90194183753 http://1ucasvb.tumblr.com/post/90194183753/what-are-your-thoughts-on-the-pi-v-tau-debate 271

mrfb said: What are your thoughts on the pi v. tau debate?

1ucasvb:

(For those unaware of the Pi. vs. Tau debate, read the Tau Manifesto and then the Pi Manifesto).

I’m actually extremely pro-tau, but only under certain conditions. I’ll explain.

Warning: The following is my personal take on these subjects. I’m no authority. This is pretty much late night armchair philosophy and ramblings of a madman. It’s just how I make sense of some of these ideas, and it’s the first time I’m trying to put these into words. Hopefully, they’ll make some sense and I won’t look like a complete nut.

On the merits of the debate

Mathematics thrives on conventions. Being able to symbolically convey a very precise idea is one of its greatest triumphs and strengths. For that, we have developed a set of (ideally unambiguous) conventional notations. Notation is pretty damn important. Learning this mathematical language takes a lot of effort, and it is a skill we should respect. A lot of important knowledge is being carried by these crazy symbols, knowledge built upon centuries of intense thought and research by some of the smartest people who have ever lived.

The use of the Greek lower-case letter pi (π) to denote a particular irrational number is one of such conventions developed in mathematics. As a convention, it is extremely valuable as it is. There’s little reason to change it. The fact we agreed with π as the ratio of a circle’s circumference to its diameter is of little consequence to any underlying mathematics, it changes nothing, so this isn’t really a point to be argued. The important thing is merely consistency.

In this respect, the tau vs. pi debate is a waste of time in my view. Saying equations are prettier because of a factor of two somewhere is missing is a bit ridiculous and non-mathematical, and entirely misses the point of having a constant defined in the first place. Are we arguing over mathematics or typography/aesthetics?

However, conceptual differences are important. This is where the debate can be fruitful, I think, so dismissing it completely can be (and I believe it is) a bad thing. Oddly nobody else seems to be making this particular point, at least not how I’m going to expose it here.

Pi

You see, π shows up everywhere in math by itself, no factors of 2 attached at all. It’s a pretty remarkable number on its own. It shows up even when things don’t seem to be related to circles. For instance, the integral of the Gaussian is √π, which is a surprising result (and it’s one of my current math animation projects). The sum of the reciprocal of squares, also known as the Basel problem, is π2/6. No circles in sight here.

But whenever tau (τ = 2π) shows up, people like to talk about circles. They’re missing the point, I think. 2π isn’t the circle constant. It’s the ANGLE constant. The circle just happens to be related to the concept of angles, and not the other way around.

The most mathematically natural way of measuring angles is in units of radians. Everything works out so simply when we use radians that it’s tempting to call it the one true way of doing so.

"Dimensions" vs. "units"

Now, before I go a little bit more into this argument, I need to clear something up. A lot of people say “radians don’t have units”, but that’s an incorrect use of terms. What these people are trying to say is that radians don’t have a dimension, that is, they are a dimensionless unit. See how we can use both terms together and still make sense? That’s because they have distinct and precise meanings.

A unit is a standard we use to measure other similar things. For instance, you can measure length in several units: meters, feet, the nearest spoon’s length, light-years, (toenail growth rate)·century, (your own name here)’s nipple-to-nipple distance, etc.

What all these units have in common is that they have the same dimension: length, or simply [L]. The other base dimensions in nature are time, [T], and mass, [M]. There are other dimensions that are used, but these are the more basic ones.

One way to understand this is to think of [L], [T] and [M] as the “real” physical quantities, or kinda like how Nature “understands” these quantities. When you read “2 meters”, you should be seeing 2 × (“1 of something we use to define a meter” × [L]).

The “meter” has a certain amount of [L] hidden in it, you see, because we defined it in terms of something else that has a length dimension. The 2 in “2 meters” is just there telling us just how much of that something we are talking about. The 2 is a pure, dimensionless number.

Using these three dimensions, we can build all sorts of quantities. Here’s a few, and some example units:

  • Force = [M][L][T]-2 → newton, pound-force
  • Energy = [M][L]2[T]-2 → joules, calories, kilotons of TNT
  • Frequency = [T]-1 → hertz, radians/seconds

Dimensions can be treated just like variables: you can multiply, divide, take powers and square roots of them, but they don’t “mix” together. You can even add different dimensions, though just like variables, you get nowhere with that: a+b is just a+b. While it makes little physical sense something with dimensions [L]+[M] (think, 1 metre + 1 kg), there’s no reason why you shouldn’t be allowed to have it, if you’re really into that kind of thing. Weirdo.

By the way, this topic is called dimensional analysis, and it’s a very interesting subject.

Radians

Radians are an unit defined as the angle enclosed by an arc around a circle that is as long as that circle’s radius. Here’s an animation I created that explains it:

Circle_radians.gif

It doesn’t matter what radius you pick (that’s why in the animation the radius is just a generic “r”), the angle is always the same because the arc’s length is also proportional to the radius, so the length of the radius always cancels itself out when you actually end up calculating radians.

But also, notice that the arc-length is a unit with dimension [L], and so is the radius. If you divide one by the other, the [L] dimensions cancel out, just like variables would. We end up with a quantity that’s just a number, a dimensionless quantity. A full turn has about 6.28 radians in it, that is, 1 turn ≈ 6.28 × (“1 of something we call radians” × [no dimension whatsoever]).

So, radians have no dimensions. We can treat them just like any other pure number. This is usually how everyone does it: they say it is a pure number, no meanings attached to it, and call it a day.

This is where my take on the subject takes a weird turn…

"Dimensions" vs. "Concepts"

But conceptually, these numbers are still measuring something. Two instances of the same number associated with the same dimension can represent two entirely different things, so there’s more to these quantities than dimensions.

For example: “1 hertz” and “1 radian per second”, while dimensionally and numerically identical (both are “1 second-1”), are totally different conceptually. Something happening once every second is completely different than something rotating one radian every second.

In the same way, a torque of 1 newton-meter is numerically and dimensionally equivalent to 1 joule of energy, but the two ideas are very different. That’s why we explicitly write torque with units of “newton-meter” instead of joules. (In fact, it can be argued that the torque would be better expressed in SI units as joules per radian.)

So, here’s where my take on all these things gets weird: I think that beyond dimensions, we also attach “concepts” to numerical quantities and units, and these are also subject to a “conceptual analysis” similar to the dimensional analysis I mentioned up there.

While dimensions have a physical meaning, “concepts” are, well, abstract. (For consistency, I’ll denote concepts in single quotes from now on. E.g.: ‘angle’)

An ‘angle’ is such a concept, attached to the unit of a radian: 1 “radian” = 1 × (“something we call radian” × ‘angle’), where “something we call radian” is the same as “ratio between length of an arc of a circle and that circle’s radius”, that is, the definition of the radian. So, hidden inside a radian, is the concept of ‘angle’ being multiplied by the number, just like a dimension such as [L] would be.

In fact, in terms of concepts, we could say: ‘angle’ = ‘circular arc’ / ‘line segment’, so that we have: 1 radian = (1 × [L] × ‘circular arc’) / (1 × [L] × ‘line segment’).

In other words, what I’m trying to say here is that even if the dimensions cancel out in the definition of a radian, the concept of ‘angle’ shouldn’t really go away with the number we’ve got. The concept ‘angle’ is intrinsically in the “radian” unit, and it is not a dimensional quantity.

I love to play with this idea of “conceptual analysis”, and it has given me some weird and accurate insights before.

Dude, just get to the point

All right, all right, here’s my point. I think we should have two definitions:

  • π = 3.14159265…
  • τ = 6.28318530… radians

Notice the difference?

π is just a pure number, like 1, 2.5 or √2. It has no concepts attached to it.

Meanwhile, τ is a number attached to the units of radians, which means τ carries the concept of ‘angle’ with it everywhere it is used, always. Seeing τ immediately implies we’re talking about angles.

This is the important conceptual difference I talked about in the beginning. This is where τ really makes a lot of sense and where it would be useful.

"The Conceptu-tau Manifesto" (groooan)

So, here’s my crazy proposal: let’s adopt tau as THE ANGLE CONSTANT.

Let’s face it, π isn’t going anywhere. It’s already well-established way beyond the scope of circles anyway. It makes no sense to fight it, and it has earned its place.

But whenever we talk about angles and rotations, there’s no question that τ is the proper constant to use, just as surely as the use of radians instead of degrees for angles. A full turn is the important idea, not a half turn.

Here’s the same animation as the one above, except this time using τ for the full turn instead of 2π.

Circle_radians_tau.gif

Notice that this time we can just keep using our unit of measure (the red arc of 1 radius in length) all the way around, counting each new whole radius (or radian) that fits, only adding a fractional bit at the end to complete the whole turn (the 0.28.. part). This makes much more sense, since that’s how we awalys used any unit of measurement: we count how many times our unit fits in the whole of the thing we’re measuring, not just in half of the thing.

With π, we are assigning a certain special name for a half-turn, even though it is the full turn the thing we are trying to measure. While this isn’t inherently a bad thing (a rotation of a half turn has a lot of importance in mathematics, hence why π exists), it is an odd special case that’s simply an arbitrary quirk of definitions.

The undeniable fact about all of this is that a full turn is more important than a half turn, so it deserves its own symbol.

However, notice that the foundation of that argument is not the numerical value of the full turn or half turn. That’s totally irrelevant, which is exactly why we’d like to use a symbol in place of these numbers! We don’t care about them! But for some reason, this is what most people seem to focus their attention on.

No. The foundation of that argument is in the word “turn”. It narrows down the single mathematical concept we are addressing in the discussion, and it’s in that context that τ really makes all the sense in the world, since it’s the one that represents a turn.

If you’re not convinced yet, just look at our language. We don’t even have a word (in common use, at least) to describe “half a turn”. We already talk about half a turn in terms of a full turn in our natural language. We all already use the definition π radians = τ/2 radians, but only when we talk about angles and rotation. It’s just how we naturally treat the concept, and it makes perfect sense that way.

If that doesn’t make it deserving of a mathematical notation, I don’t know what does.

An example of the conceptual use of τ as the angle constant

Now, imagine we live in an alternate reality where τ = 6.28… radians, as I proposed. What could math feel like?

The following is obviously incredibly biased (this is an opinion text, so that’s kind of the point here), but it’s pretty close to the thought process I had when I was trying to make sense of the same ideas.

Euler’s Identity: eτi = 1

You see that mathematical expression for the first time in your life.

You see τ in there. Your brain attaches to it the idea of a “full turn”, as you have been trained to. Your brain is now thinking of things rotating and angles.

You see a representation of a “full turn” multiplied by the imaginary unit, i. You try to make sense of that, and you fail. As you should. But now you’re thinking about the complex plane and what could a “complex full turn” possibly mean.

But your brain doesn’t give up. I hasn’t finished reading the expression yet. So it reads the exponential function. You already know the e0 = 1, that’s one of the key properties of this function. But now, the exponential of a “complex full turn” (whatever that is) is doing something new. What it is? You look at the right of the equals sign.

You see the number 1, the multiplicative identity. This is the same value as e0 that you have already thought of. So, the exponential of a “full complex turn” is doing the same thing as doing nothing.

Your brain makes the connection: the exponential of “a full complex turn” (whatever that is) is bringing you back at the same place as you started. You know something is rotating, and you know this is happening on a complex plane.

Aha! Your brain finally gets it. It’s the only idea that makes sense now: the exponential function is performing a rotation in the complex plane itself.

And if you know trigonometry and think just a little harder, you should deduce that eθi = cos(θ) + i·sin(θ), Euler’s formula.

So, call me crazy or whatever you may, but this actually sounds like a nice convention to have around.

Final words

To be honest, I feel pretty uncomfortable talking about these things. This notion of concepts attached to numbers may be a bit nutty, and I’m not familiar with this sort of approach to things anywhere. (Though a quick Google Search has brought up Bertrand Russell’s Theory of Descriptions), which sounds kinda alike)

But this is similar to the way my brain works, for better or for worse. This is how I learned to tackle math concepts, and this is the kind of approach I try to convey in all of my animations. I try to carry these ‘concepts’ around using things like matching colors and visual styles.

Since so many people are fond of my animations, perhaps this idea has some merits, and I’m not a complete lunatic.

Either way, I don’t think there’s a magic trick to it or anything. It’s just about making sure you are keeping track of what everything represents at all times. This is the key approach to learning mathematics. The more stuff you can connect and correlate, the better and deeper your understanding will be.

And best of all, it’s supposed to make sense, even when it doesn’t. Usually, when it doesn’t make sense, it’s your intuition that’s wrong. It’s an odd lesson to learn, but these are the rules we play by in math.

"But I don’t want to go among mad people," Alice remarked.

"Oh, you can’t help that," said the Cat: "We’re all mad here. I’m mad. You’re mad."

"How do you know I’m mad?" said Alice.

"You must be," said the Cat, "otherwise you wouldn’t have come here."

(from Lewis Carroll’s 1865 novel, Alice’s Adventures in Wonderland)

Well, seems appropriate to reblog this post from Pi Day today. Happy Tau Day!

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Affine transformations preserve parallel lines, and include rotations, scaling, shears and translations. Linear transformations can’t perform translations, but this can be achieved if we go to a higher dimension.

In this animation, a planar (2D) shape lying on the plane z = 1 is translated by means of a linear transformation in three dimensions: a shear along the z axis.

Rotations can be performed normally, also around the z axis. For rotations around any other axis parallel to the z axis, it’s just a matter of performing the appropriate translation that cancels out the translation of the axis performed by the rotation.

This way, all transformations are now linear in 3D, and can be represented by a single 3x3 affine transformation matrix that acts on two dimensions.

The “shadow” of the shape illustrates the relative position between the two images, on the planes z = 0 and z = 1, and was included to better visualize the shear and how it is linear in 3D space.

Affine transformations preserve parallel lines, and include rotations, scaling, shears and translations. Linear transformations can’t perform translations, but this can be achieved if we go to a higher dimension.

In this animation, a planar (2D) shape lying on the plane z = 1 is translated by means of a linear transformation in three dimensions: a shear along the z axis.

Rotations can be performed normally, also around the z axis. For rotations around any other axis parallel to the z axis, it’s just a matter of performing the appropriate translation that cancels out the translation of the axis performed by the rotation.

This way, all transformations are now linear in 3D, and can be represented by a single 3x3 affine transformation matrix that acts on two dimensions.

The “shadow” of the shape illustrates the relative position between the two images, on the planes z = 0 and z = 1, and was included to better visualize the shear and how it is linear in 3D space.

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Mathematics is full of wonderful but “relatively unknown” or “poorly used” theorems. By “relatively unknown” and “poorly used”, I mean they are presented and used in a much more limited scope than they could be, so the theorem may be well known, but only superficially, and its generality and scope can be very understated and underappreciated.

One of my favorites of such theorems was discovered around the 4th century by the Greek mathematician Pappus of Alexandria. It is known as Pappus’s Centroid Theorem.

Pappus’s theorem originally dealt with solids of revolution and their surface areas and volumes. This was all Pappus was able to figure out with the geometric proof methods available to him at the time. Later on, the theorem was rediscovered by Guldin, and studied by Leibniz, Cavalieri and Euler.

The theorem is usually split into two, one for areas and one for volumes. However, the basic principle that makes both work is exactly the same, though it is much more general in the case of areas on a plane, or volumes in space.

The invention/discovery of calculus eventually brought to the table much more general methods that made Pappus’s theorem somewhat limited in comparison.

Still, the theorem is based on a very clever idea that is quite satisfying both conceptually and intuitively, and brilliant in its simplicity. By knowing this key idea one can greatly simplify some problems involving volumes and areas, so the theorem can still be useful, especially when associated with methods from calculus.

Virtually all of the literature that mentions the theorem focus on solids of revolution only, as if the theorem was merely a curiosity, and they never really address why centroids would play a role. This is a big shame. The theorem is much more general, useful and conceptually interesting, and that’s why I’m writing this long post about it. But first, a few words on centroids.

Centroids

Centroids are the purely geometric analogues of centers of mass. They are a single point, not necessarily lying inside a shape, that defines the weighted “average position” of a shape in space. Naturally, the center of mass of any physical object with uniform density coincides with its geometric centroid, since the mass is uniformly distributed along the object’s volume.

Centers of mass are useful because they give you a point of balance: you can balance any object by any point directly under its center of mass (under constant vertical gravity). This comes from the fact the torques acting on the shape exactly cancel out in this configuration, so the object does not tilt either way.

Centroids exist for objects with any dimension you wish. A scattering of points has a centroid that is just their average position. In 1 dimension we have a line segment, whose centroid is always at its center. In 2, 3 or more dimensions, things get a bit trickier. We can always find the centroid by integrating all over the shape and weighting each point by that point’s position. This is generally a complicated enough problem by itself, but for simpler shapes this can be trivial.

Centroids, as well as centers of mass, also have the nice property of being additive: the union of two shapes has a centroid that is the weighted average of both centroids. So if you can decompose a shape into simpler ones, you can find its centroid without any hassle.

Pappus’s Centroid Theorems (for surfaces of revolution)

So, let’s imagine we have ourselves some generic planar curve, which we’ll call the generator. This can be a line segment or some arc of a circle, like in the main animation of this post, above. Anything will do, as long as it lies on a plane.

If we rotate the generator around an axis lying on its plane, and which does not cross the generator, it will “sweep” an area in space. Pappus’s theorem then states:

The surface area swept by a generator curve is equal to the length of the generator c multiplied by the length of the path traced by the geometric centroid of the generator L. That is: A = Lc.

In other words, if you have a generating curve with length c, and its centroid is at a distance a from the axis of rotation, then after a complete turn the generator will have swept an area A = 2πac, where L = 2πa is just the circumference of the circle traced by the centroid. This is what the first animation in this post is showing.

Note that in that animation we are ignoring the circular parts on top and bottom of the cylinder and cone, for simplicity. But this isn’t really a limitation of the theorem at all. If we add line segments for generating those regions we get a new generator with a different centroid, and the theorem still holds. Here’s what that setup looks like:

Try doing the math! It’s nice to see how things work out in the end.

The second theorem is exactly the same statement, but it deals with volumes. So, instead of a planar curve, we now have a closed planar shape with an area A.

If we now rotate it around an axis, the shape will sweep a volume in space. The volume is then just area of the generating shape times the length of the path traced by its centroid, that is, V = LA = 2πRA, where R is the distance the centroid is from the axis of rotation.

Why it works

Now, why does this work? What’s the big deal about the centroid? Here’s an informal, intuitive way to understand it.

You may have noticed that I drew the surfaces in the previous animations in a peculiar translucent way. This was done intentionally, not just to give the surfaces a physically “real” feel, but to illustrate the reason why the theorem works. (It also looks cooler!)

Here are the closed cylinder and cone after they were generated by a rotating curve:

Notice how the center of the top and bottom of the cylinder and the cone are a darker, denser color? This results from the fact that the generator is sweeping more “densely” in those regions. Farther out from the center of rotation, the surface is lighter, indicating a lower “density”.

To better convey this idea, let’s consider a line segment and a curve on a plane.

Below, we see equally spaced line segments of the same length placed along a red curve, perpendicular to it, in two different ways. Under the segments, we see the light blue area that would be swept by the line segment as it moved along the red curve in this way.

In this first case, the segments are placed with one edge on the curve. You can see that when the red curve bends, the spacing between the segments is not constant, since they are not always parallel to each other. This is what results in the different density in the “sweeping” of the surface’s area. Multiplying the length of those line segments by the length of the red curve will NOT give you the total area of the blue strip in this case, because the segments are not placed along the curve centered at their geometric centroids.

What this means is that the differences in areas being swept by the segment as the curve bends don’t cancel out, that is, the bits with more density don’t make up for the ones with a lower density, canceling out the effect of the bend.

However, in this second case, we place the segment so that its centroid is along the curve. In this case, the area is correctly given by Pappus’s theorem, because whenever there’s a bend in the curve one side of the segment is sweeping in a lower density and the other is sweeping at a higher density in a such a way that they both exactly cancel out.

This happens because that “density” is inversely proportional to the “speed” of each point of the line segment as it is moving along the red curve. But this “speed” is directly proportional to the distance to the point of rotation, which lies along the red curve.

Therefore, things only cancel out when you use the centroid as the anchor/pivot on the curve, which allows us to assume a constant density throughout the entire path, which in turn means there’s a constant area/volume being swept per unit of length traversed. The theorem follows directly from this result.

The same argument works in 3D for volumes.

Generalizations and caveats

As mentioned, the theorem is much more general than solids and surfaces of revolution, and in fact works for a lot of tracing curves (open or closed) and generators, as long as certain conditions are met. These are described in detail in the article linked at the end.

First, the tracing curve, the one where the generator moves along (always colored red in these illustrations), needs to be sufficiently smooth, otherwise you can’t properly define the movement of the generator along its extent. Secondly, the generator must be two dimensional, always lying on a plane perpendicular to the tracing curve. Third, in order for the theorem to remain simple, the curve and the plane must intersect at the centroid of the generator.

This means that, in two dimensions, the generator can only be a line segment (or pieces of it), as in the previous images. In this case, both the “2D volume” (the area) and “2D area” (the lateral curves that bound the area, traced by the ends of the segment) can be properly described by the theorem. If the tracing curve has sharp enough bends or crosses itself, then different regions may be covered more than once. This is something that needs to be accounted for via other means.

The immediate extension of this 2D case to 3D is perfectly valid as well, where the line segment gets replaced by a circle as a generator. This gives us a cylindrical tube of constant radius along the tracing curve in space, no longer confined to a plane. Both the lateral surface area of the tube and its volume can be computed directly by the theorem. You can even have knots as the curve!

In 3D, we could also have other shapes instead of a circle as the generator. In this case there are complications, mostly because now the orientation of the shape matters. Say, for example, that we have a square-shaped generator. As it moves along a curve it can also twist around, as seen in the animation below:

In both cases, the volume is exactly the same, and is given by Pappus’ theorem. This can be understood as a generalization of Cavalieri’s principle along the red curve, which also only works because we’re using the centroid as our anchor.

However, the surface areas in this case are NOT the same: the surface area of the twisted version is larger.

But if the tracing curve is planar (2D) itself and the generator does not rotate relative to the curve (that is, it remains “upright” all along the path), like in the case of surfaces of revolution, then the theorem works fine for areas in 3D.

So the theorem holds nicely for “2D volumes” (planar areas) and 3D volumes, but usually breaks down for surface areas in 3D. The theorem only holds for surface areas in 3D in a particular orientation of the generator along the curve (see reference).

In all valid cases, however, the centroid is the only point where you get the direct statement of the theorem as mentioned before.

Further generalizations

Since the theorem holds for 2D and 3D volumes, we can do a lot more with it. So far, we only considered a generator that is constant along the curve, which is why we have the direct expression for the volume. We actually don’t need this restriction, but then we have to use calculus.

For instance, in 3D, given a tracing curve parametrized by 0 ≤ sL, and a generator as a shape of area A(s), which varies along the curve in such a way that the centroid is always in the curve, then we can compute the total volume simply by evaluating the integral:

V = 0LA(s) ds

Which is basically a line integral along the scalar field given by A(s).

This means we can use Pappus theorem to find the volume of all sorts of crazy shapes along a curve in space, which is quite nifty. Think of tentacles, bent pyramids and crazy helices, like this one:

What we have here are five equal equilateral triangles positioned on the vertices of a regular pentagon. Their respective centroids lie on their centers, and since all of them have the same area the overall centroid (the blue dot) is exactly in the middle of the pentagonal shape, which is true no matter how you rotate the pentagon or the individual triangles.

This means we can generate a solid along the red curve by sweeping these triangles while everything rotates in any crazy way we want (like the overall pentagon and each individual triangle separately, as in the animation), and Pappus’s theorem will give us the volume of this shape just the same. (But not the area!)

If this doesn’t convince you this theorem is awesome and underappreciated, I don’t know what will.

So there you go. A nifty theorem that doesn’t get enough love and appreciation.

Reference

For a great, detailed and proper generalization of the theorem (which apparently took centuries to get enough attention of someone) see:

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mrfb said: What are your thoughts on the pi v. tau debate?

(For those unaware of the Pi. vs. Tau debate, read the Tau Manifesto and then the Pi Manifesto).

I’m actually extremely pro-tau, but only under certain conditions. I’ll explain.

Warning: The following is my personal take on these subjects. I’m no authority. This is pretty much late night armchair philosophy and ramblings of a madman. It’s just how I make sense of some of these ideas, and it’s the first time I’m trying to put these into words. Hopefully, they’ll make some sense and I won’t look like a complete nut.

On the merits of the debate

Mathematics thrives on conventions. Being able to symbolically convey a very precise idea is one of its greatest triumphs and strengths. For that, we have developed a set of (ideally unambiguous) conventional notations. Notation is pretty damn important. Learning this mathematical language takes a lot of effort, and it is a skill we should respect. A lot of important knowledge is being carried by these crazy symbols, knowledge built upon centuries of intense thought and research by some of the smartest people who have ever lived.

The use of the Greek lower-case letter pi (π) to denote a particular irrational number is one of such conventions developed in mathematics. As a convention, it is extremely valuable as it is. There’s little reason to change it. The fact we agreed with π as the ratio of a circle’s circumference to its diameter is of little consequence to any underlying mathematics, it changes nothing, so this isn’t really a point to be argued. The important thing is merely consistency.

In this respect, the tau vs. pi debate is a waste of time in my view. Saying equations are prettier because of a factor of two somewhere is missing is a bit ridiculous and non-mathematical, and entirely misses the point of having a constant defined in the first place. Are we arguing over mathematics or typography/aesthetics?

However, conceptual differences are important. This is where the debate can be fruitful, I think, so dismissing it completely can be (and I believe it is) a bad thing. Oddly nobody else seems to be making this particular point, at least not how I’m going to expose it here.

Pi

You see, π shows up everywhere in math by itself, no factors of 2 attached at all. It’s a pretty remarkable number on its own. It shows up even when things don’t seem to be related to circles. For instance, the integral of the Gaussian is √π, which is a surprising result (and it’s one of my current math animation projects). The sum of the reciprocal of squares, also known as the Basel problem, is π2/6. No circles in sight here.

But whenever tau (τ = 2π) shows up, people like to talk about circles. They’re missing the point, I think. 2π isn’t the circle constant. It’s the ANGLE constant. The circle just happens to be related to the concept of angles, and not the other way around.

The most mathematically natural way of measuring angles is in units of radians. Everything works out so simply when we use radians that it’s tempting to call it the one true way of doing so.

"Dimensions" vs. "units"

Now, before I go a little bit more into this argument, I need to clear something up. A lot of people say “radians don’t have units”, but that’s an incorrect use of terms. What these people are trying to say is that radians don’t have a dimension, that is, they are a dimensionless unit. See how we can use both terms together and still make sense? That’s because they have distinct and precise meanings.

A unit is a standard we use to measure other similar things. For instance, you can measure length in several units: meters, feet, the nearest spoon’s length, light-years, (toenail growth rate)·century, (your own name here)’s nipple-to-nipple distance, etc.

What all these units have in common is that they have the same dimension: length, or simply [L]. The other base dimensions in nature are time, [T], and mass, [M]. There are other dimensions that are used, but these are the more basic ones.

One way to understand this is to think of [L], [T] and [M] as the “real” physical quantities, or kinda like how Nature “understands” these quantities. When you read “2 meters”, you should be seeing 2 × (“1 of something we use to define a meter” × [L]).

The “meter” has a certain amount of [L] hidden in it, you see, because we defined it in terms of something else that has a length dimension. The 2 in “2 meters” is just there telling us just how much of that something we are talking about. The 2 is a pure, dimensionless number.

Using these three dimensions, we can build all sorts of quantities. Here’s a few, and some example units:

  • Force = [M][L][T]-2 → newton, pound-force
  • Energy = [M][L]2[T]-2 → joules, calories, kilotons of TNT
  • Frequency = [T]-1 → hertz, radians/seconds

Dimensions can be treated just like variables: you can multiply, divide, take powers and square roots of them, but they don’t “mix” together. You can even add different dimensions, though just like variables, you get nowhere with that: a+b is just a+b. While it makes little physical sense something with dimensions [L]+[M] (think, 1 metre + 1 kg), there’s no reason why you shouldn’t be allowed to have it, if you’re really into that kind of thing. Weirdo.

By the way, this topic is called dimensional analysis, and it’s a very interesting subject.

Radians

Radians are an unit defined as the angle enclosed by an arc around a circle that is as long as that circle’s radius. Here’s an animation I created that explains it:

Circle_radians.gif

It doesn’t matter what radius you pick (that’s why in the animation the radius is just a generic “r”), the angle is always the same because the arc’s length is also proportional to the radius, so the length of the radius always cancels itself out when you actually end up calculating radians.

But also, notice that the arc-length is a unit with dimension [L], and so is the radius. If you divide one by the other, the [L] dimensions cancel out, just like variables would. We end up with a quantity that’s just a number, a dimensionless quantity. A full turn has about 6.28 radians in it, that is, 1 turn ≈ 6.28 × (“1 of something we call radians” × [no dimension whatsoever]).

So, radians have no dimensions. We can treat them just like any other pure number. This is usually how everyone does it: they say it is a pure number, no meanings attached to it, and call it a day.

This is where my take on the subject takes a weird turn…

"Dimensions" vs. "Concepts"

But conceptually, these numbers are still measuring something. Two instances of the same number associated with the same dimension can represent two entirely different things, so there’s more to these quantities than dimensions.

For example: “1 hertz” and “1 radian per second”, while dimensionally and numerically identical (both are “1 second-1”), are totally different conceptually. Something happening once every second is completely different than something rotating one radian every second.

In the same way, a torque of 1 newton-meter is numerically and dimensionally equivalent to 1 joule of energy, but the two ideas are very different. That’s why we explicitly write torque with units of “newton-meter” instead of joules. (In fact, it can be argued that the torque would be better expressed in SI units as joules per radian.)

So, here’s where my take on all these things gets weird: I think that beyond dimensions, we also attach “concepts” to numerical quantities and units, and these are also subject to a “conceptual analysis” similar to the dimensional analysis I mentioned up there.

While dimensions have a physical meaning, “concepts” are, well, abstract. (For consistency, I’ll denote concepts in single quotes from now on. E.g.: ‘angle’)

An ‘angle’ is such a concept, attached to the unit of a radian: 1 “radian” = 1 × (“something we call radian” × ‘angle’), where “something we call radian” is the same as “ratio between length of an arc of a circle and that circle’s radius”, that is, the definition of the radian. So, hidden inside a radian, is the concept of ‘angle’ being multiplied by the number, just like a dimension such as [L] would be.

In fact, in terms of concepts, we could say: ‘angle’ = ‘circular arc’ / ‘line segment’, so that we have: 1 radian = (1 × [L] × ‘circular arc’) / (1 × [L] × ‘line segment’).

In other words, what I’m trying to say here is that even if the dimensions cancel out in the definition of a radian, the concept of ‘angle’ shouldn’t really go away with the number we’ve got. The concept ‘angle’ is intrinsically in the “radian” unit, and it is not a dimensional quantity.

I love to play with this idea of “conceptual analysis”, and it has given me some weird and accurate insights before.

Dude, just get to the point

All right, all right, here’s my point. I think we should have two definitions:

  • π = 3.14159265…
  • τ = 6.28318530… radians

Notice the difference?

π is just a pure number, like 1, 2.5 or √2. It has no concepts attached to it.

Meanwhile, τ is a number attached to the units of radians, which means τ carries the concept of ‘angle’ with it everywhere it is used, always. Seeing τ immediately implies we’re talking about angles.

This is the important conceptual difference I talked about in the beginning. This is where τ really makes a lot of sense and where it would be useful.

"The Conceptu-tau Manifesto" (groooan)

So, here’s my crazy proposal: let’s adopt tau as THE ANGLE CONSTANT.

Let’s face it, π isn’t going anywhere. It’s already well-established way beyond the scope of circles anyway. It makes no sense to fight it, and it has earned its place.

But whenever we talk about angles and rotations, there’s no question that τ is the proper constant to use, just as surely as the use of radians instead of degrees for angles. A full turn is the important idea, not a half turn.

Here’s the same animation as the one above, except this time using τ for the full turn instead of 2π.

Circle_radians_tau.gif

Notice that this time we can just keep using our unit of measure (the red arc of 1 radius in length) all the way around, counting each new whole radius (or radian) that fits, only adding a fractional bit at the end to complete the whole turn (the 0.28.. part). This makes much more sense, since that’s how we awalys used any unit of measurement: we count how many times our unit fits in the whole of the thing we’re measuring, not just in half of the thing.

With π, we are assigning a certain special name for a half-turn, even though it is the full turn the thing we are trying to measure. While this isn’t inherently a bad thing (a rotation of a half turn has a lot of importance in mathematics, hence why π exists), it is an odd special case that’s simply an arbitrary quirk of definitions.

The undeniable fact about all of this is that a full turn is more important than a half turn, so it deserves its own symbol.

However, notice that the foundation of that argument is not the numerical value of the full turn or half turn. That’s totally irrelevant, which is exactly why we’d like to use a symbol in place of these numbers! We don’t care about them! But for some reason, this is what most people seem to focus their attention on.

No. The foundation of that argument is in the word “turn”. It narrows down the single mathematical concept we are addressing in the discussion, and it’s in that context that τ really makes all the sense in the world, since it’s the one that represents a turn.

If you’re not convinced yet, just look at our language. We don’t even have a word (in common use, at least) to describe “half a turn”. We already talk about half a turn in terms of a full turn in our natural language. We all already use the definition π radians = τ/2 radians, but only when we talk about angles and rotation. It’s just how we naturally treat the concept, and it makes perfect sense that way.

If that doesn’t make it deserving of a mathematical notation, I don’t know what does.

An example of the conceptual use of τ as the angle constant

Now, imagine we live in an alternate reality where τ = 6.28… radians, as I proposed. What could math feel like?

The following is obviously incredibly biased (this is an opinion text, so that’s kind of the point here), but it’s pretty close to the thought process I had when I was trying to make sense of the same ideas.

Euler’s Identity: eτi = 1

You see that mathematical expression for the first time in your life.

You see τ in there. Your brain attaches to it the idea of a “full turn”, as you have been trained to. Your brain is now thinking of things rotating and angles.

You see a representation of a “full turn” multiplied by the imaginary unit, i. You try to make sense of that, and you fail. As you should. But now you’re thinking about the complex plane and what could a “complex full turn” possibly mean.

But your brain doesn’t give up. I hasn’t finished reading the expression yet. So it reads the exponential function. You already know the e0 = 1, that’s one of the key properties of this function. But now, the exponential of a “complex full turn” (whatever that is) is doing something new. What it is? You look at the right of the equals sign.

You see the number 1, the multiplicative identity. This is the same value as e0 that you have already thought of. So, the exponential of a “full complex turn” is doing the same thing as doing nothing.

Your brain makes the connection: the exponential of “a full complex turn” (whatever that is) is bringing you back at the same place as you started. You know something is rotating, and you know this is happening on a complex plane.

Aha! Your brain finally gets it. It’s the only idea that makes sense now: the exponential function is performing a rotation in the complex plane itself.

And if you know trigonometry and think just a little harder, you should deduce that eθi = cos(θ) + i·sin(θ), Euler’s formula.

So, call me crazy or whatever you may, but this actually sounds like a nice convention to have around.

Final words

To be honest, I feel pretty uncomfortable talking about these things. This notion of concepts attached to numbers may be a bit nutty, and I’m not familiar with this sort of approach to things anywhere. (Though a quick Google Search has brought up Bertrand Russell’s Theory of Descriptions), which sounds kinda alike)

But this is similar to the way my brain works, for better or for worse. This is how I learned to tackle math concepts, and this is the kind of approach I try to convey in all of my animations. I try to carry these ‘concepts’ around using things like matching colors and visual styles.

Since so many people are fond of my animations, perhaps this idea has some merits, and I’m not a complete lunatic.

Either way, I don’t think there’s a magic trick to it or anything. It’s just about making sure you are keeping track of what everything represents at all times. This is the key approach to learning mathematics. The more stuff you can connect and correlate, the better and deeper your understanding will be.

And best of all, it’s supposed to make sense, even when it doesn’t. Usually, when it doesn’t make sense, it’s your intuition that’s wrong. It’s an odd lesson to learn, but these are the rules we play by in math.

"But I don’t want to go among mad people," Alice remarked.

"Oh, you can’t help that," said the Cat: "We’re all mad here. I’m mad. You’re mad."

"How do you know I’m mad?" said Alice.

"You must be," said the Cat, "otherwise you wouldn’t have come here."

(from Lewis Carroll’s 1865 novel, Alice’s Adventures in Wonderland)

79580708802 http://1ucasvb.tumblr.com/post/79580708802/1ucasvb-happy-pi-day-this-is-just-the-first 756

1ucasvb:

Happy Pi day!

This is just the first post for today. There’ll be two more, so be sure to check them out later if you miss them!

Here’s an arc-length parametrization of a closed curve for the Greek lowercase letter pi, famously used for the circle constant, π = 3.1415926535897932384626… (that’s what I bothered memorizing!)

Arc-length parametrizations are also called unit-speed parametrizations, because a point moving along the path will move with speed 1: the point moves 1 unit of arc-length per 1 unit of time.

It is generally very hard, if not impossible, to find this parametrization in closed form. But it always exists for nice continuous curves. Since it has some pretty cool uses, just knowing it exists is a powerful enough tool for mathematicians to use it on other cool theorems.

Using computers, we can usually approximate it numerically to any degree of accuracy we desire. The basic algorithm is pretty simple: just make a table of arc-length for each value of t. Then, the unit parametrization is just reading the table in reverse: find t given arc-length. Some interpolation is usually necessary.

1ucasvb:

Happy Pi day!

This is just the first post for today. There’ll be two more, so be sure to check them out later if you miss them!

Here’s an arc-length parametrization of a closed curve for the Greek lowercase letter pi, famously used for the circle constant, π = 3.1415926535897932384626… (that’s what I bothered memorizing!)

Arc-length parametrizations are also called unit-speed parametrizations, because a point moving along the path will move with speed 1: the point moves 1 unit of arc-length per 1 unit of time.

It is generally very hard, if not impossible, to find this parametrization in closed form. But it always exists for nice continuous curves. Since it has some pretty cool uses, just knowing it exists is a powerful enough tool for mathematicians to use it on other cool theorems.

Using computers, we can usually approximate it numerically to any degree of accuracy we desire. The basic algorithm is pretty simple: just make a table of arc-length for each value of t. Then, the unit parametrization is just reading the table in reverse: find t given arc-length. Some interpolation is usually necessary.


79557434791 http://1ucasvb.tumblr.com/post/79557434791/the-sine-and-cosine-functions-for-the-circle-as 1972

The sine and cosine functions for the circle, as every student should see them.

(Edit: the animation is also available, without watermark, at higher resolution and slower frame rate at Wikimedia Commons.)

HAPPY PI DAY! To celebrate, here’s this long-due animation of the usual trigonometric functions, sine and cosine, geometrically defined in terms of the unit circle.

I know this animation is a bit of   the same   as several   others   of my previous   animations  , but this is THE version that I should have done ages ago, if not done first of all.

This is what the sine and cosine functions, the ones you are taught, really are in terms of the unit circle.

First, we have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.

In the middle of this circle, in yellow, is represented the angle theta (θ), that we’re going to plug in our trigonometric functions. This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as you can see. An exact copy of this little angle is shown at the top right, visually helping us define what θ is.

At this angle, and starting at the origin, we trace a (faint) green line outwards. This line intersects the unit circle at a single point, which is the green point you see spinning around at a constant rate as the angle θ changes, also at a constant rate.

Now, we take the vertical position of this point and project it straight (along the faint red line) onto the graph on the left of the circle. This gets us the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:

   y coordinate of green point = sin θ

As the angle θ changes, we can see the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines you see passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) = -1

Now, we do a similar thing with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual way we plot graphs (where y = f(x), with y vertical and x horizontal), we have to “untilt” it in order to repeat the process above in the same orientation. This was represented by that “bend” you see on the top right.

So, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, as you can see due to our “bend”, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:

   x coordinate of green point = cos θ

The blue curve traced by this point, as it moves up and down with changing θ, is the the graph of the cosine function. Notice again how it behaves at it crosses every quadrant, reflecting the fact cos(0) = 1, cos(π/2) = 0, cos(π) = -1 and cos(3π/2) = 0.

And there you go. That’s all there is to it. That’s what sine and cosine are. Simple, huh?

Now, while the concept itself is pretty simple, a lot of people get confused about what the sine and cosine functions actually represent, because visualizations such as this are not presented to them when they are first taught trigonometry.

A lot of teachers, and plenty of school books, fail to mention any of this in detail, as I tried to do here, instead throwing a bunch of formulas in front of students. But the geometric intuition, as presented here, is much simpler to grasp, much more useful in general, and will stick to you for life once you get it. The formulas and important values for sine and cosine don’t need to be memorized anymore, because now you should understand what these values should be, given the underlying logic of things. And that’s what math is all about: making sense of things so they are plainly evident to anyone.

In my most popular post to date (over 360 thousand notes as of now, holy crap!), I saw a lot of people commenting that seeing the top graph, which is the sine function for the circle, made all that trigonometry stuff click.

I was baffled. People were angry that no teacher has ever showed anything like that to them before. That’s crazy! At this age where computers are everywhere, this sort of thing should be in every classroom, and be seen by every student.

So, in order to do justice to the unit circle and these immensely important trigonometric functions, and in order to fill an obvious pedagogical hole in math classrooms and textbooks everywhere, I decided to finally make this animation. No fancy or crazy alternative takes on the sine and cosine this time, just the good ol’ pair of trigonometric functions we all should understand and love.

Happy Pi Day, everyone!

The sine and cosine functions for the circle, as every student should see them.

(Edit: the animation is also available, without watermark, at higher resolution and slower frame rate at Wikimedia Commons.)

HAPPY PI DAY! To celebrate, here’s this long-due animation of the usual trigonometric functions, sine and cosine, geometrically defined in terms of the unit circle.

I know this animation is a bit of   the same   as several   others   of my previous   animations  , but this is THE version that I should have done ages ago, if not done first of all.

This is what the sine and cosine functions, the ones you are taught, really are in terms of the unit circle.

First, we have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.

In the middle of this circle, in yellow, is represented the angle theta (θ), that we’re going to plug in our trigonometric functions. This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as you can see. An exact copy of this little angle is shown at the top right, visually helping us define what θ is.

At this angle, and starting at the origin, we trace a (faint) green line outwards. This line intersects the unit circle at a single point, which is the green point you see spinning around at a constant rate as the angle θ changes, also at a constant rate.

Now, we take the vertical position of this point and project it straight (along the faint red line) onto the graph on the left of the circle. This gets us the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:

   y coordinate of green point = sin θ

As the angle θ changes, we can see the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines you see passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) = -1

Now, we do a similar thing with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual way we plot graphs (where y = f(x), with y vertical and x horizontal), we have to “untilt” it in order to repeat the process above in the same orientation. This was represented by that “bend” you see on the top right.

So, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, as you can see due to our “bend”, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:

   x coordinate of green point = cos θ

The blue curve traced by this point, as it moves up and down with changing θ, is the the graph of the cosine function. Notice again how it behaves at it crosses every quadrant, reflecting the fact cos(0) = 1, cos(π/2) = 0, cos(π) = -1 and cos(3π/2) = 0.

And there you go. That’s all there is to it. That’s what sine and cosine are. Simple, huh?

Now, while the concept itself is pretty simple, a lot of people get confused about what the sine and cosine functions actually represent, because visualizations such as this are not presented to them when they are first taught trigonometry.

A lot of teachers, and plenty of school books, fail to mention any of this in detail, as I tried to do here, instead throwing a bunch of formulas in front of students. But the geometric intuition, as presented here, is much simpler to grasp, much more useful in general, and will stick to you for life once you get it. The formulas and important values for sine and cosine don’t need to be memorized anymore, because now you should understand what these values should be, given the underlying logic of things. And that’s what math is all about: making sense of things so they are plainly evident to anyone.

In my most popular post to date (over 360 thousand notes as of now, holy crap!), I saw a lot of people commenting that seeing the top graph, which is the sine function for the circle, made all that trigonometry stuff click.

I was baffled. People were angry that no teacher has ever showed anything like that to them before. That’s crazy! At this age where computers are everywhere, this sort of thing should be in every classroom, and be seen by every student.

So, in order to do justice to the unit circle and these immensely important trigonometric functions, and in order to fill an obvious pedagogical hole in math classrooms and textbooks everywhere, I decided to finally make this animation. No fancy or crazy alternative takes on the sine and cosine this time, just the good ol’ pair of trigonometric functions we all should understand and love.

Happy Pi Day, everyone!

76812811092 http://1ucasvb.tumblr.com/post/76812811092/given-two-vectors-in-three-dimensions-one-can 608

Given two vectors in three dimensions, one can define their vector or cross product as new vector, perpendicular to both original vectors, and with magnitude proportional to the sine of the angle from the first to the second vector.

In this animation, the cross product of two vectors a (blue) and b (red) are used, and their cross-product (vertical, in purple) is shown varying as the angle between both vectors changes.

As you can see, when both vectors are separated by a right angle (90° = π/2 radians or τ/4 radians), their cross product’s vector reaches a maximum length, and when both vectors are parallel, their cross product is zero. (Here, both vectors a and b are unit vectors, so the magnitude of their cross product doesn’t grow beyond 1 either)

Similar to the way the dot product can be used to find if two vectors are perpendicular, one can use the cross product to find out if two vectors are parallel. You just have to check if these products are zero in each case.

The mathematics

In mathematical notation, we write |v| or ||v|| as the norm or magnitude of a vector v. With that notation, we can say that |a × b| = |a| |b| sin(θ), where θ is the angle from a to b.

The actual value of the magnitude |a × b| has a nice geometric interpretation: it is the area of the paralellogram made by the vectors a and b.

The cross product is said to be “anticommutative”, that is, the order used is important (so it is not commutative, in which case the order wouldn’t matter), and the “anti-” bit says that switching the order switches the sign of the product. Mathematically, a × b = -(b × a)

To be precise, the vector that results from the cross product is said to be a pseudovector, as it is not invariant through reflections: if you see your coordinate axes from a mirror, your right-handed coordinate system would become left-handed, so the definition of the vector breaks down. Vectors are extremely useful because they don’t depend on your system of coordinates, but pseudovectors do, and that’s why they are a caveat worthy of note. This is usually not a problem as long as you stick within the same handed-ness in different system of coordinates, by avoiding such reflections.

There are several ways to actually compute the value of the cross product between two vectors, but the most common one and easier to remember is by finding the determinant of a particular matrix.

Curiously enough, the definition of a vector product returning another vector is unique to 3 and 7 dimensions. More general but similar objects can be defined for other dimensions.

In physics

The cross product is very useful in physics for describing things such as torques (the rotational equivalent of a force), angular momentum (the rotational equivalent of linear momentum), and magnetic forces on charged particles (which act perpendicular to both the velocity of the particle and the magnetic field).

The physical nature of the vector quantity in such cases as torque or angular momentum can be tricky to understand without the proper insight, which is something that is rarely addressed by physics text books.

It is common for students to get stuck to the idea that a vector, as represented by the arrow, points to the direction the force or whatever it is acting or “going towards”. But for angular momentum and torque, this intuition breaks down.

Angular momentum

The proper way to think about the vector for angular momentum is that the vector gives you an axis of rotation. The way the arrow is pointing tells you which of two possible ways the rotation is going (clockwise or counterclockwise, depending on your choice of coordinates and point of view). Using the right-hand rule (for a right-handed coordinate system), you can figure this out easily.

The magnitude of the vector is the actual magnitude of the momentum. So the vector is just a compact way to merge both bits of information on angular momentum in a single mathematical object. The consistency in all of the definitions is what makes it all work nicely, not coincidence.

Torque

The idea of a vector representing the axis and direction of rotation is the same here, but you can also, alternatively, consider it a plane in which the torque is acting on. The vector is normal to this plane.

But another tricky idea here is the dimensions of the torque vector. Remember that in physics, we say length, time and mass, for instance, are dimensions for a physical quantity. We say a meter, a second and a kilogram are units with the dimensions described before, respectively. This difference in terms (units vs. dimensions) is very important, and a lot of people don’t get it right the first time.

So, the dimensions of the torque vector are pretty weird: Newton-meter. A lot of students realize that this is the same thing as a Joule, which is a unit of energy. So why not say torque has the same dimension as energy, call it Joule, and get rid of the Newton-meter thing?

The answer is that while the dimensions match, the concepts don’t. Torque and energy are entirely different concepts, entirely different physical quantities, so they shouldn’t be treated the same even though their dimensions seem to match. But in my opinion, this difference is dogmatic if taken as Newton-meter vs. Joule, because it hides a very important detail.

I think torque makes more sense in units of Joules per radian. The radian is a dimensionless unit, which means it was hidden in there all along in our dimensional analysis. We were not comparing Joules with Joules, but Joules per radians with Joules! The radians bit comes from the fact torques act along an arc.

This is easy to see if we consider the work done by a torque τ: W = τθ, where θ is the angle the torque acted around, rotating an object. In this case, if you consider the dimensions of radians as non-disposable, you can easily see that it all works out.

Wrapping up

The cross product is a very handy tool for defining some more complicated physical quantities. It may seem arbitrary at first, but the reasoning behind its definition is mathematically sound and extremely useful in practice.

In order to fully appreciate it, one must first get rid of a few intuitions on what vectors represent in physics. Vectors can represent a lot of things that are not explicitly directional, as you first start getting used to them, so the sooner you abandon that intuition the better.

Given two vectors in three dimensions, one can define their vector or cross product as new vector, perpendicular to both original vectors, and with magnitude proportional to the sine of the angle from the first to the second vector.

In this animation, the cross product of two vectors a (blue) and b (red) are used, and their cross-product (vertical, in purple) is shown varying as the angle between both vectors changes.

As you can see, when both vectors are separated by a right angle (90° = π/2 radians or τ/4 radians), their cross product’s vector reaches a maximum length, and when both vectors are parallel, their cross product is zero. (Here, both vectors a and b are unit vectors, so the magnitude of their cross product doesn’t grow beyond 1 either)

Similar to the way the dot product can be used to find if two vectors are perpendicular, one can use the cross product to find out if two vectors are parallel. You just have to check if these products are zero in each case.

The mathematics

In mathematical notation, we write |v| or ||v|| as the norm or magnitude of a vector v. With that notation, we can say that |a × b| = |a| |b| sin(θ)

, where θ is the angle from a to b.

The actual value of the magnitude |a × b| has a nice geometric interpretation: it is the area of the paralellogram made by the vectors a and b.

Cross product parallelogram

The cross product is said to be “anticommutative”, that is, the order used is important (so it is not commutative, in which case the order wouldn’t matter), and the “anti-” bit says that switching the order switches the sign of the product. Mathematically, a × b = -(b × a)

To be precise, the vector that results from the cross product is said to be a pseudovector, as it is not invariant through reflections: if you see your coordinate axes from a mirror, your right-handed coordinate system would become left-handed, so the definition of the vector breaks down. Vectors are extremely useful because they don’t depend on your system of coordinates, but pseudovectors do, and that’s why they are a caveat worthy of note. This is usually not a problem as long as you stick within the same handed-ness in different system of coordinates, by avoiding such reflections.

There are several ways to actually compute the value of the cross product between two vectors, but the most common one and easier to remember is by finding the determinant of a particular matrix.

Curiously enough, the definition of a vector product returning another vector is unique to 3 and 7 dimensions. More general but similar objects can be defined for other dimensions.

In physics

The cross product is very useful in physics for describing things such as torques (the rotational equivalent of a force), angular momentum (the rotational equivalent of linear momentum), and magnetic forces on charged particles (which act perpendicular to both the velocity of the particle and the magnetic field).

The physical nature of the vector quantity in such cases as torque or angular momentum can be tricky to understand without the proper insight, which is something that is rarely addressed by physics text books.

It is common for students to get stuck to the idea that a vector, as represented by the arrow, points to the direction the force or whatever it is acting or “going towards”. But for angular momentum and torque, this intuition breaks down.

Angular momentum

The proper way to think about the vector for angular momentum is that the vector gives you an axis of rotation. The way the arrow is pointing tells you which of two possible ways the rotation is going (clockwise or counterclockwise, depending on your choice of coordinates and point of view). Using the right-hand rule (for a right-handed coordinate system), you can figure this out easily.

The magnitude of the vector is the actual magnitude of the momentum. So the vector is just a compact way to merge both bits of information on angular momentum in a single mathematical object. The consistency in all of the definitions is what makes it all work nicely, not coincidence.

Torque

The idea of a vector representing the axis and direction of rotation is the same here, but you can also, alternatively, consider it a plane in which the torque is acting on. The vector is normal to this plane.

But another tricky idea here is the dimensions of the torque vector. Remember that in physics, we say length, time and mass, for instance, are dimensions for a physical quantity. We say a meter, a second and a kilogram are units with the dimensions described before, respectively. This difference in terms (units vs. dimensions) is very important, and a lot of people don’t get it right the first time.

So, the dimensions of the torque vector are pretty weird: Newton-meter. A lot of students realize that this is the same thing as a Joule, which is a unit of energy. So why not say torque has the same dimension as energy, call it Joule, and get rid of the Newton-meter thing?

The answer is that while the dimensions match, the concepts don’t. Torque and energy are entirely different concepts, entirely different physical quantities, so they shouldn’t be treated the same even though their dimensions seem to match. But in my opinion, this difference is dogmatic if taken as Newton-meter vs. Joule, because it hides a very important detail.

I think torque makes more sense in units of Joules per radian. The radian is a dimensionless unit, which means it was hidden in there all along in our dimensional analysis. We were not comparing Joules with Joules, but Joules per radians with Joules! The radians bit comes from the fact torques act along an arc.

This is easy to see if we consider the work done by a torque τ: W = τθ, where θ is the angle the torque acted around, rotating an object. In this case, if you consider the dimensions of radians as non-disposable, you can easily see that it all works out.

Wrapping up

The cross product is a very handy tool for defining some more complicated physical quantities. It may seem arbitrary at first, but the reasoning behind its definition is mathematically sound and extremely useful in practice.

In order to fully appreciate it, one must first get rid of a few intuitions on what vectors represent in physics. Vectors can represent a lot of things that are not explicitly directional, as you first start getting used to them, so the sooner you abandon that intuition the better.

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Anonymous said: Love your work, please keep it up! Hopefully your busy and not done with blogging.

Yes, I’ve been really busy this semester. Rest assured, I haven’t given up on the posts, and there’s a lot more to come!


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My animation on line integrals on scalar fields is today’s (Nov 12th 2013) featured picture on the English Wikipedia’s front page! :D

I’m quite proud to be featured once again, and hope there’s more to come!

My animation on line integrals on scalar fields is today’s (Nov 12th 2013) featured picture on the English Wikipedia’s front page! :D

I’m quite proud to be featured once again, and hope there’s more to come!

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The additive synthesis, through Fourier series, of square, sawtooth and triangle waves

I just updated these three animations. The old ones I had on Wikipedia were among the first I’ve ever done, and were ugly, tiny and outdated.

What you’re seeing here is how a periodic function (in blue) can be approximated by a Fourier series (in red). The number N shows how many terms are being used.

A Fourier series is just a sum (usually infinite) of sines and cosines of different frequencies and amplitudes that approximates a desired function. These frequencies and amplitudes constitute what we call the frequency domain of the function, though this is more useful when we consider a continuous spectrum of frequencies.

The function doesn’t need to be continuous, as you can see in the case of the square and the sawtooth wave.

However, the Fourier series approximation does get a bit wacky around the discontinuities. The wavyness near those points never really goes away, but it usually stays within a certain limit. This is known as Gibbs phenomenon, and it’s a familiar problem in signal processing.

Due to the simplicity of sines and cosines, Fourier series are a great tool when studying the behavior of more complicated periodic functions, a common problem in differential equations with extremely wide applications in physics and engineering.

An important aspect of all of this that gets brushed over in most classes is that you don’t really need both sines and cosines in a Fourier series, since the sum of a sine and a cosine of the same frequency is just a senoidal function with a different phase and amplitude. In other words, you only use sine and cosines in order to encode a phase.

This is why Fourier series are much more elegantly handled using complex numbers, as the complex exponential can handle both phase and amplitude very succintly.

In the future I’m hoping to make a post explaining, in terms most people with a basic understanding of math can understand, why all of this Fourier analysis stuff works in the first place, and why you should think it is awesome. Because it is very awesome.

Previous posts on Fourier analysis

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I’m now open for donations

I’ve been making these educational animations for Wikipedia for years now. It has always been a completely volunteer effort, and I never really got anything from it other than the joy of learning new things about math and physics, and being able to show them to others in a memorable, intuitive and visually pleasing way.

I just wanted people to understand how cool math and physics are, and how simple some complicated looking things can be if you look at them the right way. And to my great pleasure, it has worked really well so far!

Over the years, the reactions have been very positive, especially since I created this tumblr blog. It’s been very exciting to see so many people getting enthusiastic about math and physics because of my work.

But lately, things have been a bit rough, and free time has been scarce. Being a college student takes its toll on anyone, and living off a tiny student grant and my personal savings (one of the reasons why I started college so late) is pretty tough.

So I decided to open up for donations. If you like my work and feel that it is worth something to the world, then consider making a donation. Anything would help me worry less about money and survival, and more about learning stuff and getting me motivated to spend more time on making more cool animations to help others.

You can find the donate button at the top right, on the blog header, or you can follow this link. Any help would be immensely appreciated.

Thank you all for the love and support so far! It means the world to me.

55672229355 http://1ucasvb.tumblr.com/post/55672229355/the-golden-ratio-ah-the-golden-ratio-no-other 2787

The Golden Ratio

Ah, the Golden Ratio. No other number has ever received so much mystical devotion. Old texts on aesthetics even call this number the “Divine Proportion”, such is its reputation in arts.

Also known by the Greek letter φ (phi), this curious irrational number has a closed-form given by:

φ = (√5 + 1)/2 = 1.61803398875…

Since it is one of the solutions of the polinomial equation x² - x - 1 = 0, this number is considered an algebraic number, as opposed to being trasncendental, like π or e are.

From nautilus shells, the human body to spiral galaxies, the Golden Ratio seems to be everywhere in Nature, right?

Well, not really.

A very large portion of what you have probably heard about this number is just hype, widely propagated myths, extremely far-fetched analysis of data or, putting it mildly, just made-up nonsense.

Now, don’t get me wrong here. The Golden Ratio really is a very interesting number with a number of outstanding mathematical properties. This is why it saddens me to see so many people praising it for all the wrong (and wildly innacurate) reasons.

For instance, several spirals in nature are logarithmic spirals because they are the same independent of the scale. This sort of thing is bound to show up whenever you have exponential growth in a circular fashion, two phenomena that are extremely common in nature. In the end, logarithmic spirals are really just exponential functions in polar coordinates.

However, not all logarithmic spirals are Fibonacci spirals. In fact, what it is known as the Fibonacci or Golden spiral, derived from the famous construction using nested squares and golden rectangles (shown below), is a very gimmicky geometric construction that really shouldn’t be expected to show up in nature at all. Nature doesn’t work with squares and rectangles!

Golden spiral

In the study of aesthetics, the Golden Ratio is often praised as being the most beautiful ratio for things, a dogma that gets passed around a lot in design circles. Several studies have shown no correlation between the Golden Ratio and a sense of beauty or aesthetics. (check links at the end of the post for more on this)

I could list most of these myths here, but I would just be repeating what has already been said by many others. So if you want to find out what’s true and what isn’t about the Golden Ratio, I recommend that you watch this talk by Keith Devlin or read this article by Donald Simanek. More links and resources can be found a the end of the post.


With that usual Golden Ratio crap out of the way, I can now finally talk about why this number is REALLY cool.

φ - The most irrational of all numbers

Irrational numbers are numbers that cannot be expressed as the ratio of two integers. Note that the keyword here is integers. This little important detail gets a lot of people confused, usually because of π.

While π is usually defined as the ratio between the circumference of a circle by its diameter, you cannot have both of those quantities being whole numbers, because π happens to be irrational. You can approximate an irrational number with rational approximations, such as 22/7 = 3.142857142857… or 3141592/1000000 = 3.141592, but no matter how large the two numbers of the ratio are, you’ll never find a ratio that is exactly π. The same is true for any other irrational number, φ included.

That animated infinite fraction you see at the top is an example of what we call an infinite continued fraction. Continued fractions are a powerful way to represent irrational numbers because they show you how good a rational approximation is: larger terms in the continued fraction mean you are adding smaller corrections, which tells you the approximation is good. Additionally, all irrational numbers have unique infinite continued fraction representations, a very useful property.

But since we know the larger terms mean “better approximations”, we can think of what would be the worst approximation ever for any number. This would be the infinite continued fraction where the terms are the smallest integer available: 1.

And, it turns out, this infinite continued fraction represents the number φ! This is what the animation is representing.

Think about that for a second. There are an infinite number of irrational numbers, and of all of them, φ is the absolute worst number to approximate using a ratio of two whole numbers. In a sense, φ can be said to be the “most irrational” of all irrational numbers!

This makes me wonder why we even call φ the “Golden Ratio” to begin with, as it is the one number that is as far from being a ratio as it is mathematically possible.

φ and Nature

This “super-irrationality” of φ can be pretty useful, and it is one of the reasons (if not the only one, other than those related to pentagonal symmetries) why approximations of φ show up in Nature, for real this time.

Imagine you have a periodic process, such as leaves growing on a plant stem. If one leaf grows directly on top of another, the leaf below will not be exposed to the Sun due to the shadow cast by the leaf above, so the leaf below will be pretty much useless.

Evolution would favor plants that add an offset between leaves, perhaps by having the stem twist as it grows. This would improve the amount of sunlight each leaf is exposed to, making the plant more efficient and giving it an evolutionary advantage.

However, if the amount of twist between consecutive leaves is a nice ratio of full turns, say 2/3, you would get an overlap between every 3rd leaf. So in this case, you don’t really want nice ratios. You want the leaves to be as spaced as possible, that is, you want the worst ratio you can think of.

As we already know, φ would be that ratio. However, φ cannot really exist out there in the real world, so approximations are as good as we can get.

And guess what? The rational approximations available for φ are the ratios between two consecutive Fibonacci numbers. But you probably knew that already.

This explains why Fibonacci numbers may show up in Nature. Whenever you have a periodic process that would benefit from being “as irregular as possible”, Fibonacci numbers are bound to show up as approximations for φ.

The “real” golden spiral

Let’s say you have a bunch of points that you want to distribute evenly on a disk, as efficiently as possible. This sort of problem shows up in Nature, like in the case of sunflower seeds.

The easiest way to do this, in terms of a set of basic rules, is by placing the points along a spiral, adding layer after layer of points.

But the BEST way to do it uses a very special spiral known as Fermat’s spiral, in which the radius is proportional to the square root of the angle, that is, r(θ) = k√θ, for some constant k.

Since the area of a disk grows with the square of its radius, this spiral has the property of “covering” equal amounts of area for the same amount of rotation.

If you pair this property with the irregular spacing mentioned previously, by picking points along this spiral in multiples of φ (in terms of full turns), you have a very simple rule to achieve the goal of distributing these points along the disk.

You can play around with this idea in the applet below. Apart from the sliders, you can also change the ratio using the left and right keys. Hold shift and/or control to increase the rate of change. You can also type in a fraction like 22/7 in the ratio text box and hit enter.

Go nuts!

To be clear, the x and y coordinates of the n-th point will be: x = cos(2πkn)·r(n) and y = sin(2πkn)·r(n), where r(n) is the radius function (that is, the polar function for the chosen spiral) and k is the ratio being used to place the points around the spiral. Only the fractional part of k matters in this model.

You’ll see that most irrational numbers produce some pretty obvious patterns right away. φ and its reciprocal (in fact, the entire family of numbers sharing that same fractional part) are the only numbers that get everything as evenly spaced as possible, no matter how large the spiral is or how many points you use. In fact, even tiny variations from these ratios already ruin the whole pattern.

Picking different functions for the radius will reveal how Fermat’s spiral is special regarding the radial spacing between dots.

For fun, I also decided to plot lines connecting two consecutive points. You can get some pretty neat images with this, showing the patterns even more clearly. As expected, φ gets you the most messy and irregular of all images, as shown in the second image in this post. For comparison, I also included some other irrational numbers as ratios.

In three dimensions

Now imagine that instead of a disk, you wanted to distribute points uniformly on the surface of a sphere. This problem shows up every now and then, and it cannot be solved so easily. The usual algorithms to solve it involve physical simulations of repelling particles with friction. After a long simulation time, the system will achieve a somewhat decent equilibrium state. This method is particularly troublesome if we’re talking about thousands of points, as we’d have to simulate the interaction between every possible pair of points.

However, we can do better than that. A spiral similar as the one for the disk can be used to distribute points across the surface of a sphere, in a way that makes them relatively uniform.

So thanks to φ and its irrational properties, we can tackle a hard problem in a relatively straightforward and direct way. Pretty clever stuff!

Find out more of the truth about φ

Well, there’s a lot of other cool stuff I could say about φ, but this post is already pretty long as it is and the links below are full of more stuff.

  1. "Math Encounters — Fibonacci & the Golden Ratio Exposed" by Keith Devlin. Accessible to all ages. (I liked it, though from the comments, a lot of people seem to hate this lecture.)
  2. "Fibonacci Flim Flam" by Donald E. Simanek, which criticizes most of the myths about the golden ratio
  3. "The golden ratio and aesthetics" by Mario Livio, a skeptic look at the claims about phi in arts
  4. "The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number" by Mario Livio, if you are looking for all the real, no-nonsense mathematical coolness behind φ
  5. "Doodling in Math: Spirals, Fibonacci, and Being a Plant", ViHart’s series on Fibonacci numbers in nature.

52482862675 http://1ucasvb.tumblr.com/post/52482862675/villarceau-circles-on-a-torus-next-time-you-have 615

Villarceau circles on a torus

Next time you have a donut or bagel at hand, give this a try. Slice it diagonally in a way that the cutting plane is tangent to the donut both below and above its inner hole. If you do it just right, what you get is two perfectly symmetric pieces whose boundaries are the union of two perfect circles (for a sufficiently round donut).

Mathematically speaking, if you have a torus (what mathematicians call that donut shape) and cut it diagonally at the correct angle, you will reveal a pair of perfect circles on the surface of the torus, known as Villarceau circles.

For every point on the surface of a torus, you can trace exactly 4 distinct perfect circles, on the surface of the torus, that pass through that point: one is around the hole of the torus, and the other around its circumference. The other two are Villarceau circles, but not really the pair shown in the animation.

Villarceau circles play an important role in Hopf fibrations. Roughly speaking, you can fill the entire 3D space with an infinite number of such circles. Apparently, this sort of thing even shows up in quantum physics, but I can’t offer any information on that.

I was first introduced to the concept of Villarceau circles through this great POV-Ray render by Tor Olav Kristensen. 

This was one of the first animations I did for Wikipedia, and I’m still rather proud of it.

On a side note, I’ve been very busy with college lately, that’s why I haven’t been posting much. But I’m working on a series of animations on vector calculus and electromagnetism, and I think they’ll turn out great. Stay tuned!

Villarceau circles on a torus

Next time you have a donut or bagel at hand, give this a try. Slice it diagonally in a way that the cutting plane is tangent to the donut both below and above its inner hole. If you do it just right, what you get is two perfectly symmetric pieces whose boundaries are the union of two perfect circles (for a sufficiently round donut).

Mathematically speaking, if you have a torus (what mathematicians call that donut shape) and cut it diagonally at the correct angle, you will reveal a pair of perfect circles on the surface of the torus, known as Villarceau circles.

For every point on the surface of a torus, you can trace exactly 4 distinct perfect circles, on the surface of the torus, that pass through that point: one is around the hole of the torus, and the other around its circumference. The other two are Villarceau circles, but not really the pair shown in the animation.

Villarceau circles play an important role in Hopf fibrations. Roughly speaking, you can fill the entire 3D space with an infinite number of such circles. Apparently, this sort of thing even shows up in quantum physics, but I can’t offer any information on that.

I was first introduced to the concept of Villarceau circles through this great POV-Ray render by Tor Olav Kristensen.

This was one of the first animations I did for Wikipedia, and I’m still rather proud of it.

On a side note, I’ve been very busy with college lately, that’s why I haven’t been posting much. But I’m working on a series of animations on vector calculus and electromagnetism, and I think they’ll turn out great. Stay tuned!

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saxpride100:

Hey, we’re doing line integrals over vector field right now, but it’s kind of the same idea. Congrats on the image of the day!

Thanks! By the way, I also made a vector field version:

saxpride100:

Hey, we’re doing line integrals over vector field right now, but it’s kind of the same idea. Congrats on the image of the day!

Thanks! By the way, I also made a vector field version:

(via symphonies-of-a-storm)

47736801235 http://1ucasvb.tumblr.com/post/47736801235/line-integral-of-a-scalar-field-this-animation 245

Line integral of a scalar field

This animation of mine is today’s Picture of the Day on Wikimedia Commons and several Wikipedia languages.

A scalar field has a value associated to each point in space. Examples of scalar fields are height, temperature or pressure maps. In a two-dimensional field, the value at each point can be thought of as a height of a surface embedded in three dimensions. The line integral of a curve along this scalar field is equivalent to the area under a curve traced over the surface defined by the field.

In this animation, all these processes are represented step-by-step, directly linking the concept of the line integral over a scalar field to the representation of integrals familiar to students, as the area under a simpler curve. A breakdown of the steps:
The color-coded scalar field f and a curve C are shown. The curve C starts at a and ends at b
The field is rotated in 3D to illustrate how the scalar field describes a surface. The curve C, in blue, is now shown along this surface. This shows how at each point in the curve, a scalar value (the height) can be associated.
The curve is projected onto the plane XY (in gray), giving us the red curve, which is exactly the curve C as seen from above in the beginning. This is red curve is the curve in which the line integral is performed. The distances from the projected curve (red) to the curve along the surface (blue) describes a “curtain” surface (in blue).
The graph is rotated to face the curve from a better angle
The projected curve is rectified (made straight), and the same transformation follows on the blue curve, along the surface. This shows how the line integral is applied to the arc length of the given curve
The graph is rotated so we view the blue surface defined by both curves face on
This final view illustrates the line integral as the familiar integral of a function, whose value is the “signed area” between the X axis (the red curve, now a straight line) and the blue curve (which gives the value of the scalar field at each point). Thus, we conclude that the two integrals are the same, illustrating the concept of a line integral on a scalar field in an intuitive way.

Line integral of a scalar field

This animation of mine is today’s Picture of the Day on Wikimedia Commons and several Wikipedia languages.

A scalar field has a value associated to each point in space. Examples of scalar fields are height, temperature or pressure maps. In a two-dimensional field, the value at each point can be thought of as a height of a surface embedded in three dimensions. The line integral of a curve along this scalar field is equivalent to the area under a curve traced over the surface defined by the field.

In this animation, all these processes are represented step-by-step, directly linking the concept of the line integral over a scalar field to the representation of integrals familiar to students, as the area under a simpler curve. A breakdown of the steps:

  1. The color-coded scalar field f and a curve C are shown. The curve C starts at a and ends at b
  2. The field is rotated in 3D to illustrate how the scalar field describes a surface. The curve C, in blue, is now shown along this surface. This shows how at each point in the curve, a scalar value (the height) can be associated.
  3. The curve is projected onto the plane XY (in gray), giving us the red curve, which is exactly the curve C as seen from above in the beginning. This is red curve is the curve in which the line integral is performed. The distances from the projected curve (red) to the curve along the surface (blue) describes a “curtain” surface (in blue).
  4. The graph is rotated to face the curve from a better angle
  5. The projected curve is rectified (made straight), and the same transformation follows on the blue curve, along the surface. This shows how the line integral is applied to the arc length of the given curve
  6. The graph is rotated so we view the blue surface defined by both curves face on
  7. This final view illustrates the line integral as the familiar integral of a function, whose value is the “signed area” between the X axis (the red curve, now a straight line) and the blue curve (which gives the value of the scalar field at each point). Thus, we conclude that the two integrals are the same, illustrating the concept of a line integral on a scalar field in an intuitive way.