2 = (\mathbb R^3-0)/\sim`} as equivalence classes, or more geometrically as the space of lines through the origin (1-dimensional subspaces). It is customary to use the notation ${tex`[x:y:z]`} for homogeneous coordinates, and if ${tex`z\neq 0`}, it is an *ordinary* point, namely ${tex`(x/z, y/z)`}; but now we have a whole "line" of new points, those that have 0 as the last coordinate, and these are precisely the points at infinity (or *ideal points*) that one could intuit about. On the other hand, lines are also represented by three numbers: ${tex.block` \alpha x + \beta y + \gamma = 0 `} and scaling the three numbers simultaneously doesn't change the line. When we use homogenous coordinates, the equation becomes ${tex.block` \alpha x + \beta y + \gamma z = 0 `} and a point lies on a line if and only if the "dot product" of their homogeneous coordinates is zero. `3a:[] 2e:{"title":"Geometric Algebra — a guided tour","cells":[{"id":0,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":1150,"value":"md`There are many reasons why geometric algebra is not more widely taught, not the least for the fact that the name itself may be perceived as \"overselling it\": it sounds too similar to *algebraic geometry*, which is a vast and deep field at the center of modern mathematics. As far as I can see, geometric algebra is a convenient language that could and should replace Gibbs' vector calculus (especially the cross product), but it's not nearly as \"deep\" a concept as, say, the Lie algebra, which by the way has many connections with geometry (broadly construed) so is also entitled to be a geometric algebra. For this reason, I'm inclined to refer to it as Clifford algebra.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":52,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":74,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":97,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":598,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":182,"value":"md`The other thing you can do with differential forms is, of course, that you may integrate them. Naturally, you are only allowed to integrate a 1-form over a 1-dimensional curve, a 2-form over a 2-dimensional surface, etc. The Fundamental Theorem of Calculus, which bridges integration with differentiation, now takes the elegant form \n${tex.block` \\int_C d \\omega = \\int_{\\partial C} \\omega `}\nwhere ${tex`\\partial C`} denotes the boundary of ${tex`C`}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":161,"value":"md`One could redo electromagnetism in terms of differential forms. The electric and magnetic fields combine to form a 2-form, i.e., an element of ${tex`\\Omega^2(\\mathbb R^4)`}. It's not as popular because you lose much of the visual intuition of vector fields in ${tex`\\mathbb R^3`}, and some have hoped that geometric algebra might be a better alternative; maybe I'll do another notebook just for that.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":209,"value":"md`All this may be overkill for understanding geometric algebra as used in computer graphics. But I couldn't resist.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":31,"value":"md`## Dirac matrices\n\nPerhaps one of the first *natural* places one would encounter a genuine geometric algebra is the algebra of Dirac matrices. Dirac was specifically looking for four matrices, labelled ${tex`\\gamma_0, \\gamma_1, \\gamma_2, \\gamma_3`}, associated with the *four* dimensions of time and space, so that they would satisfy the conditions that ${tex`\\gamma_0^2=I`} (the identity matrix), ${tex`\\gamma_1^2=\\gamma_2^2=\\gamma_3^2=-I`} and that they \"anti-commute\" with one another. (It may be encapsulated in the *anti-commutation relations* \n${tex.block` \\{ \\gamma_\\mu, \\gamma_\\nu \\} = 2g_{\\mu\\nu} `}\nwhere ${tex`\\{A, B\\} = AB + BA`} is the *anti-commutator*, and ${tex`g_{\\mu\\nu}`} is the Minkowski metric of space-time: ${tex`g_{00}=1`}, ${tex`g_{ii} = -1`} for ${tex`i=1,2,3`}, and ${tex`g_{\\mu\\nu}=0`} otherwise.) The smallest matrices Dirac found that worked were ${tex`4\\times 4`} (which confusingly doesn't actually have to do with the dimension of space-time).\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":196,"value":"md`So, as physicists went about calculating with Dirac matrices, they find themselves only ever having to work with products in \"increasing\" order up to 4 matrices, much like the business with ${tex`dx\\wedge dy\\wedge dz`}. For instance, if you have ${tex`\\gamma_0\\gamma_1\\gamma_0\\gamma_2`}, you can rewrite it as \n${tex.block` \\gamma_0 (-\\gamma_0\\gamma_1)\\gamma_2 = -\\gamma_0 \\gamma_0\\gamma_1\\gamma_2 = -I\\gamma_1\\gamma_2 = -\\gamma_1\\gamma_2 `}\nwithout having to write down any ${tex`4\\times 4`} matrix explicitly. `","pinned":false,"hidden":false,"mode":"ojs"},{"id":240,"value":"md`It then seems natural to regard these ${tex`\\gamma_\\mu`} as abstract objects satisfying certain relations, and the fact that they admit a *representation* as ${tex`4\\times 4`} matrices is convenient (or not) for certain purposes. This algebra, sometimes known as the Dirac algebra, is spanned by \n${tex.block`\n I \\\\\n \\gamma_0 \\quad \\gamma_1 \\quad \\gamma_2 \\quad \\gamma_3 \\\\\n \\gamma_0 \\gamma_1 \\quad \\gamma_0 \\gamma_2 \\quad \\ldots \\\\\n \\gamma_0 \\gamma_1 \\gamma_2 \\quad \\ldots \\\\\n \\gamma_0 \\gamma_1 \\gamma_2 \\gamma_3 `}\nfor a total of ${tex`1+4+6+4+1=2^4=16`} symbols. The algebra structure may be expressed in a ${tex`16\\times 16`} multiplication table, which might be easier for a computer, if not us, to handle.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":256,"value":"md`The difference with the exterior algebra is that here, the grading is not preserved, in the sense that an ostensibly \"degree-4\" expression like the one above may turn out to be \"degree-2\" (but at least the parity—i.e., even/oddness—is preserved). A mathematician would call it a *filtration* instead of a grading, though it doesn't hurt to still use the word \"degree\" (or \"grade\") from time to time. (This is very similar to the universal enveloping algebra of a Lie algebra, whatever that means.)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":649,"value":"md`As we'll see, this example might give a wrong impression because elements in different degrees are commensurable: they are all ${tex`4\\times 4`} matrices, and you can just add them up to get another matrix.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":264,"value":"md`# Geometric algebra\n\nHopefully this gets us prepared for the full generalization of the geometric algebra—also known as Clifford algebra—associated with *any* metric on a (finite-dimensional) vector space ${tex`V`} (over the real numbers). A metric here just means an inner product, but not necessarily positive definite as we are familiar with in Euclidean geometry, nor even \"non-degenerate\" (so certain directions may have zero length). The word \"metric\" comes from the fact that it can be used to measure lengths and angles (and also areas and volumes), and in general it may vary from point to point on an arbitrary \"manifold\" (curved space), but for us will be fixed.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":446,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":448,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":830,"value":"md`Next, we need to extend this geometric product to other elements of ${tex`\\bigwedge V`}, and one of the properties we wish to have is *associativity*.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":366,"value":"md`If we fix any basis of ${tex`V`}, we then have a natural basis of the exterior algebra, and we might as well use it for the geometric product too. With ${tex`\\mathbf{e}_1, \\ldots, \\mathbf{e}_n`} the standard basis of ${tex`V`}, it is convenient to denote ${tex`\\mathbf{e}_{12}=\\mathbf{e}_1\\wedge\\mathbf{e}_2, \\ldots`} for the basis vectors of the degree-2 piece, ${tex`\\mathbf{e}_{123}=\\mathbf{e}_1\\wedge\\mathbf{e}_2\\wedge\\mathbf{e}_3, \\ldots`} for the degree-3 piece, and so forth. Then one could write down a multiplication table for the basis vectors. So, at the most mundane level, the geometric algebra is merely a vector space of dimension ${tex`2^n`}, with the basis \"vectors\" labelled by subsets of ${tex`\\{1, \\ldots, n\\}`}, endowed with a particular multiplication table. An arbitrary element might look like\n${tex.block`1 + 3\\mathbf{e}_1 - 4 \\mathbf{e}_{12}`}\nand with the multiplication table for the basis (and the distributive law) one could multiply in this vector space easily (or a computer would).\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":731,"value":"md`Now let's play with \\`ganja.js\\` for multiplication in this algebra. The code should be self-explanatory.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2,"value":"Algebra = require('ganja.js')","pinned":true,"hidden":false,"mode":"ojs"},{"id":741,"value":"Algebra(2,1,()=>(1+3e2-2e12)*(2e3))","pinned":true,"hidden":false,"mode":"ojs"},{"id":749,"value":"md`Can you figure out what order \\`ganja\\` uses for the basis?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":508,"value":"$37","pinned":false,"hidden":false,"mode":"ojs"},{"id":833,"value":"md`## Example\n\nLet's work it out for a concrete example: ${tex`V = \\mathbb R^3`} with signature (2,0,1). `","pinned":false,"hidden":false,"mode":"ojs"},{"id":836,"value":"tex.block`\n \\begin{matrix}\n & 1 & \\mathbf{e}_1 & \\mathbf{e}_2 & \\mathbf{e}_3 & \\mathbf{e}_{12} & \\mathbf{e}_{13} & \\mathbf{e}_{23} & \\mathbf{e}_{123} \\\\\n 1 & 1 & \\mathbf{e}_1 & \\mathbf{e}_2 & \\mathbf{e}_3 & \\mathbf{e}_{12} & \\mathbf{e}_{13} & \\mathbf{e}_{23} & \\mathbf{e}_{123} \\\\\n \\mathbf{e}_1 & \\mathbf{e}_1 & 1 & \\mathbf{e}_{12} & \\mathbf{e}_{13} \\\\\n \\mathbf{e}_2 & \\mathbf{e}_2 & -\\mathbf{e}_{12} & 1 & \\mathbf{e}_{23}\\\\\n \\mathbf{e}_3 & \\mathbf{e}_3 & -\\mathbf{e}_{13} & -\\mathbf{e}_{23} & 0 \\\\\n \\mathbf{e}_{12} & \\mathbf{e}_{12} \\\\\n \\mathbf{e}_{13} & \\mathbf{e}_{13} \\\\\n \\mathbf{e}_{23} & \\mathbf{e}_{23} \\\\\n \\mathbf{e}_{123} & \\mathbf{e}_{123} \\\\\n\n \\end{matrix}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":873,"value":"md`What would ${tex`\\mathbf{e}_1 \\mathbf{e}_{12}`} be? By the table above and associativity,\n${tex` \\mathbf{e}_1 \\mathbf{e}_{12} = \\mathbf{e}_1 \\left( \\mathbf{e}_1 \\mathbf{e}_2 \\right) = \\left( \\mathbf{e}_1 \\mathbf{e}_1 \\right) \\mathbf{e}_2 = \\mathbf{e}_2 `}\nso we can fill more of our table (feels like playing sudoku):\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":894,"value":"tex.block`\n \\begin{matrix}\n & 1 & \\mathbf{e}_1 & \\mathbf{e}_2 & \\mathbf{e}_3 & \\mathbf{e}_{12} & \\mathbf{e}_{13} & \\mathbf{e}_{23} & \\mathbf{e}_{123} \\\\\n 1 & 1 & \\mathbf{e}_1 & \\mathbf{e}_2 & \\mathbf{e}_3 & \\mathbf{e}_{12} & \\mathbf{e}_{13} & \\mathbf{e}_{23} & \\mathbf{e}_{123} \\\\\n \\mathbf{e}_1 & \\mathbf{e}_1 & 1 & \\mathbf{e}_{12} & \\mathbf{e}_{13} & \\mathbf{e}_2 & \\mathbf{e}_3 \\\\\n \\mathbf{e}_2 & \\mathbf{e}_2 & -\\mathbf{e}_{12} & 1 & \\mathbf{e}_{23} & -\\mathbf{e}_1 & & \\mathbf{e}_3 \\\\\n \\mathbf{e}_3 & \\mathbf{e}_3 & -\\mathbf{e}_{13} & -\\mathbf{e}_{23} & 0 && 0 & 0 \\\\\n \\mathbf{e}_{12} & \\mathbf{e}_{12} & -\\mathbf{e}_2 & \\mathbf{e}_1 & & -1 & -\\mathbf{e}_{23} & \\mathbf{e}_{13} \\\\\n \\mathbf{e}_{13} & \\mathbf{e}_{13} & -\\mathbf{e}_3 & & 0 & \\mathbf{e}_{23} & 0 & 0 \\\\\n \\mathbf{e}_{23} & \\mathbf{e}_{23} & & -\\mathbf{e}_3 & 0 & -\\mathbf{e}_{13} & 0 & 0 \\\\\n \\mathbf{e}_{123} & \\mathbf{e}_{123} \\\\\n\n \\end{matrix}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":913,"value":"md`But what is ${tex`\\mathbf{e}_1 \\mathbf{e}_{23}`}?\n${tex.block` \\mathbf{e}_1 \\mathbf{e}_{23} = \\mathbf{e}_1 \\left( \\mathbf{e}_2 \\mathbf{e}_3 \\right) = \\left( \\mathbf{e}_1 \\mathbf{e}_2 \\right) \\mathbf{e}_3 = \\mathbf{e}_{12} \\mathbf{e}_3 `}\n\n${tex.block` \\mathbf{e}_1 \\mathbf{e}_{23} = \\mathbf{e}_1 (\\mathbf{e}_{13} \\mathbf{e}_{12}) = (\\mathbf{e}_1 \\mathbf{e}_{13}) \\mathbf{e}_{12} = \\mathbf{e}_3 \\mathbf{e}_{12} `}\nnot helping much; and none of the others yields to pure reasoning like this. You might guess that ${tex` \\mathbf{e}_1 \\mathbf{e}_{23}`} ought to be ${tex`\\mathbf{e}_{123}`},\nbut we'd want to make sure that's the only possible choice. But for now, let's suppose that (as you would a game of sudoku), and see what all the consequences are and if it leads to any contradiction.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1084,"value":"$38","pinned":false,"hidden":false,"mode":"ojs"},{"id":1208,"value":"md`In fact, the clean way is not to start with defining the geometric product *in terms of* the wedge product, but to start from scratch (i.e. only scalars and vectors): a single condition that ${tex`\\mathbf{a}\\mathbf{a}=\\mathbf{a}\\cdot\\mathbf{a}`} generates the whole algebra, and we shall simply make elements like ${tex`\\mathbf{e}_1\\mathbf{e}_2`}, etc. the basis, denoted ${tex`\\mathbf{e}_{12}`}, etc. (We can then define the wedge product from the geometric product.)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":668,"value":"md`# What's it all good for?\n\nHere's where the fun begins, and I'm afraid most mathematicians miss out on. One could start by realizing complex numbers and quaternions in terms of geometric algebra (which is what Clifford did), but I believe it really is the good old *geometry* that justify all this.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1180,"value":"md`## Vector calculus & linear algebra\n\nThe ordinary way to find the projection of a vector $\\\\mathbf{a}$ in the direction of another vector ${tex`\\mathbf{b}`} is via the inner product. Using geometric product, this is easy to remember:\n${tex.block`\\mathbf a = \\mathbf{a}\\mathbf{b}\\mathbf{b}^{-1} = \\left(\\mathbf{a}\\cdot\\mathbf{b} + \\mathbf{a}\\wedge\\mathbf{b}\\right)\\mathbf{b}^{-1} `}\nwhere of course \n${tex.block`\\mathbf{b}^{-1} = \\frac{\\mathbf{b}}{| \\mathbf{b}|^2} = \\frac{\\mathbf{b}}{\\mathbf{b}\\cdot \\mathbf{b}}`}\n(Not all elements in geometric algebra have an inverse, but they are vey useful when they do.) This easily generalizes to projection onto a subspace, say ${tex`B=\\mathbf{b}_1\\wedge \\mathbf{b}_2`}:\n${tex.block`\\mathbf a = \\mathbf{a} B B^{-1} `}\nIt might remind you of the Gram-Schmidt process.\n\nWith the projection, we can talk about reflection, and by composing two reflections we get a rotation. All very neat, but we won't dwell on it. \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":549,"value":"$39","pinned":false,"hidden":false,"mode":"ojs"},{"id":689,"value":"md`All this can be put together very nicely in geometric algebra, and the metric we'll use is ${tex`(3,0,0)`}. Let's first list out the basis:\n ${tex.block` (1) \\\\\n \\mathbf{e}_1 \\quad \\mathbf{e}_2 \\quad \\mathbf{e}_3 \\\\\n \\mathbf{e}_{12} \\quad \\mathbf{e}_{13} \\quad \\mathbf{e}_{23} \\\\\n \\mathbf{e}_{123} `}\nAs you may guess, a point ${tex`[x:y:z]`} is represented by ${tex`p=x\\mathbf{e}_1 + y\\mathbf{e}_2 + z\\mathbf{e}_3 `}, and a line ${tex`[\\alpha : \\beta : \\gamma`} shall be represented by ${tex`l=\\alpha\\mathbf{e}_{23} - \\beta\\mathbf{e}_{13} + \\gamma\\mathbf{e}_{12}`}. The fact that the point lies on the line translates to the simple equation ${tex`p \\wedge l=0`} (which justifies the ${tex`-`} sign in front of ${tex`\\mathbf{e}_{13}`}.\n\nThe wedge product ${tex`p \\wedge q`} of two points is (purely) in degree-2, so it does represent a line; and by the fact that ${tex`p\\wedge(p\\wedge q) = 0 = q\\wedge(p\\wedge q)`}, it must be the line passing through the two points.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":713,"value":"md`That's all very neat, but it didn't actually use the geometric product, so it doesn't justify using the metric (2,0,1) instead of the other choices. Can we calculate the intersection of two lines using the geometric product?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":551,"value":"md`## Conformal geometry\n\nConformal geometry (in 3D, say) is one in which circles and spheres take the center stage. The basic operation is inversion with respect to a sphere, and angles between geometric objects are preserved. At infinity lies a single (ideal) point, which \"compactifies\" the space into the 3-sphere ${tex`S^3`}, as opposed to the projective space ${tex`\\mathbb{RP}^3`}.\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"md`# \\`ganja.js\\` playground`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"Algebra(0).graph(x=>Math.sin(x*5)) ","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"Algebra(0,1).inline(()=>(3+2e1)*(1+4e1))()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1247,"value":"Algebra(3,0,1,()=>{\n var gridX = Array.apply([],{length:21}).map((x,i)=>[1e123+2e012+1e013+(i-10)*.2e023,1e123-2e012+1e013+(i-10)*.2e023]);\n var gridY = Array.apply([],{length:21}).map((x,i)=>[1e123+2e023+1e013+(i-10)*.2e012,1e123-2e023+1e013+(i-10)*.2e012]);\n var orig = 1e123-1e013;\n var p = 1e123+1e013+.65e023-.9e012;\n var p2 = 1e123+1e013-1.35e023+.8e012;\n var p3 = 1e123+1e013+1.35e023+1.1e012;\n var p4 = p.slice();\n var camera=1+0e0, t=0;\n console.log('line : ',(orig&p2&p3)^(p&p2&p3));\n var el=this.graph(()=>{\n t = t+ 0.001;\n camera.set(Math.cos(t)+Math.sin(t)*1e13);\n return [orig,\"origin in 3D\",gridX,gridY,0x4444cc, p,\"point in 2D\",orig&p,\"vector in 3D\",0xffcccc,[orig,p2,p3],0xCC4444,\"bivector in 3D\",p2&p3,\"line in 2D\"]},{gl:true, animate:true,camera}); //\n el.style.width='100%'; return el;\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":8,"value":"Algebra(2,0,1,()=>{\n var point = (x,y)=>1e12-x*1e02+y*1e01;\n \n var A=point(-1.5,0.5), B=point(0.5,-0.2), C=point(1,0.5),\n a=()=>B&C,b=()=>C&A,c=()=>A&B,\n oa=()=>a<b<c<oa^ob, //orthocenter\n ca=()=>c<<(A+B), cb=()=>a<<(B+C), cc=()=>b<<(C+A), circum=()=>ca^cb, //circumcenter\n centroid = ()=>A+B+C,\n Euler = ()=>ortho&circum;\n \n return this.graph([\n 0xFFFFEE,[A,B,C], // the triangle\n 0xFFAAAA,oa,ob,oc,ortho,\n 0x88AA88,ca,cb,cc,circum,\n 0x8888FF,centroid,\n Euler,\"Euler Line\",\n 0x444444,A, 'A', B, 'B', C, 'C' \n ],{gl:false, grid:false});\n})","pinned":true,"hidden":false,"mode":"ojs"},{"id":1091,"value":"function b(str) {\n const i = str.indexOf('_')\n if (i == -1) {\n return '\\mathbf{' + str + '}'\n } else {\n return '\\mathbf{' + str.slice(0, i) + \"}\" + str.slice(i)\n }\n} ","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W3a","stdlib":"1","titling":"auto"}