Pi Day 2018

This is the third installment in my series of articles on π.  The earlier articles can be found here and here.  Due to lack of time, I am going to make this short.

In my previous articles, I built a case for switching to using diameter in most places we currently use radius.  The math and science communities have noticed a disconnect in how we use π, and many are now advocating for replacing the very commonly used 2π with τ.  In studying this problem, I have traced the cause to the common use of π in equations that use radius.  The problem with this is that π is a function of diameter, thus using it with radius (d/2) requires an additional factor of 2 to correct for that.  Instead of switching to τ, we should use the natural measure of a circle, which is diameter, where we are using radius.  In the second article linked to above, you can find the reduced equations for basic circle and sphere math, using d instead of r, and it is very clear that d is superior in those equations.

Unfortunately, doing the same for trig functions is far more challenging.  This is because trig defines circles as fans of infinitesimally thin triangles, where two vertices reside on the edge of the circle and one resides at the center.  Treating circles this way requires the use of the radius as the fundamental measure, because the radius is the unit of measure for the triangles.  This means that attempting to use diameter in trig instead of radius is likely to make the math far more complicated.  Of course, we could just switch to diameter everywhere that it works and stay with radius for trig and related fields.

There may be a better solution, though.  I have no faith that this solution will ever be adopted, but it is at least worth consideration, as it may help us to advance and improve trig and geometry.  Instead of using triangles, we could use rectangles.  Triangles have this fascinating property, where a triangle is half the area of a rectangle with the same height and width dimensions.  What is fascinating about this is that if we extend two of the lines out so they have the same length on either side of their common vertex, we get what is a sort of split rectangle.  The shape looks like a bow tie, and if we measure one half of it as if it were a rectangle, we can multiply the height and width of that half to get the area of the entire thing.  Alternatively, we can measure it like a triangle, multiplying the height and width of the whole thing, then divide by two.  This will also get us the area.  And in fact, this is what we would get, if we used diameter instead of radius.

It gets better though.  If we use diameter to treat circles as these double-triangles, we divide the distance traversed to get the full area by two, and now we have the diameter based unit circle I drew up for my second article linked above.

Of course, the problem still exists that we would ultimately have to redefine all of the trig functions based on double the hypotenuse, and the math and science communities are not going to buy into a change like that.  Doing the algebraic conversions might reveal useful patterns though, just as it did with simple circle geometry.

Perhaps next year, I will write an article with some results on the conversions of trig functions to using diameter.  That is, assuming I have time work any of that out.

Pi Day 2016

I recently wrote an article on the subject of whether we should use π (pi) or τ (tau) in circle related math.  As a refresher, π is the ratio of the circumference of a circle to its diameter.  It turns out that in a majority of equations that use π, it is multiplied by 2.  A growing group of mathematicians and scientists is pushing to replace π with τ, which is just 2π.  They argue that this would simplify a lot of math and make geometry and trig significantly easier to learn.  This is probably true, but it addresses the problem from the wrong side.  I argue that π is not the problem.  The problem is that we have this unreasonable attachment to radius.  The reason for having to apply a multiple of 2 everywhere is not that π is the wrong value.  The reason is that π is a ratio of the diameter of a circle, but we always pair π with the radius, which requires a multiple of 2 to correct this.

There are a lot of different arguments for why we should use τ instead of π, but in the end, they are based on the idea that the radius is the natural way to measure a circle.  This comes from the idea that a circle is composed of an infinite number of infinitely small triangles, but while that is a valid representation, it is not actually true.  A circle is composed of a continuous, regular curve, with no straight lines anywhere.  Just as we would not try to define any other shape as the distance from the center to any point along its perimeter, radius is not the natural way to measure a circle.  Outside of mathematics and science, where it has become tradition to use radius, measuring merely half the width of an object is not something that is often useful.

The answer is not to switch from π to τ.  The answer is to use the natural measure of the size of a circle: diameter.  Some "tauists" claim that the decision of ancient civilizations to use diameter instead of radius was arbitrary.  If that is true though, then why did every civilization that set out to measure the ratio of the circumference of a circle to its size choose to use diameter instead of radius?  The answer is not chance.  The answer is that the choice was not arbitrary.  Diameter is the natural way to measure a circle.  To someone that has not been taught to measure a circle by radius, diameter is the obvious measure of size.

The fact is, replacing radius with diameter in circle equations simplifies them just as much as replacing π with τ, with the added benefit that new students will not be confused as to why we suddenly only care about half of the size of the circle.  Besides that, how does the average person measure the radius of a circle?  They measure the diameter and divide by two, because half a width is not a natural measurement to try to take!

In celebration of Pi Day, I bring you the following:

This is the real unit circle.  You will notice that one full turn is not equal to 2π.  This is because we are measuring in diameters instead of radians.  There is no need to redefine the circle constant when using diameter, because diameter is what it was made from!  Unfortunately, "diameterians" does not sound as good as "radians," so I propose just calling them "diameters."  That works just fine, since the circumference of a circle is, by definition, π diameters.  Besides that, I think "diameters" would be far less confusing to new students, as it does not sound like some kind of new unit like "radians" does.  (Be honest, how many of you struggled with radians, because it was not initially clear that radians are literally just the distance around the circle measured in radii?  Now, think about π diameters.  Without the fancy sounding name, it is much more clear what it means.)

Now, with my nice new diameter based unit circle in mind, here are some basic circle and sphere equations using diameter instead of radius:

Circle

Definition: d² = 4(x² + y²)
Circumference: πd
Area: ¹⁄₄πd²

Sphere

Definition: d² = 4(x² + y² + z²)
Surface Area: πd²
Volume: ¹⁄₆πd³

I considered adding some trig to this, but there is so much, and most of it is already so complex, that it would have taken a lot of algebra to reduce things down.  In short, I gave up.  This is a good taste though.

Here are some things I notice with the above.  First, in the circle equations, you can replace π with diameters (see the unit circle above) to find the values for partial circles.  For example, ^π⁄₂d will give you the circumference of half a circle (^π⁄₂ diameters is halfway around the circle).  You can do something similar using τ with radius, but I just wanted to point out τ does not have an advantage here.  Now take a look at the circle circumference equation and the sphere surface area equation.  Notice the logical step from the first to the second?  Merely squaring the diameter promotes the equation to the analogous equation of the equivalent shape one dimension higher.  Area to volume is not so pretty, but sphere volume is simpler than its radius based version, and the additional factor of 4 on the circle area equation adds trivial complexity. (I made an algebra mistake in circle area.  It is fixed now.)  I also notice a similar progression with circle area to sphere volume, where the multiplier is 1/(2 * dimensions), and like circumference to surface area, the exponent is the number of dimensions of what we are measuring.  The sphere volume is simpler with diameter, and the circle area is only trivially more complex.  In fact, the only places that complexity is increased noticeably are the two standard form equations that are typically used as the definitions for circles and spheres, and outside of education, more general forms are typically used for these, which are so much more complex that the extra factor of 4 would not make any difference.

The big advantage τ has is that replacing a bunch of "2π"s in textbooks is much easier than doing the algebra required to simplify the equations when you swap r for ^d⁄₂.  If you aren't going to do it right though, what is the point of doing it at all?  It is true that teaching τ with radius will be easier than what we are doing now, but many students will still start off confused that we are measuring only half of the circle (and isn't one of the big "tauist" arguments that it is absurd that 1 π only gets us halfway around the traditional unit circle?).  Sticking with π, but using diameters instead of radians, means that we don't have to overthrow a constant that has been ingrained in mathematics over the course of many millenia.  We don't have to deal with teaching two constants just so students will be able to understand even recently written papers, not to mention all of the classical mathematics treatises.  It is a lot easier to teach students that r = d/2 than it is to get them to memorize two circle constants out to n digits just so they can work with math from different eras.  We also don't have to try to explain to students why we only care about half the width of a circle.  The only advantage τ has is convenience in fixing textbooks.  Using π with diameters just makes sense, even to those without a heavy mathematical background.

I hope you like my Pi Day celebration!  Give a few minutes to celebrate diameter today as well.

Tau vs Pi

This is part of a series republished from another blog.  I wrote this in February 2016.  The others were originally published on Pi Day of 2016 and 2018, and can be found here and here.

Pi (π) is an amazing number that defines the ratio of the circumference of a circle to its diameter.  It has important historical significance, because many civilizations knew that it existed, but until recently, none had been able to discover what it was to any degree of precision.  In the modern world, we use π all over the place.  It turns out though, that there are some places where π just does not make sense.  For example, the unit circle, used in trig and other more advanced mathematics, ends up with a lot of complicated and unintuitive fractions, and a single revolution around the circle is equal to 2π in distance.  This is confusing to new students and makes a lot of the math more complicated.  In fact, it turns out that 2π is used all over the place, and it may even be used significantly more often than π by itself.

A group of people have started advocating the use of an alternative to π.  Tau (τ) is equal to 2π.  It could be defined as the ratio of the circumference of a circle to its radius.  A unit circle using τ makes more sense, because a single revolution is equal to τ.  A quarter revolution is τ/4 (using π, it is π/2), a half a revolution is τ/2 (π), and 3/4s of a revolution is 3τ/4 (3π/2).  With τ, the fraction of the circle is the fraction of τ, but with π, it is twice the fraction of the circle, which makes it more difficult to understand and complicates the math.  So, this group supporting τ says we should switch from using π to using τ, because it simplifies the math.

It turns out that this is only part of the story though, and I am writing this, because I have not seen any evidence that anyone else fully understands the issue.  The problem is not that τ always makes more sense than π.  The problem is that we are using π wrong.  To fully understand this, we need to define both π and τ, without reference to the other.  In most debates on the subject, τ is defined as 2π, which is technically true but also misleading.   τ is not merely 2π.  Both τ and π are ratios relating the width of a circle to its circumference.  Mathematically, τ = c/r, while π = c/d, where c = circumference, r = radius, and d = diameter.  If you look closely, you might see why we end up using 2π all over the place.  Look at it this way: π = c/2r -> 2π = c/r.  It should be obvious by now.  π is a ratio of the circumference to the diameter, but we are doing all of our math using the radius.  Of course we have to multiply π by 2 all the time, because we are implicitly dividing the diameter by 2 nearly everywhere we use it!

The most obvious solution is to replace 2π with τ.  This is certainly a valid solution, but it is not the only solution, and it is not necessarily the best or most sensible solution either.  The other solution is to keep using π, but use d instead of r.  This would even fix the unit circle, as radians are defined as the distance from 0 multiplied by r.  π radians is not actually π.  It is the distance πr, but the r is not written, because it is implied.  If we replaced r with d, a full revolution would be exactly π (the d is implied this time).  The only problem is that we could no longer call them radians, because the name comes from "radius," and we would be using the diameter.

So, why do we use r instead of d?  This question stumped me when I took geometry in highschool.  It did not make much sense.  The reason is simple: When we find the area or perimeter of a square or rectangle, we use width (w) and height (h).  We don't use w/2 or h/2.  So why, with circles, are we always using d/2?  It just does not make much sense.  Even when we are calculating the area of a triangle, we use (w * h)/2, not w/2 * h or w * h/2.  It does not make sense to use half the width of the circle when finding perimeter (circumference) or area, when don't use half lengths anywhere else.  I recently realized why we use r instead of d, and the answer is rather disappointing.  The definition of a circle is the following equation: r^2 = x^2 + y^2.  This is the only place I can find where it makes significantly more sense to use half width over whole width.  So why is it that we are using this one equation to define the normal case instead of using all of the others and defining this one as a special case?  Honestly, I can see no reason for doing it this way, except perhaps that this is how it was done in the past.  I don't happen to subscribe to the theory that tradition trumps logic and reason.  If tradition does not make sense, it is time to trade it for something that does.

It turns out that this entire argument is almost pointless though.  Using 2π all over the place works, and it is pretty entrenched in our mathematics.  Outside of education, math and science are not going to suddenly change because someone decides it is better to do things a bit differently.  Within education is where the "almost" comes in though.  I am not the only person who noticed that using r instead of d does not make sense in the context of all of the rest of geometry.  I am certainly not the only person who noticed that the unit circle does not make sense using π as we do.  We could just replace 2π with τ, but that would only fix the unit circle and part of the math.  It does not fix the underlying problem with using only half of the distance for circles, while we use the entire distance for everything else.  If we really want to make education easier, we should keep using π, but switch to using d instead of r.