An Example of a Parallelogram

Today I want to run through an example of how we use our new tools to read geometric information out of a parallelogram.

I’ll work within $\mathbb{R}^3$ with an orthonormal basis $\{e_1, e_2, e_3\}$ and an identified origin $O$ to give us a system of coordinates. That is, given the point $P$ , we set up a vector $\overrightarrow{OP}$ pointing from $O$ to $P$ (which we can do in a Euclidean space). Then this vector has components in terms of the basis:

$\displaystyle\overrightarrow{OP}=xe_1+ye_2+ze_3$

and we’ll write the point $P$ as $(x,y,z)$ .

So let’s pick four points: $(0,0,0)$ , $(1,1,0)$ , $(2,1,1)$ , and $(1,0,1)$ . These four point do, indeed, give the vertices of a parallelogram, since both displacements from $(0,0,0)$ to $(1,1,0)$ and from $(1,0,1)$ to $(2,1,1)$ are $e_1+e_2$ , and similarly the displacements from $(0,0,0)$ to $(1,0,1)$ and from $(1,1,0)$ to $(2,1,1)$ are both $e_1+e_3$ . Alternatively, all four points lie within the plane described by $x=y+z$ , and the region in this plane contained between the vertices consists of points $P$ so that

$\displaystyle\overrightarrow{OP}=u(e_1+e_2)+v(e_1+e_3)$

for some $u$ and $v$ both in the interval $[0,1]$ . So this is a parallelogram contained between $e_1+e_2$ and $e_1+e_3$ . Incidentally, note that the fact that all these points lie within a plane means that any displacement vector between two of them is in the kernel of some linear transformation. In this case, it’s the linear functional $\langle e_1-e_2-e_3,\underline{\hphantom{X}}\rangle$ , and the vector $e_1-e_2-e_3$ is perpendicular to any displacement in this plane, which will come in handy later.

Now in a more familiar approach, we might say that the area of this parallelogram is its base times its height. Let’s work that out to check our answer against later. For the base, we take the length of one vector, say $e_1+e_2$ . We use the inner product to calculate its length as $\sqrt{2}$ . For the height we can’t just take the length of the other vector. Some basic trigonometry shows that we need the length of the other vector (which is again $\sqrt{2}$ ) times the sine of the angle between the two vectors. To calculate this angle we again use the inner product to find that its cosine is $\frac{1}{2}$ , and so its sine is $\frac{\sqrt{3}}{2}$ . Multiplying these all together we find a height of $\sqrt{\frac{3}{2}}$ , and thus an area of $\sqrt{3}$ .

On the other hand, let’s use our new tools. We represent the parallelogram as the wedge $(e_1+e_2)\wedge(e_1+e_3)$ — incidentally choosing an orientation of the parallelogram and the entire plane containing it — and calculate its length using the inner product on the exterior algebra:

$\displaystyle\begin{aligned}\mathrm{vol}\left((e_1+e_2)\wedge(e_1+e_3)\right)^2&=2!\langle(e_1+e_2)\wedge(e_1+e_3),(e_1+e_2)\wedge(e_1+e_3)\rangle\\&=2!\frac{1}{2!}\det\begin{pmatrix}\langle e_1+e_2,e_1+e_2\rangle&\langle e_1+e_2,e_1+e_3\rangle\\\langle e_1+e_3,e_1+e_2\rangle&\langle e_1+e_3,e_1+e_3\rangle\end{pmatrix}\\&=\det\begin{pmatrix}2&1\\1&2\end{pmatrix}\\&=\left(2\cdot2-1\cdot1\right)=3\end{aligned}$

Alternately, we could calculate it by expanding in terms of basic wedges. That is, we can write

$\displaystyle\begin{aligned}(e_1+e_2)\wedge(e_1+e_3)&=e_1\wedge e_1+e_1\wedge e_3+e_2\wedge e_1+e_2\wedge e_3\\&=e_2\wedge e_3-e_3\wedge e_1-e_1\wedge e_2\end{aligned}$

This tells us that if we take our parallelogram and project it onto the $y$ - $z$ plane (which has an orthonormal basis $\{e_2,e_3\}$ ) we get an area of ${1}$ . Similarly, projecting our parallelogram onto the $x$ - $y$ plane (with orthonormal basis $\{e_1,e_2\}$ we get an area of $-1$ . That is, the area is ${1}$ and the orientation of the projected parallelogram disagrees with that of the plane. Anyhow, now the squared area of the parallelogram is the sum of the squares of these projected areas: $1^2+(-1)^2+(-1)^2=3$ .

Notice, now, the similarity between this expression $e_2\wedge e_3-e_3\wedge e_1-e_1\wedge e_2$ and the perpendicular vector we found before: $e_1-e_2-e_3$ . Each one is the sum of three terms with the same choices of signs. The terms themselves seem to have something to do with each other as well; the wedge $e_2\wedge e_3$ describes an area in the $y$ - $z$ plane, while $e_1$ describes a length in the perpendicular $x$ -axis. Similarly, $e_1\wedge e_2$ describes an area in the $x$ - $y$ plane, while $e_3$ describes a length in the perpendicular $z$ -axis. And, magically, the sum of these three perpendicular vectors to these three parallelograms gives the perpendicular vector to their sum!

There is, indeed, a linear correspondence between parallelograms and vectors that extends this idea, which we will explore tomorrow. The seemingly-odd choice of $e_3\wedge e_1$ to correspond to $e_2$ , though, should be a tip-off that this correspondence is closely bound up with the notion of orientation.

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November 5, 2009 - Posted by John Armstrong | Analytic Geometry, Geometry

6 Comments »

Looks very interesting. Especially since for the past few months I have been studying Geometric algebra (aka Clifford algebra) which subsumes most of the geometric concepts you have been dealing with. I am actually a physics student and without this self-study I wouldn’t have understood a thing of what you wrote. Pity they don’t teach these advanced math concepts.

I would like to know whether you’re familiar with GA and if so what is your opinion of it?

Comment by Rie-mann | November 6, 2009 | Reply
Sure, Clifford algebras are useful for various things.. I’m not sure what exactly you want my opinion about.

Comment by John Armstrong | November 6, 2009 | Reply
There is a geometry to the Ideocosm.

* Metaphor: a parallelogram in the space of ideas.
* “A is to B as C is to D” locates four points in the Ideocosm (Zwicky’s name for the space of all possible ideas). Sometimes, in literature, one of these points is implicit.
* “A is to B” is a vector, with tail at A and head at B (I note that metaphors occur in Mathematics). The vector has a direction; it points in a particular way.
* “C is to D” is a vector.
* “A is to B… AS… C is to D” tells us that those two vectors are parallel.
* When one says “figure of speech,” one may analyze the laws of figure (Geometry), as well as the laws of speech.

I’ve discussed this at greater length in other blaths. For example, is the Ideocosm really a Metric Space? What is the topology of the space off all possible ideas; mathematical ideas; physical system ideas?

Ia a metaphor between metaphors a parallopiped? I’m very serious.

Comment by Jonathan Vos Post | November 6, 2009 | Reply
Jonathan, these ideas about the ideocosm are not obviously stupid, but at the same time it’s hard for me to see how they could be a fruitful line of inquiry. Is there a tiniest germ of a testable hypothesis about the space of ideas (much less any sort of research program)? Is it even a coherent concept, or might it crumble into self-referential paradox (“ideocosm” being itself an idea, an idea about ideas)?

The parallelogram idea is sort of suggestive, and we’ve talked at the Cafe about how spans crystallize part of what we generally mean by an analogy (I would say “analogy”, not “metaphor”). There are also spans between spans, which you can read about also on the blog you’re reading now. Exact lines of inquiry, partly linguistic and partly philosophic, might be possible here and also interesting. There is certainly a rich n-categorical mathematics of spans.

But “ideocosm” itself strikes me as too wild and woolly, pitched at a wrong level as it were, and not at the stage of anything approaching a science. Something for late-night bull sessions perhaps, fueled by intoxicants.

Comment by Todd Trimble | November 7, 2009 | Reply
Todd Trimble is right, of course. Since he has a full-time job and I do not, he is more than welcome to provide the intoxicants when we meet f2f.

So… Limits and Convergence, in applying my claim to Mathematics itself.

The paper arXiv:0810.5078
Title: Demonstrative and non-demonstrative reasoning by analogy
Authors: Emiliano Ippoliti

analyzes a set of issues related to analogy and
analogical reasoning, namely: 1) The problem of analogy and its duplicity; 2) The role of analogy in demonstrative reasoning; 3) The role of analogy in non-demonstrative reasoning; 4) The limits of analogy; 5) The convergence, particularly in multiple analogical reasoning, of these two apparently distinct aspects and its methodological and philosophical consequences. The paper, using example from number theory, argues for an heuristc conception of
analogy.

This paper seems to be addressing interesting points, some of which have a categorical flavor, such as:

“Furthermore, analogy exhibits dynamical limits: it can start from fruitfulness and end in nonsense. Quantum mechanics is an example of such dynamical limits, in which an initial analogical success becomes a failure: “in particular analogy between quantum systems and classical particles and waves become a stumbling block preventing a
consistent interpretation of the theory.” The result is that the double analogy between classic physics and quantum physics has to be abandoned in order to gain a ‘real’ understanding of quantum mechanics: “if we want to build or learn new theory then we are likely to use analogy as a bridge between the known and the unknown. But as soon as the new theory is on hand it should be subjected to a critical examination with a view to dismounting its heuristics scaffolding and reconstructing the system in a literal way.”

“Although Bunge’s criticisms of analogy is the consequence of a logical empiricist and realist conception of analogy I disagree with (i.e. analogy is an obstacle because it can’t provide a literal and objective description of the quantum world), he points out some important limits (both static and dynamical) of analogy, which not only affect both the demonstrative and the non-demonstrative role of analogy, but should also be taken in account every time analogy is
used or analysed.”

So, to mkae the Ideocosm less fuzzy, what CAN we say about its topology?

Comment by Jonathan Vos Post | November 8, 2009 | Reply
[...] last week I said that I’d talk about a linear map that extends the notion of the correspondence between [...]

Pingback by The Hodge Star « The Unapologetic Mathematician | November 9, 2009 | Reply

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