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Posts tagged with angle

playing along with Elias Wegert in R:

X <- matrix(1:100,100,100)                  #grid
X <- X * complex(imaginary=.05) + t(X)/20    #twist & shout
X <- X - complex(real=2.5,imaginary=2.5)     #recentre
plot(X, col=hcl(h=55*Arg(sin(X)), c=Mod(sin(X))*40 ) ,        pch=46, cex=6)

Found it was useful to define these few functions:

arg <- function(z) (Arg(z)+pi)/2/pi*360     #for HCL colour input
ring <- function(C) C[.8 < Mod(C) &   Mod(C) < 1.2]        #focus on the unit circle
lev <- function(x) ceiling(log(x)) - log(x)
m <- function(z) lev(Mod(z))
plat <- function(domain, FUN) plot( domain, col= hcl( h=arg(FUN(domain)), l=70+m(domain)), pch=46, cex=1.5, main=substitute(FUN) )           #say it directly

NB, hcl's hue[0,360] so phase or arg needs to be matched to that.










Oh! This one only took me 17 years or so to figure out. This was a “fact” I had committed to memory in school but never thought about why.

 

From The Symplectization of Science by Mark Gotay and James Isenberg:

image

There are some connections to circles and homogeneous coordinates (v/‖v‖) but let’s leave those for another time.

Gotay & Isenberg’s exposition using the metric makes it clear that the
/‖v‖ part of the definition of cosine isn’t where the right-angle concept comes from. It comes from the v₁ w₁ + v₂ w₂.

image

 

So if the slope of my starting line is m, why is the slope of its perpendicular line −1/m?

First I could draw some examples.

imageimageimage

I drew these with http://www.garrettbartley.com/graphpaper.html which is a good place to count out the “rise over run” and “negative run over rise” Δx & Δy distances to make sure they really do look perpendicular.

The length and the (affine or “shift”) positioning of perpendicular line segments doesn’t matter to their perpendicularity. So to make life easier on myself I’ll centre everything on zero and make the segments equal length.

image

The metric formula is going to work if let’s say my first vector v is (+1,+1) (one to the right and one up) and my second vector goes one down and one to the right. Then the metric would do:

+1 • +1 (horizontal) + +1 • −1 (vertical)

which cancels.

 

What if it were a slope of 9.18723 or something I don’t want to think about inverting?

This is a case where it’s probably easier to think in terms of abstractions and deduce, rather than using imagination in the conventional way.

If I went over +a steps to the right and +b steps to the up (slope=b/a), then the metric would do:

a•? + b•¿

What is that missing? If I plugged in (?←−b, ¿←a) or (?←b, ¿←−a), the metric would definitely always cancel.

And in either of those cases, the slope of the question marks (second line) would be −a/b.

So the multiplicative inverse (flipping) corresponds to swapping terms in the metric so that the two parts anti-match. And the additive inverse (sign change) means the anti-matched pairs will “fold in” to zero each other (rather than amplifying=doubling one another).




In philosophical debates about absolute truth, people cite “the truths of pure mathematics” as beyond reproach—eternal and universal things discovered/invented by us fallible mortals. But the more deeply I look into these issues myself, the more I see evidence that mathematics is not as stable as I’d supposed:

  • constructivists and intuitionists argue that the foundations of mathematics don’t make sense
  • logicians accuse mathematicians of not being rigorous enough
  • mathematicians themselves admit they totally ignore foundational issues and just concentrate on getting interesting results that make sense within their set of assumptions and could probably be “straightened up” to satisfy the logicians
  • Bill Thurston referred to mathematics itself as a social entity — it is the dynamical creation of a community, it lives inside the heads of the people who prove these things and not on paper.
  • John L Bell and Geoffrey Hellman: “Contrary to the popular (mis)conception of mathematics as a cut-and-dried body of universally agreed upon truths and methods, as soon as one examines the foundations of mathematics, one encounters divergences of viewpoint and failures of communication that can easily remind one of religious, schismatic controversy.
  

Norman Wildberger thinks real numbers have been a wrong turning in mathematics. He also claims, here in the video above, that angles θ are illogical. (Or maybe I should say, certain angles are used illogically.) Some angles, like 60°, can be constructed via ruler and compass. But other angles like 34° and 26° are not constructible.

So although “I know what you mean” when you talk about a real number or an angle that measures 90.1°, maybe we should both recognise that they don’t really make sense and speak in air quotes.

 

Related but different. On the topic of left-brains, right-brains, closed-minds, and open-minds in science. You can see youtube user njwildberger being beaten up on the XKCD forums for suggesting such unconventional and—ick!—philosophical ideas. Listen to these self-satisfied, smarter-than-thou sabelotodos savaging the “ridiculousness” of someone who would undercut this Well Established Knowledge.

I find that incredible because XKCD’s vision of science seems to be about open-mindedness, learning from data, and accepting the truth based on logic rather than tradition or popularity.

The Data So Far
Significant

Citogenesis
Beliefs

OK, “data” needs to be replaced with something else in theoretical maths. But you could at least listen to what the guy’s saying rather than his credentials or his sweatpants. (Conversely: if John Conway says it, does that make it true? He gives talks in sweatpants as well.)

I bet ≥ some of these know-it-alls have lauded Galileo for smashing the accepted wisdoms handed down from Aristotle with cold, hard logic. What’s the difference to making fun of njwildberger because he’s suggesting something weird or unconventional? Prima facie it makes sense to me.

Maybe you don’t care about the foundational issues (isn’t that called hand-waving elsewhere?), or maybe you can disprove what he’s saying—but this PageRank 7 site is just attacking him rather than his idea. (For example they look at his publication record to see if he’s “someone we should take seriously”.)

You want to know why people aren’t interested in science? I think it’s in part because science and maths is associated with such stuck-up, judgmental people—putting down everyone who’s less “intelligent" than they are.




The word ‘space' has acquired several meanings, which is what you would expect of such a sexy, primitive, metaphorically rich, eminently repurposeable concept.

  1. Outer space, of course, is where cosmonauts, Hubble telescopes, television satellites, and aliens reside. It’s ℝ³, or something like that.
    image 
  2. Grammatical spaces keep words apart. The space bar got a little more exercise than the backspace key while I was writing this list.
    image 
  3. Non-printable area (space) is also free from ink or electronic text in newspapers: ad space. Would you like to buy one?
    image 
  4. Closely related is the negative area in sculpture, architecture, and other visual arts.
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  5. Or in music. Don’t forget to “play” the notes you don’t play, Thelonious!
  6. Or the space you need to give someone in a relationship, if you want to allow them to be themselves whilst also being with you.
  7. Space on my hard drive to store an exact digital replica of all my vinyl? This kind of space also applies to human memory capacity, computer RAM, and other electronic pulsings which seem rather more time-based than spatial & static.
    image 
  8. Businessmen refer to competitive neighbourhoods: the online payments space; the self-help books category; the $99-and-under motel space; and so on.
    image
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  9. Space as distinct from time. Although cosmologists will tell you that spacetime is a pseudo-Riemannian manifold which looks locally like ℝ⁴, a geographer or ecologist will tell you that locally space looks like ℝ² (since we live solely on the surface of the Earth).
    imageimageimageimageimageimage
    I believe the ℝ² view is also taken by programmers who geotag things (flickr photos, twitter tweets, 4square updates): second basement = 85th floor and canopy = rainforest floor as far as that’s concerned.

    Both perspectives are valid. They’re just different ways of modelling “the world” with tuples. Is it surprising that cold, rigid, soulless mathematics allows for different, contradictory viewpoints? Time is like space in the grand scheme of things, but for life on Earth time-averages and space-averages are very different.
    Europe, upside-down. 
  10. Parameter space. The first graphs one learns in school plot input x versus output ƒ(x).
    image
    But another kind of plot — like a solid liquid gas diagram
    image
    — plots input a versus input b, with the area coloured or labelled by output ƒ(x). (In the case of matter’s phases, the codomain of ƒ is the set {solid, liquid, gas, plasma} rather than the familiar .)

    • When I push this lever, what happens? What about when I push that one?
    • There are connections to Fourier spectrum.
  11. Phase space. Paths, orbits, and trajectories taken through other spaces. Like the string of (x₍ᵤ₎,y₍ᵤ₎,z₍ᵤ₎)-coordinates that a water rocket takes across the lawn. Or the path of temperature (temp₍ᵤ₎) during a year in Bloomington.
    image
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    Or the trajectory of the dynamical system (your feelings₍ᵤ₎, your partner’s feelings₍ᵤ₎) representing your marriage.
    image

    Roger Penrose uses the example of the configuration space of a belt to explain that phases can happen on non-trivial manifolds. (A belt can take on as many configurations as a string, plus it can be twisted into a Moebius band, but if it’s twisted twice that’s the same as twisted zero times.) 
    image 

[Sorry, I don’t have a Unicode character for subscript t, so I used u to represent the time-indexing of path variables. Maybe that’s better anyway, because time isn’t the only possible index.]

  1. Personal space. I forgot personal space. Excuse me; pardon me.



  2. All of the spaces above are like an existing nothing. The space between your arm and your chest, the space where I draw—all of these are conceptually “empty” but impinge on and interact with the rest of reality.

    All of those senses of the word are completely nothing alike to how mathematicians use the word. Mathematicians mean “stuff plus structure to the stuff” which is not at all like the other spaces.


    Abstract spaces.
    These are best understood as ordered tuples, i.e. “Things plus the relationships and desired interpretation of those things.” The space—more like “the entire logical universe I’m going to be talking about here”—is supposed to contain EVERYTHING you need, in order to work with any of the parts. So for example to use a division sign ÷, the space must include numbers like and . (Or you could just do without the ÷ sign. You can make a ring that’s not a division ring; look it up.)

    • A Banach space is made up of vectors (things that can be added together), is complete (there are enough things that infinite limit sequences make sense), with a notion of distance (norm), but not necessarily angle. Also two things can be 0 distance away from each other without being the same thing. (That’s unlike points in Euclidean space: (2,5,2) is the only thing 0 away from (2,5,2)).
    • A group is complete in the sense that everything you need to do the operation is included. (But not complete in the way that Banach space is complete with respect to sequences converging. Geez, this terminology is overloaded with meanings!)
      image 
    • A vector space is complete in the same way that a group is. In the abstract sense. Again, a vector is “anything that can be added together”. The vectors’ space completely brings together all the possible sums of any combination of summands.

      For example, in a 2-space, if you had (1,0) and (0,1) in the space, you would need (1,1) so that the vector space could be complete. (You would also need other stuff.)

      And if the vector space had a and b, it would need to contain a+b — whatever that is taken to mean — as well as a+b+b+(a+b)+a and so on. In jargon, “closed under addition”.
    • A topological space (confusingly, sometimes called “a topology”) is made up of things, bundled together with the necessary overlap, intersection, union, superset, subset concepts so that “connectedness” makes sense.
      image 
    • A Hilbert space has everything a Banach space does, plus the notion of "angle". (Defining an inner product is as good as defining an angle, because you can infer angle from inner multiplication.) ℂ⁷ is a hilbert space, but the pair ({0, 1, 2}, + mod 2) is not.
      image
       imageimageimageimageimageimageimageimageimageimage
    • Euclidean space is a flat, rigid, stick-straight, all-joins-square Hilbert space.
    • To recap that: vector space  Banach space  Hilbert space, where the  symbol means “is less structured than”.

      Topological spaces can be even more unstructured than a vector space. Wikipedia explains all of the T0 T1 ⊰ T2 ⊰ T2.5  T3  T3.5  T4 ⊰ T5 ⊰ T6 progression which was thoroughly explored during the 20th century. (Those spaces differ in how separated “neighbours” are taken to be.)

I don’t mean to imply that these spaces can only be thought of as tuples: ({things}, operations). There are categorical ways to understand them which may be better. But don’t look at me; ask the ncatlab!

  1. Lastly, sometimes ‘space’ just means a collection of related things, without necessarily specifying, like above, the tools and viewpoints that we take to their relationships.
    • The space of all possible faces.
    • The space of all possible boyfriends.
    • The space of all possible songs.
    • The space of all possible sentences.
    • Qualia space, if you’re a theorist of consciousness.
    • The space of all possible romantic relationships.
    • The space of all possible computer programs of length 17239 bytes.
    • Whatever space politics occupies. (And we could debate about that.)
    • (consumption, leisure, utility) space
    • The space of all possible strategy pairs.
    • The space of all possible wealth distributions that sum to W.
    • The space of all bounded functions.
    • The space of all 8×8 matrices over the field ℤ₁₁.
    • The space of all polynomials.
    • The space of all continuous functions from [0,1] → [0,1].
    • The space of all square integrable functions.
    • The space of all bounded linear operators.
    • The space of all possible models of ______.
    • The space of all legal configurations of the Rubik’s cube.

(Some of these may be assumed to come packaged with a particular set of interpretations as in the previous ol:li.)

this,1




This is trippy, and profound.

The determinant — which tells you the change in size after a matrix transformation 𝓜 — is just an Instance of the Alternating Multilinear Map.

image

(Alternating meaning it goes + − + − + − + − ……. Multilinear meaning linear in every term, ceteris paribus:

\begin{matrix} a \; f(\cdots  \blacksquare  \cdots) + b \; f( \cdots \blacksquare \cdots) \\ = \shortparallel | \ | \\ f( \cdots a \ \blacksquare + b \ \blacksquare \cdots) \end{matrix}    \\ \\ \qquad \footnotesize{\bullet f \text{ is the multilinear mapping}} \\ \qquad \bullet a, b \in \text{the underlying number corpus } \mathbb{K} \\ \qquad \bullet \text{above holds for any term } \blacksquare \text{ (if done one-at-a-time)})

 

Now we tripThe inner product — which tells you the “angle” between 2 things, in a super abstract sense — is also an instantiation of the Alternating Multilinear Map.

In conclusion, mathematics proves that Size is the same kind of thing as Angle

Say whaaaaaat? I’m going to go get high now and watch Koyaanaasqatsi.

image

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This is for my homies in maths class.

Mathematical matrices are blocks of numbers, arrayed in 2-D. (Higher-dimensional array-verbs are called tensors.)

  1. image
    Left “times” right equals target. Each entry in the target is the result of a series of +'s and ×'s along the red and blue. A long sum of pairwise products.

  2. Your left hand goes across and your right hand goes up/down.
    imagewhere
    image.
  3. There need to be as many abcdefg's as there are 1234567's or else the operation can't be done.
  4. Also you can tell how big the output matrix will be. There can be three blue rows so the output has three rows. There can be four red columns so the output has four columns.
  5. This is the “inner product” because multiplying vector-shaped blocks (tall blocks) like Aᵀ•B results in an equal or smaller sized output.

    (There is also an “outer product" which is a different way of combining the info from the two matrices. That gives you an equal or larger shaped result when you multiply vector/list-shaped tall blocks A∧B.)
  6. Try playing around with this one or that one.

Matrix multiplication is the simplest example of a linear operator, the broad class of which explains quantum mechanics and ODE’s. You can also apply different matrices at different points as in a vector field — on a flat surface or a curvy, holey surface.




A circle is made up of points equidistant from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.

But what if you’re on a hill? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (equiforce or equiwork or equi-effort curve) then it would look different — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.

Here’re some brain-wrinkling pictures of “circles”, under different L_p metrics:

astroid p=⅔
p = ⅔

The subadditive “triangle inequality” A→B→C > A→C no longer holds when p<1.

p = 4p = 4 

 p = 1/2
p
= ½
. (Think about a Poincaré disk to see how these pointy astroids can be “circles”.)
 p = 3/2 p = 3/2 

 workin on my ♘ ♞ movesThe moves available to a knight ♘ ♞ in chess are a circle under L1 metric over a discrete 2-D space.




Vectors, concretely, are arrows, with a head and a tail. If two arrows share a tail, then you can measure the angle between them. The length of the arrow represents the magnitude of the vector.

The modern abstract view is much more interesting but let’s start at the beginning.

Force vectors

Originally vectors were conceived as a force applied at a point.

As in, “That lawn ain’t mowing itself, boy. Now you git over there and apply a continuous stream of vectors to that lawnmower, before I apply a high-magnitude vector to your bee-hind!”

Thanks Galileo, totally gonna get you back, man

The Galilean idea of splitting a point into its x-coordinate, y-coordinate and z-coordinate works with vectors as well. “Apply a force that totals 5 foot-pounds / second² in the x direction and 2 foot-pounds / second² in the y direction”, for instance.

Therefore, both points and vectors benefit from adding more dimensions to Galileo’s “coordinate system”. Add a w dimension, a q dimension, a ξ dimension — and it’s up to you to determine what those things can mean.

If a vector can be described as (5, 2, 0), then why not make a vector that’s (5, 2, 0, 1.1, 2.2, 19, 0, 0, 0, 3)? And so on.

4th Dimension Plus

So that’s how you get to 4-D vectors, 13-D vectors, and 11,929-D vectors. But the really interesting stuff comes from considering -dimensional vectors. That opens up functional space, and sooooo many things in life are functions.

(Interesting stuff also happens when you make vectors out of things that are not traditionally conceived to be “numbers”. Another post.)

Abstractions

In the most general sense, vectors are things that can be added together. The modern, abstract view includes as vectors:

Things you can do with vectors

Given two vectors, you should be able to take their outer product or their inner product.

The inner product allows you to measure the angle between two vectors. If the inner product makes sense, then the space you are playing in has geometry. (Not all spaces have geometry — some just have topology.)

And — this is weird — if the concept of angle applies, then the concept of length applies as well. Don’t ask me why; the symbols just work that way.

Magnitude

But the “length” of a song (one of my for-instances above) would not be something like 2:43. The magnitude of a song vector would be the total amount of energy in the sound wave | compression wave.

\| \text{song} \| = \int \text{compression wave}

What is the angle between two songs, two spike-trains, two security prices? What is the angle between two heartbeats? It’s the correlation between them.

Linear Algebra

Also, you can do linear algebra on vectors — provided they’re coming out of the same point. Some might say that the ability to do linear algebra on something is what makes a vector.

That can mean different things in different spaces — like maybe you’re superposing wave-forms, or maybe you’re converting bitmap images to JPEG. Or maybe you’re Photoshopping an existing JPEG. Oh, man, Photoshop is so math-y.

shearing the mona lisa

Shearing the mona lisa (linear algebra on an image — from the Wikipedia page on eigenvectors, one of which is the red arrow)