The most pressing issue with taking "lengths" between two points is taking into account the orientation of the displacement between them. The orthodoxy is to correct this problem by taking the absolute value of the displacement between two points as the "length" between them; this is not only clumsy but also highly lacking in foresight. If, at this point, you think I'm talking rubbish, I urge you to ponder the following: why, in statistics, do we take the square of residuals rather than their absolute values? Taking the absolute value of displacements inherently requires a framework of absolute value arithmetic, which at best requires some understanding of the dreaded "triangle inequality" and at worst producing the simplest of logical errors (e.g. if two points have a separation of \(2\) units and a third collinear point is separated from one of them by \(1\) unit, calculating the separation between this point and the other point gives two different answers; RED FLAG ALERT!).

Consider the following alternate explanation: rather than taking the absolute value of displacements, why not square them? Sounds easy enough, doesn't it? In fact, it removes a lot of the issues that we faced earlier; first, and most importantly, of all, the orientation problem is gone (HOORAY!). If you have a basic understanding of linear algebra, you then would understand that squaring displacements is equivalent to taking the dot product of the displacement with itself. In foresight, a geometer would realise instantly that Pythagoras' theorem is heavily simplified to just a simple arithmetical result. So, you may ask now, what price do we pay for living in this "parallel universe"?

Let me just shatter your so-called mathematical bravado first by saying that in this domain which you insist on calling it an "alternate reality" (what utter BS, let's be frank), all our logical flaws are gone. Now, this "price" we pay; it's not really a big price to pay, unless you're infuriatingly lazy with computations and just looking for quick outs to proofs by clever tricks. Let me put it this way: a big benefit of working with squares of displacements is that all the geometrical results we can obtain are algebraic in nature, i.e. in polynomial form, but the (not-so-)big drawback is that the degree of these polynomial results can get exponentially large (that's if you're not careful).

In fact, let us illustrate this with one key result. We know that we can "add" "lengths" by a simple arithmetical relation; given three points \(A,B,C\) with "lengths" \(l_1 \equiv l(A,B)\), \(l_2 \equiv l(A,C)\) and \(l_3 \equiv l(B,C)\), then without knowledge of the orientation of the three points, we can deduce that one of the following three results are true:

\( l_1 + l_3 = l_2 \\ l_1 + l_2 = l_3 \\ l_2 + l_3 = l_1 \)

What happened if we squared these "lengths"? Suppose \(Q_i \equiv l_i^2\) for \(i=1,2,3\); then squaring both sides of one of these equations gives

\( Q_1 + Q_2 - Q_3 = 2l_1l_2 \),

which, when squared again, gives

\( (Q_1 + Q_2 - Q_3)^2 = 4Q_1Q_2 \).

We can expand the left-hand side of this equation and rearrange the whole equation to obtain a more symmetric form of this relation:

\( (Q_1 + Q_2 + Q_3)^2 = 2(Q_1^2 + Q_2^2 + Q_3^2) \).

This relation is known in RT circles as the **Triple quad formula**, and is the relation that links the pairwise (squared) separation between three points on a line, which we will now call the **quadrance** (which can be thought of classically as the squared "length"). The symmetric form of this relation implies and confirms that the framework of squaring "lengths" is orientation-free and (in foresight) opens the door to creating more general frameworks of geometry just by altering the definition of our dot product (understanding of linear algebra required; will explain in another time).

Wait, so the relationship between the pairwise separation of three points is a QUADRATIC one? But we were able to just add two "lengths" together; it's the easy, and intuitive thing to do... Well, yeah, but I just explained all the logical issues. Sure, it's a price to pay, but just look at the formula: IT'S SYMMETRIC! I know you won't get that by taking absolute values of displacements, that's for sure. So ultimately, it's not a big price to pay, since we actually GAIN information by squaring the "lengths", instead of losing information; I refer to Wildberger's (text)book on Rational Trigonometry to illustrate more of this beauty.

One more thing I should mention is whether there is a derivation of this formula without any reference to "lengths". Of course there has to be, right? It's not like Norman just plucked these formulas on a whim... I will say, for the record, that to prove this relation we can just cook up three collinear points, compute their quadrances and substitute them into the equation and let magic work on itself. But, of course, there has to be some sort of geometrical intuition to this result. For this, I refer you to his video on this stuff, linked here. And, at this point (if you have the galls to get here), I shall leave you to ponder your thoughts on this. Until next time...

(Stay tuned for the next installment, where I talk about the Quadruple quad formula & higher-dimensional variations of it!)

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## Comments

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TopNewestOne thing that might be interesting in the future is to think about Projective Geometries place with respect to Affine Geometries place.

A possible wiki-page could be helpful here.

As well, seeing these formulas at work with finite fields will be necessarily useful and interesting!

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Is the n-tuple quad formula still an open problem? Can't wait to your next installment on this.

Why should the elementary units of geometry be quadratic? Adding distances has worked well up until this century; but now we have computers, the uses of which will benefit greatly from making our geometry more general and powerful, and most importantly algebraic. But WHY is it that geometry has or should have this quadratic nature? Does it have something to do with how our eyes and brains work? Does it have something to do with the analogies between 3D affine geometry and 2D projective geometry?

You still haven't posted the video about a proof for Pascal's theorem.

This is Boaz btw. Hope you're doing well.

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Hi John,

With respect to: "Why should elementary units of geometry be quadratic?"

Abstraction and mathematics. If we consider ourselves life long learners and mathematics the language of education then we will need our tools to be as abstract as possible for them to carry such an important message to the future(and to the past). Most of us are not focused on any particular goal but do acknowledge the power of mathematics in helping to guide us to somewhere meaningful.

With this in mind: Do you learn mathematics simply to learn mathematics? No other reason? To create purpose out of nothing is a strength of mathematics. When we begin to relate it to "real life"(distance and angle) we lose a lot of the power that comes with it. Not to say real life application is not an asset(clearly it can be) but it need not be primary or fundamental.

Remember mathematics is not about us.(my argument that proceeds this statement could be better structured to support this).

Mathematics is simply about mathematics. Nothing more and nothing less. We simply get the privilege to explore it. i.e. much to learn... for all of us.

Not all mathematics teaches and yet we learn about it. Think of the joys of exploring mathematics as an economist of number theory. One who might simply look at the truth and how one might encourage the world to breathe alongside it.

There is a much better mathematical argument but at this moment I think the best one is: Geometry is quadratic because that is the truth. It is not a way of doing it. It is simply what is observed, and not forced.

Great comment John(Boaz)

-Peter

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Hi Gennady,

This is wonderful. One critique I have is not to try to "convince" the reader. Don't yell(use caps), remember you can't change anyones mind by being louder. As you have stated before: The beauty and magic at work here is something incredibly remarkable.

Let your awesome work speak for itself. It already does simply by you thinking about it. Nevermind the actual physical typing and formatting of it.

The finite fields, especially the "trivially"(which I think we will soon all find out is not so trivial) small ones will be the most important in the context of educating the rest of the world, and the world to come.

The finite field of 3 is where we need to see, not just rational trigonometry at work, but universal hyperbolic geometry. (Think of how powerful Rubik's cube is in popular culture... why might that be...)

Great and wonderful post, I can't wait to read what comes next! Keep it coming!!!

-Peter

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I don't think I was intending to yell using caps; I wanted to emphasise some key points without using boldface (which I tend to use to define terms). As for your comment re finite fields, I 100% agree that the field \(\mathbb{F}_3\) is a very important field in geometry, as it is the smallest field not of characteristic 2, and would be the natural starting point into finite field geometry.

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I would love to see a through exploration of \(\mathbb{F}_3\) in which points and duals are discussed and proven. This intuition might lead to a new way to think about solving a Rubik's cube.

What are your thoughts on using \(\mathbb{F}_3^3\) to be able to map an intuition (a person's thoughts) to solving a Rubik's cube? Do you think people can intuitively take the theorems and concepts to heart to turn the Rubik's cube into the mathematics manipulative to learn universal hyperbolic geometry?

This is my hope.

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One of the weird things that has been done as the effect of Norman's work is those who still "believe"(whatever that means) in the "real numbers"(whatever that means) is that "fact" and "opinion" will begin to have much more profound and interesting dichotomies.

Being alive just got a whole lot more interesting.

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