
Geometry, Algebra, and Intuition (2017) - sebg
http://www.shapeoperator.com/2017/02/28/geometry-algebra-intuition/
======
formalsystem
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...](https://www.amazon.com/Byrne-Six-Books-Euclid-
Multilingual/dp/3836559382/ref=sr_1_3?keywords=euclid+elements&qid=1571798232&sr=8-3)

~~~
posterboy
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.

~~~
indigochill
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](https://www.youtube.com/watch?v=ztsi0CLxmjw)).

------
avodonosov
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.

~~~
mkl
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.

~~~
soVeryTired
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.

~~~
avodonosov
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.

~~~
soVeryTired
> 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.

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

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

~~~
avodonosov
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?

------
georgewsinger
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.

~~~
mlyle
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.

~~~
jacobolus
> _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 _a_ ² + _b_ ² = ( _a_ \+ _b_ )² ⇔ _ab_ \+ _ba_ =
0 are obviously equivalent, so taking either of them as a definition for
“perpendicular” immediately proves the other.

~~~
rsj_hn
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.

~~~
jacobolus
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](https://observablehq.com/d/5e75dd8e56fe255f)

~~~
rsj_hn
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.

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

Also Lakatos:
[https://math.berkeley.edu/~kpmann/Lakatos.pdf](https://math.berkeley.edu/~kpmann/Lakatos.pdf)

~~~
rsj_hn
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.

------
mindgam3
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.

~~~
dmix
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.

------
29athrowaway
[https://abstrusegoose.com/376](https://abstrusegoose.com/376)

------
adamnemecek
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](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](https://observablehq.com/@enkimute/animated-orbits)

~~~
jwmerrill
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...](https://www.shapeoperator.com/2019/02/10/proving-theorems-about-angles-
without-angles/)

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

~~~
yiyus
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!

