Geometry, Algebra, and Intuition (2017) | Hacker News Reader

Geometry, Algebra, and Intuition (2017)

shapeoperator.com

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If you enjoyed this post, you'll probably love "The Elements of Euclid"[1] by Byrne which provides entirely visual proof for ALL the basic proofs of euclidean geometry.

I actually first came across the book when I saw it mentioned in Beautiful Explanations by Tufte. The beauty of the images is just on another level, the book will just make you feel good when you stare at it and after staring at it you'll absorb a proof accidentally with barely any effort on your part.

There is a mistaken belief that visual proofs are less serious than algebraic ones but I believe this is mostly due to a lack of imagination when it comes coming up with good visual proofs. Byrne's book will help you see just how powerful pictures can be. There's lots of good work happening in the Category Theory community to turn diagrams into first class objects in constructing proofs so I'm very optimistic about a boom in visual proof construction.

[1] https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/d...

The Taschen reprint of Byrne's Euclid I consider one of the finer books I have. I have spent so much time with it.

But there is now also this web version which is made with so much love it in many ways even improves on the reprint in quality: https://www.c82.net/euclid/

diagrams invariably show only 2 dimensions, so you can't reasonably show anything that has complexity in more than three dimensions, which means any problem with three independent variables is out. Animation can add the missing dimension; well color can, too.

I'm cautiously optimistic about VR as a tool for teaching and understanding math up to three dimensions. You may have seen the "Non-euclidian virtual reality" video floating around YouTube (https://www.youtube.com/watch?v=ztsi0CLxmjw).

One pitfall I'm wary of when introducing visual proofs is not being able to make the leap of how to formalize the proof, i.e. how to turn it into a purely mechanical process that a computer could understand.

It can make these sorts of proofs overly convincing. https://math.stackexchange.com/questions/743067/visually-dec.... My favorite is the approximation of the circle one, because it doesn't rely on tricky, underhanded drawing inaccuracies, but instead demonstrates a need to truly formalize what it is you're talking about.

For category theory, most people approaching it already have some experience with mathematical proofs and could probably sketch out how to boil a diagram chasing proof down into tedious set of logical statements. If anyone hasn't, I'd recommend doing so for a simple example.

Note a version of this can occur for the "algebraic" style of proofs as well. Occasionally students can't really explain why they're "allowed" to cancel out terms (it can be a minor leap to see that really what's being relied on here is injectivity).

The other tricky thing about intuitions, visual or otherwise, at least in my experience, is that I often hold multiple mutually incompatible visualizations/intuitions about a mathematical object or process and the most crucial component of my intuition is knowing when to discard one and use the other when they conflict. To actually harmonize all of them requires, well, fully formalizing everything. Otherwise you end up mistaking your intuition for the object itself and going down a logically incoherent path (the evergreen target for this always seems to be Godel's incompleteness theorems).

You still need intuition though, because otherwise coming up with the creative spark for a proof is nigh impossible. But it's not a substitute for the formal object itself.

More fundamentally, I think both approaches, visual and "algebraic" in the sense of the article make it seem like mathematics is about getting the "correct" answer, when really the part of pure mathematics that resonates most with me is about running wild with "what if" and then rigorously chasing down the implications thereof.

For example, the commonly asked playground question "is infinity number?" is not best answered with a "no" or a "yes", but rather an exploration of what no and yes would entail, which first requires the formalization of infinity, which could have many different, mutually incompatible forms! Another fun one is coming up with a world where infinity plus one is larger than infinity (this often leads to an exploration of the ordinals).

I also enjoy how relevant your username is.

Could you please provide some links to these efforts for turning diagrams into proofs, please?

1. This was one of my favorite intro to category theory books, will make you familiar with the general language http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf

2. After skimming 1, check out Algebra Chapter 0 which uses Category Theory and diagrams to prove various theorems from Abstract Algebra (Hard read) https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Math...

I'm suspicious of vector algebraic proof of the the Pythagorean Theorem.

Don't vector operation properties themselves follow from the Pythagorean Theorem (at least in their application to space and geometric objects)? If so, using them to prove the theorem doesn't make sense.

I'm not sure right now whether such circularity exists, but one should be careful.

It seems circular to me. It relies on ||x||² = x·x, but the fact that ||x||² = x₁² + x₂² = x·x comes from the Pythagorean theorem.

I don't think it's quite circular, but in a sense you're right.

if c = -a + b, then for any inner product you have

<c,c> = <a,a> + <b,b> - 2<a,b> as a result of bi-linearity of the inner product.

By definition if <a,b> = 0 then a and b are orthogonal and a version of pythagoras' theorem holds.

But you're right in saying the choice of inner product

<x,y> = x_1y_1 + ... + x_ny_n

is motivated by pythagoras' theorem in Euclidean space.

By definition?

In the abstract world of algebra we are free to choose any definition (different rules will give different algebras).

But if we want our algebraic manipulations to prove the theorem about geometric objects we need to prove isomorphism between our algebra and the geometric objects and operations on them.

I doubt distributivity and other properties of operations on geometric vectors can be proven without the the Pythagorean theorem.

> By definition?

Yes, in a Hilbert space (i.e. an abstract vector space with an inner product), the definition of orthogonality is that the inner product of two nonzero vectors is zero.

I'm not sure I really know what you mean by geometric vectors.

Well... think harder, what can I say :)

From that comment I'm not sure you do either.

I suspected trolling in your question about geometric vectors.

Sides of a triangle and elements of your algebra are different domains. In order to translate results between them one needs to prove this makes sense.

In the article the author only shows that inner product of c by itself equals to sum of inner product squires of a and b, if a and b are orthogonal.

Who told you this has anything to do with lengths of triangle sides?

there are hundreds of proofs for the pythagorean theorem, surely some of them using the axioms of geometry

This isn't just using the axioms of geometry, though, it's trying to prove the Pythagorean theorem using the Pythagorean theorem as an axiom. Hence circularity.

no, not exactly. the euclidean norm can be defined to fulfill a couple basic properties or equations (like composibility, or schwarz' inequality, but I don't recall exactly), and it's pure coincidence, if you will, that the norm is equal to the root mean square. That's not circular reasoning.

The name euclidean norm implies that the geometric angle (no pun intended) was the motivation, but what's really central is a question of epistemology. There's not much of a point to prescribe a certain approach over another, without a good argument.

Excellent post.

In my experience, algebraic thinkers absorb information much faster than visual thinkers, but they more often make silly conceptual errors that visual thinkers don't make. For example, an algebraic thinker might accidentally add a vector to a scalar, since their symbols look identical on paper. But a visual thinker would be much less likely to do this, since their visual representations for scalars and vectors would likely be so distinct.

A rule of thumb: when short-term speed is crucial, think algebraically. When long-term understanding is crucial, think visually.

One problem with algebraic intuition is that it leave ideas "unhooked" in your mind. I mean this in the following sense:

> While you are leaning things you need to think about them and examine them from many sides. By connecting them in many ways with what you already know.... you can later retrieve them in unusual situations. It took me a long time to realize that each time I learned something I should put "hooks" on it. This is another face of the extra effort, the studying more deeply, the going the extra mile, that seems to be characteristic of great scientists. -- Richard Hamming

Algebraic proofs are stored as symbolic/syntactic movies in your head. But syntactic movies resemble other syntactic movies, causing algebraic proofs to blend together with all of the other symbolic/syntactic theorems. Visualizing proofs, on the other hand, makes each theorem significantly more distinguished from each other. You are much more likely to recall and understand important facts this way, in my opinion. You are therefore more likely to apply them in novel ways to solve new problems.

Here Einstein famously describes visual vs. syntactic thinking in a letter to Jacques S. Hadamard:

> (A) The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined.

> There is, of course, a certain connection between those elements and relevant logical concepts. It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of this rather vague play with the above-mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought — before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.

> (B) The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will.

> (C) According to what has been said, the play with the mentioned elements is aimed to be analogous to certain logical connections one is searching for.

> (D) Visual and motor. In a stage when words intervene at all, they are, in my case, purely auditive, but they interfere only in a secondary stage, as already mentioned.

A good friend of mine has a PhD in physics, and is a classic algebraic thinker. He is so, so much faster than me. But he once remarked that he forgets the proofs of almost everything he learned in grad school, and has to get back into the concrete exercises to regain his algebraic intuition. Visual thinkers may be slow, but they never forget.

IMO the post is a little circular. If we rely upon the projection product of Euclidean vectors, we've already granted Pythagorean theorem in our assumptions.

There's a lot of ways to arrange things visually, but knowing we want c^2 really cuts the options down; knowing that we also will have a and b be degree two in the relation pretty much constrains us to that shape. We need a and b in some form on the sides, and we need a square with sides c.

If we prefer, once we have the "4 triangles" model, it's easy for us to proceed to elementary algebra if we want, rather than relying on a geometrical transformation:

  (a+b)(a+b)       area of the big square
  1/2 (ab)         area of each of the triangles
  (a+b)(a+b) - 4 * 1/2 (ab) = c^2
                   take the area of the big square, take the
                   little triangles out, only the c^2 square remains
  a^2 + 2ab + b^2 - 4 * 1/2 ab = c^2
                   distribute
  a^2 + b^2 = c^2
                   simplify

If you hate 4 triangles, you can easily do it with 2 of the a by b triangles, and a c by c right triangle forming a trapezoid. There's myriad geometrical constructions to start with before we get to the algebra. But we need to have some kind of geometric construction that leads to the algebra to conclude geometrical relations from algebraic relations.

> If we rely upon the projection product of Euclidean vectors, we've already granted Pythagorean theorem in our assumptions.

The Pythagorean theorem in particular (and any theorem in general) is always a little bit “circular”; the relation is inherent in any definition of perpendicular in a model of Euclidean space.

In geometric algebra (where multiplication of vectors distributes over addition), the two statements a² + b² = (a + b)² ⇔ ab + ba = 0 are obviously equivalent, so taking either of them as a definition for “perpendicular” immediately proves the other.

You need more than the definition of perpendicular, you need the definition of "angle" in such a way as to ensure flatness of your space, otherwise a triangle might not correspond to three vectors that sum to zero. In this case, you cannot obtain the pythagorean identity from the algebraic identities.

Remember a triangle is a set of 3 curves living in your space that meet back up. But angles between curves are measured as angles between the tangent vector to the curves and do not live in your space, they live in the tangent space.

A triangle on the sphere, for example, has angles that don't sum to 180 degrees and the three tangent vectors do not sum to zero even though the three curves meet back at the same point.

So what's crucial here is an assumption of flatness, which allows you to associate a tangent space to the underlying space in a way consistent with the underlying metric. This allows you to make the association between geodesics (distance minimizing curves in your geometry) and vectors in your tangent space so that you can pretend that the straight lines actually live in your space and are also distance minimizing. This is what you need for the pythagorean theorem.

This is not something that you can get just from the distributive law, you need the distributive plus the property that a triangle has angles that sum to 180, or equivalently that the tangent vectors to your triangle can be embedded in your space and sum to zero.

Notice I mentioned “Euclidean space”. That inherently involves flatness. Obviously there are several premises/axioms needed to set it up, and a variety of ways to do so.

In Euclid, we have the famous parallel postulate which helps us establish flatness.

> angles that don't sum to 180 degrees

Note that Euclid’s Elements nowhere mentions angle measures. It only describes the concept of a right angle (and angles more or less than right). The Pythagorean theorem does not depend on angle measures. If you ask me angle measures are a quite poor/confusing tool to introduce in introductory Euclidean geometry courses, since they are a type of logarithm, and much more inherently complicated than the rest of a typical geometry course.

> triangle is a set of 3 curves

This is one possible definition of “triangle”. For Euclid a “trilateral figure” is contained by three straight lines, and “A straight line is a line which lies evenly with the points on itself.” (Which has been rather hard for readers to interpret throughout time.)

* * *

Tangentially, I’ve been working a lot with spherical triangles in my ongoing project. :-) https://observablehq.com/d/5e75dd8e56fe255f

How you are going to define euclidean space without the pythagorean theorem? That's basically the definition of euclidean. But the advantage of the pythagorean theorem is it allows you to measure how you deviate from flatness by comparing the difference of c^2 with a^2 + b^2.

That was my point upthread (any proof of the Pythagorean identity is somewhat circular, since it is inherent in the structure).

The way Euclid does it is to set up various axioms which imply flat space without explicitly declaring the Pythagorean identity to be an axiom. But you could easily do it the other way around. Euclid’s axioms (and other alternatives proposed over the years) were chosen specifically to make the Pythagorean identity true.

Cf. Feynman https://www.youtube.com/watch?v=hxKw4xEEFHQ (proximately relevant bit starts at about 22 minutes)

Also Lakatos: https://math.berkeley.edu/~kpmann/Lakatos.pdf

I see, yes, Euclid's axioms are not the most intuitive approach to different geometries.

What is nice is to have the tools to examine what the properties of a given geometry are, and given that geometry is a matter of curvature, it's not going to be decided by the tangent plane, it's going to be decided by the second derivative. You can get at that explicitly by embedding your space in a flat space like R^N and looking at the second derivative, or you can do intrinsic operations like parallel transport. E.g. look at small variations in the tangent plane from point to point. But the second derivative is key. Geometric algebra lives in the cotangent plane so it alone is not going to detect issues of curvature in your underlying space. This is true even though a lot of important calculations about differentials and volume elements are happening in that cotangent plane, so it's an important thing to get right, but it can't detect issues of curvature and thus it can't 'prove' the pythagorean theorem, which is a flatness statement.

Eh, I guess any proof is circular, but it is, IMO, "more" circular when the thing we're trying to prove was one of the preconditions of how we defined the system--- properties that we wanted to obtain a priori.

It's like defining negative exponents based on the properties we want for commutation, identity, etc... and then afterwards proving that something raised to a negative exponent times something raised to a positive exponent is 1 and patting ourselves on the back.

> thing we're trying to prove was one of the preconditions

This entirely depends on what you consider to be a definition of the dot product. There are many possible ways this could be set up. (e.g. if you wanted you could develop this whole theory of vector algebra within the system of Euclid’s axioms. Or you could set it up based on explicit coordinates and concrete arithmetic of numbers with no geometrical basis per se. Or ...)

The hard work leading up to this proof is showing that the algebraic definition a·b = 0 corresponds to the usual notion of perpendicularity in Euclidean space. After that, the algebraic proof of the Pythagorean identity is trivial.

This seems almost exactly backwards to me.

“Geometric” proofs (i.e. based on spatial reasoning) are fast and fluid, relying on an imagined spatial configuration which can be seen all at once.

“Algebraic” proofs (i.e. based on symbol manipulation) are serialized and low-bandwidth, and working them out takes a significant amount of laborious fiddling with symbols on paper, and is almost impossible to work out mentally except in nearly trivial cases.

The benefit of the “algebraic” version is often that putting in that labor can often yield a result even when the prover has no special insight. That is, the method can be more reliably carried out by non-geniuses (assuming that the prover has sufficient patience and stamina), because the work can be broken into small individually opaque steps and written down on paper instead of needing to be seen all at once.

To the extent algebraic proofs are fast/elegant/obvious, they generally rely on a clear 2-dimensional notation on paper where it is easy to see how to simplify the parts based on visual patterns known via extensive past experience. Cf. https://arxiv.org/abs/math/9205211

Algebraic reasoning is not “low-bandwidth,” because there is a high degree of “compression,” which is much of the modern algebra is all about, and which is why, for example, algebraic topology has taken over the point-set topology as the main tool in the topological research.

That Hamming quote reflects on something I've felt in my gut to be important for a long time: there is nothing more intuitive than a cohesive internal world model, where all conclusions are "trivial" because these hooks which you reference carry us through the logical deductions automatically. Everything makes sense because there's a basal (axiomatic) set of rules with which to interpret the things we encounter, and a helpful set of heuristic functions which help us quickly decide further actions. Learning is the process of turning a datum sans context into another node in the great graph database in our head, and learning rules over which to operate on this graph.

> In my experience, algebraic thinkers absorb information much faster than visual thinkers, but they more often make silly conceptual errors that visual thinkers don't make. For example, an algebraic thinker might accidentally add a vector to a scalar, since their symbols look identical on paper. But a visual thinker would be much less likely to do this, since their visual representations for scalars and vectors would likely be so distinct.

It's too bad these algebraic thinkers aren't working in a nice IDE with type inference.

This is great. Makes me want to do a similar post for chess. Whenever I try to explain to people what chess thinking involves I compare it geometry. Finding visual patterns on the board. But there’s also calculation and tactics which is very similar to algebraic thinking in terms of how you can derive a solution by following rules.

I tried reading chess strategy books but it was always the mobile puzzle apps that really trained me to be a better player.

Like programming books which demand you actually type out examples (which IMO is really useful) the same is true for almost all learning. Especially for something like math.

Khan Academy mixed short instructional videos with quick tests which I found quite useful. but nothing beats thinking it out from scratch and building your own stuff.

https://abstrusegoose.com/376

Good thing Geometric Algebra exists, where the difference between the two is really blurred.

Check out bivector.net, a new community for GA enthusiasts.

Join us on discord https://discord.gg/vGY6pPk

The revolution is nigh.

Check out this demo of ganja.js, a GA JS framework affiliated with bivector

https://observablehq.com/@enkimute/animated-orbits

You might like some of the other posts on the blog! I’ve written several other posts showing how Geometric Algebra can be used in plane geometry problems that would typically be treated with lengths and angles.

Most recent post, which links to other parts of the series: https://www.shapeoperator.com/2019/02/10/proving-theorems-ab...

An older post applying GA to the geometry of sunsets: https://www.shapeoperator.com/2016/12/12/sunset-geometry/

Your sunset post was my first introduction to GA. What a rabbit hole! It has been one of the most fascinating subjects I have got into in the last years, and very useful for my work. Thank you!

I find myself needing to move between both to really understand something, however, once I understand the problem I am working on I love using geometric algebra to play with it. I love Pablo Colapinto's Versor library and most of his writings and work [1].

[1] http://versor.mat.ucsb.edu/

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