# Relativistic pedantry

I must say first off, that I teach math and computer science, and was never qualified to teach physics. But I am interested in physics, and got drawn into in a physics discussion about how time does not stretch or compress in the visible world, and this is why in most of science, time is always the independent variable, stuck for most practical purposes on the x axis.

In the macroscopic world, time and mass are pretty reliable and so close to Einstein’s formulas (or those associated with the Special and General Theories of Relativity) at the macroscopic level that we prefer to stick to simpler formulas from classical mechanics, since they are great approximations, so long as things move well below the speed of light.

I am not sure (is anyone?) about how time is influenced by things like gravity and velocity (in particluar, the formulas stating how time is a dependent varable with respect to these things), but I remember an equation for relative mass, which doesn’t use time that would provide some insight into relativity:

$\displaystyle{m(v) = lim_{v \to c^-} \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} = \infty}$

Here, the independent variable is velocity, and it is evident that even for bodies that appear to move fast (on the scale of 10 to 20,000 km/h), it doesn’t have much impact on this equation. Rest mass and relative mass are essentially the same, and a body would have to move at nearly the speed of light for the mass of the moving body to change significantly. Indeed, as velocity v gets closer to the speed of light c, mass shoots up to infinity. I understand that Einstein stated that nothing can move faster than light, and this is supported by the above equation, since that would make it negative under the radical.

It does not escape my notice that velocity is supposed to depend on time, making the function $m(v(t))$, but time warps under things like high velocity also (as well as high gravity), so that time depends on … ? This is where I tell people to “go ask your physics prof” about anything more involved.

Sattelites move within the range of 10,000 to 20,000 km/h, hundreds of kilometres above the Earth’s surface. My assertion that there is not much change here in relativity terms. But this is still is large enough to keep makers of cell phones up at night, since not considering Einstein equations in time calcluations can cause GPS systems to register errors in a person’s position on the globe on the order of several kilometres, rendering the GPS functions on cell phones essentially useless.

My companion was trying to make the latter point, where I was thinking much more generally. We stick to classical mechanics, not because the equations are necessarily the correct ones, but instead because they are simple and lend a great deal of predictive power to the macroscopic world around us.

# Statistical factoids which are actually true

I have often discussed with my students about the hilarity that ensues when one conflates correlation with causation. The lesson, of course, is that they should never be confused, otherwise you can conclude things like “World hunger is caused by a lack of television sets.” Get it? Our first world has lots of TV sets and not much hunger. The third world lacks TV sets and has a great deal of world hunger. So, the solution is that we can ship our TV sets to the third world, and world hunger can be eradicated. While I don’t have the numbers, I would assume the correlation exists. Whether causation exists is a whole other matter, and is much more difficult to prove.

So, to explain the title of this blog, I mean “actually true” in the sense that the correlations are for real; but not the causation necessarily.

I am proposing here to make a few (not too many) posts in honor of a website called Spurious Correlations.

Currently, its front page reassures us that while it is true that government spending on science and technology correlates positively, and strongly, with death by suicide, we ought to fall short on curbing spending on science research, as some of that science might be about how to reduce the incidents of suicide.

Further down that page, I am not sure what to make of the high correlation shown between per capita cheese consumption, and the number of people who died after being tangled in their bedsheets. Or of the relationship between the number of people who fell and drowned out of their fishing boat, and the marriage rate in Kentucky.

But one of the perks of the website is in its ability to conjure up statsistics based on user choices. Here was the result of some playing around I did on their website:

They appear to prefer to show all of their graphs in time series, which still shows the data more or less rising and falling together, but linear correlations are nicer. They offered the data that was used to plot this graph, and I was able use that data to make my own scatterplot relating the data to each other rather than against time, showing the data has the same r value:

Now I can feel confident in saying that if there are, say, a total of 84,000 deaths due to cancer on the 52 Thursdays of any given year, there will be 15,650 Lawyers practising in Tennessee that year also. I have also worked out that if you got rid of all of the Tennessee lawyers, we would save the lives of just over 20,000 cancer patients per year. Isn’t statistics great?

When you take the square root of the coefficient of determination, you get 0.971299, which agrees with the r value offered on their website. The data, according to the website, originates from the Centre for Disease Control and the American Bar Association.

# A short problem

This math problem had me going for a bit. Looked at from a distance, it looked like one thing; and when I had the occasion to sit down and hash it out, it was quite another.

A student submitted a project that was of her game which was played with just two dice. If you roll a 2 or 12 you win; but if you roll any sum from 5 to 9 you lose; and if you roll a 3, a 4, a 10, or an 11, you survive to another round. You are limited to a maximum of three turns to roll a winning total.

It was not the normal Bernoulli trial, since this doesn’t just have the two states of success and failure; but it introduces a third state, which we will call “survival”. While you don’t “win” if you survive, you can still play again, but you can’t go past 3 turns. Three survivals in a row gets you nothing. You need to get a 2 or a 12 in three turns or less.

$P(2 or 12) = \frac{1}{18}$, the probability of winning on your first try. Even though there are three states, we can still discuss wait time. In this context, it can be 0, 1, or 2. With a wait time of 1, $P(2 or 12) = P(3, 4, 10, or 11)\times (\frac{1}{18})=(\frac{5}{18})\times(\frac{1}{18}) \approx 0.015432$.

It means that to even make it to the second turn you can only get there with a 3, 4, 10, or 11. If you got 2 or 12 on the first try, there would be no need for a second turn. Similarly, to get to the third turn, you needed to survive twice and win the third time: $(\frac{5}{18})^2\times(\frac{1}{18}) \approx 0.00428669$.

It made sense, but I still wasn’t sure about this. What about the probability of losing completely? Those are the numbers from 5 to 9, which has a probability of $\frac{2}{3}$. Why wasn’t I making use of that information?

I don’t think it was necessary in computing the probabilities I did, since the winning conditions preclude rolling any sum between 5 and 9. But it can come in handy as a check. A good indicator that I am on the right track is to see if expectations of winning and losing for 1000 trials, add up to 1000. For winning, I need to add up the probabilities for all 3 wait times: $\frac{1}{18} + \frac{5}{324}+\frac{25}{5832} = \frac{439}{5832}\approx 0.0752743$. For 1000 games, the wait time of winning is: 75.2743 games.

The expectation of losing is similarly calculated, based on a single-turn probability of 2/3: $\frac{2}{3} + \frac{2}{3}\times \frac{5}{18} + \frac{2}{3}\times \left(\frac{5}{18}\right)^2 = \frac{439}{486}\approx 0.9033$. This means you will lose, on average 903 out of every 1000 games, or an expectation of 903.3. Notice that for 1000 games, the winning and losing conditions don’t add to 1000. What we didn’t add was survival until the third turn: $\left(\frac{5}{18}\right)^3 = \frac{125}{5832}\approx 0.02143$. This means that you will “survive” (but not win)21.43 out of every 1000 games. With enough decimal precision, we do indeed get 1000 games when we add all these numbers up, or at the very least add the expectation using the fractions: $1000 \times \left(\frac{439}{5832} + \frac{439}{486} + \frac{125}{5832}\right) = 1000 \times \left(\frac{439}{5832} + \frac{5268}{5832} + \frac{125}{5832}\right) = 1000$.

# Technology in Mathematics: Chalk

First off, May the Fourth be with you! (Had to get that in there…)

Chalk still stands out as the technology of choice of mathematicians everywhere. Those of us forced to use whiteboards realize the wisdom of the ancients once we have to do without chalkboards and chalk.

Whiteboard markers are much more expensive than chalk; it is hard to know when markers run out of ink (whereas there is no question when you are running out of chalk); and after many uses, they leave a residue on the whiteboard which is hard to clean off. Also, whiteboard markeers have many components which make it difficult to recycle, and produces more waste when they go dry. The markers easily go dry, and one has to be always in the habit of ensuring the cap is on tightly when not in use.

Chalk, on the other hand, just runs out. It can produce dust. And though I have had a dust (house dust) allergy since I was a child, I have taught at a blackboard for 15 or more years, and have not experienced breathing problems with chalk. Chalk stands out on a dark background. A chalkdust-clogged blackboard can simply be cleaned off with a wet sponge and a squeegee.

This is in contrast to using  the alcohol-based cleaner for whiteboards. While the sponge can be rinsed after use, a cloth or paper towel soon becomes clogged with the ink wiped from the whiteboards from using the cleaner, and are soon disposed of. This just contributes to a landfill problem.

For educators concerned with the vertical classroom, students can work on these non-permanent black vertical surfaces just like they do with whiteboards.

# The infinite sum which equals -1/12

On 16 January, 1913, Srinivasa Ramanujan, a clerk living in Madras, India, sent a letter to Oxford professor Godfrey Harold Hardy, which contained many of his intuitive mathematical musings, one of which led to the conclusion:

*** QuickLaTeX cannot compile formula:
\displaymath \sum_{n=1}^{\infty} n = 1 + 2 + 3 + 4 + 5 + ... = \frac{-1}{12}

*** Error message:
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In normal mathematics, this bizarre result would be just that — a bizarre result. In high school math, as well as in most university math courses, a response of

*** QuickLaTeX cannot compile formula:
\frac{-1}{12}

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would get a mark of zero. On the face of it, the series is clearly divergent, and there is no reason to doubt that the actual answer should be

*** QuickLaTeX cannot compile formula:
\infty

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.

What is also clear is that Ramanujan and Hardy were not dummies, having helped to revolutionize early 20th century mathematics in England, as well as making people begin to take notice that India has, for a long time, made significant advancements in mathematics. It is also important to note that this bizarre result is used today in string theory, in the context of the Riemann Zeta function.

*** QuickLaTeX cannot compile formula:
\displaystyle \zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z} = \frac{1}{1^z}+\frac{1}{2^z}+\frac{1}{3^z}+...

*** Error message:
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The difference is that the Zeta function works with complex numbers. The zeta function does not replace ordinary summation. Also, a finite sum still exists in the real numbers if their partial sums are shown to converge.

The zeta function comes from a well-known summation which goes something like:

*** QuickLaTeX cannot compile formula:
\displaystyle \frac{1}{1-z} = 1 + z + z^2 + z^3 + z^4 + ... ; -1 < z < 1

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The restriction on z is important if the series is to converge. When z wanders outside those boundaries, the series diverges, and the sytem breaks down, and we can’t find the sum. The zeta function, however, attempts to generalize this equation for any value of z.

The value of these sums have meaning in a particular mathematical context. It is as if you encountered the imaginary number

*** QuickLaTeX cannot compile formula:
i = \sqrt{-1}

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, and said that the result is meaningless, since square roots are supposed to only return positive numbers. But in math involving the complex plane, it is quite meaningful, and enjoys actual use in the design of integrated circuits, for instance.

To see how Ramanujan encountered -1/12 as the sum of 1 + 2 + 3 + …, you can do normal summation, pretending that the series we are about to encounter always converge (and none of the following do):

*** QuickLaTeX cannot compile formula:
\displaystyle \frac{1}{1 + 1} = 1 - 1 + 1 - 1 + 1 - 1 + ... = \frac{1}{2}

*** Error message:
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As you calculate partial sums, you get 1, 0, 1, 0, 1, …, and this never settles on any single value. This series is divergent. The 1/2 “sum” can be seen as an average of 0 and 1, and this actually is called a Cesaro sum.

Now suppose we squared this:

*** QuickLaTeX cannot compile formula:
\displaystyle \left(\frac{1}{1 + 1}\right)^2 = (1 - 1 + 1 - 1 + 1 - 1 + ...)^2 = 1 - 2 + 3 - 4 + ... = \frac{1}{4}

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The sum 1 – 2 + 3 – 4 … is the product of  the square of

*** QuickLaTeX cannot compile formula:
(1 - 1 + 1 - 1 + 1 ...)^2

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, which can be seen by using a grid to help you square them:

 × 1 -1 1 -1 1 … 1 1 -1 1 -1 1 … -1 -1 1 -1 1 1 … 1 1 -1 1 -1 1 … -1 -1 1 -1 1 1 … 1 1 -1 1 -1 1 … … … … … … … …

Then, you may add numbers from that table belonging to the same diagonal. Above, they happen to be the same colour. The 1’s and -1’s in gray are incomplete diagonals due to the limitations of a finite table.

You get: 1 – 2 + 3 – 4 + 5 – … =

*** QuickLaTeX cannot compile formula:
\displaystyle \left(\frac{1}{2}\right)^2 = \frac{1}{4}

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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, which we will refer to as

*** QuickLaTeX cannot compile formula:
S_2

*** Error message:
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.

Now, let’s get back to the original “sum”:

*** QuickLaTeX cannot compile formula:
S_1 = 1 + 2 + 3 + 4 + 5 + ... = -1/12

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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. We can re-generate

*** QuickLaTeX cannot compile formula:
S_2

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using only

*** QuickLaTeX cannot compile formula:
S_1

*** Error message:
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as follows:

*** QuickLaTeX cannot compile formula:
\begin{aligned}
S_1   & = & 1 + & 2 + & 3 + & 4 + & 5 + ... \\
4S_1 & = &    &    4  +&        & 8  + &  ... \\
S_1 - 4S_1 & = & 1 - & 2 + & 3 -  &  4 + & 5 - ...
\end{aligned}

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



… and finally, using our finding that S2 = 1/4, we arrive at the original incredible sum:

*** QuickLaTeX cannot compile formula:
\begin{aligned}
-3S_1 & = & \frac{1}{4} \\
S_1 & = & \frac{-1}{12}
\end{aligned}

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



The sum is verified by the Riemann Zeta function for

*** QuickLaTeX cannot compile formula:
\zeta(-1)

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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.

The feeling I get when I see a result like this is like when reading the Douglas Adams book Hitchhiker’s Guide To The Galaxy, and then being told by their whizzbang computer that the answer to life, the universe and everything was 42. You feel no more enlightened than before you asked the question. But then that is because the answer is not understood. And for that matter, even the question may need some work.

Mathematicians such as Euler and Ramanujan had endeavoured to unlock the mysteries of these strange results.

We know that

*** QuickLaTeX cannot compile formula:
\zeta(z)

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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works best for z > 1. For all z < 1 (especially negative numbers),

*** QuickLaTeX cannot compile formula:
\zeta(z)

*** Error message:
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does not return a direct value, but instead the mathematician needs to base a conjecture of what the value could be, based on analytical continuation. It turns out, this is far from guesswork since, for any negative z in the set of complex numbers, there is always exactly one answer that analytic continuation ever offers.

*** QuickLaTeX cannot compile formula:
\zeta(-1) = -1/12

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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is the result when

*** QuickLaTeX cannot compile formula:
z = -1

*** Error message:
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.

The zeta function also equals zero in strange places, such as when z is a negative even integer. The reader is reminded that the set of complex numbers is utterly vast, subsuming the entire set of real numbers as a subset of the complex numbers. This is because a complex number C is defined as

*** QuickLaTeX cannot compile formula:
C = a + bi

*** Error message:
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, where a and b belong to the set of real numbers, and i, once again, is

*** QuickLaTeX cannot compile formula:
\sqrt{-1}

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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. If you allow

*** QuickLaTeX cannot compile formula:
b = 0

*** Error message:
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, then C is a real number.

According to Riemann’s Hypothesis, the other set of zeroes for the zeta function lie entirely in the set of complex numbers where

*** QuickLaTeX cannot compile formula:
C = \frac{1}{2} + bi

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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, for some value of b. In other words, all other zeroes for the Riemann zeta function are of the form

*** QuickLaTeX cannot compile formula:
\zeta\left(\frac{1}{2} + bi\right)

*** Error message:
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for some real number b.

Now, this is a hypothesis, which hasn’t been proved wrong yet, but it also hasn’t been proved to be correct generally. The Clay Mathematics Institute has made this one of their millenium prize problems, offering one million dollars to anyone who can solve it. If this is proven, other connected conjectures are proven along the way, such as the Goldbach Conjecture, and the Twin Prime Conjecture.

The locations of the zeroes of the zeta function appear to bear some relationship to the number of primes in an arithmetic progression with a fixed difference k. Proving Riemann will also validate the zeta function as useful for studying prime numbers, as had been done by Leonard Euler.

# Update on Tex editors

Nearly three years ago, I wrote about a comparison of LaTeX editors. Soon after, I began to use a third editor which, if you are a latex expert, you almost certaintly would have heard about, and are probably in fact using TeXStudio, an editor that has been around for close to a decade, but never appeared to show up on Linux installation packages. The editors that showed up, at least for me, were LyX and TeXmacs.

TeXstudio, once I discovered it, I installed it everywhere I could: on my Windows 10 and 7 machines, on my Linux installations, and even on Cygwin, even though they already had a Windows installation. To this day I have not seen any difference in output or functionality. All invocations of TeXstudio require a lot of time and packages for an installation of enough features.

First thing’s first: the editor. In LyX and TeXmacs, I needed to bail out of the editor, and export the code to LaTeX whenever I needed to do any serious equation editing or table editing or the like. In contrast, TeXstudio leaves me with no reason to ever leave the editor. First of all, the editor allows for native latex code to be entered. If there are pieces of Latex code that you don’t know, or have a fuzzy knowledge about, there is probably an icon or menu item that covers it. For document formatting, a menu item leads to a form dialog where you can fill in the form with sensible information pertaining to your particular document, default font size, paper size, margins, and so on. The ouput of this dialog is the preamble section to the LaTeX source file. To the rest of that source file, you add your document and formatting codes.  It is a kind of “notepad” for LaTeX, with syntax highlighting and shortcut buttons, menus and dialogs. It comes close to being WYSIWYG, in that “compiling” the code and pressing  the green “play” button brings up a window with the output of the existing code you are editing. It is not a live update, but it saves you the agony of saving, going on the command line compiling the code, and viewing in seeminly endless cycles. Now you can view the formatted document at the press of the play button.

# A brief note on Pythagorean Triples

And I decided today to share what I learned about an algorithm for generating Pythagorean triples for any

*** QuickLaTeX cannot compile formula:
m

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



and

*** QuickLaTeX cannot compile formula:
n

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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, where

*** QuickLaTeX cannot compile formula:
m, n \in

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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Z. A Pythagorean triple are any three whole numbers which satisfy the equation

*** QuickLaTeX cannot compile formula:
a^2 + b^2 = c^2

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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. Let

*** QuickLaTeX cannot compile formula:
a = m^2 - n^2

*** Error message:
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;

*** QuickLaTeX cannot compile formula:
b = 2 m n

*** Error message:
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, and you will obtain a solution to the relation

*** QuickLaTeX cannot compile formula:
a^2 + b^2 = c^2

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
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. It is therefore not that hard, if we allow

*** QuickLaTeX cannot compile formula:
m

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



and

*** QuickLaTeX cannot compile formula:
n

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



to be any numbers from 1 to 100, and

*** QuickLaTeX cannot compile formula:
m \ne n

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



, to write a computer program to generate the first 9800 or so Pythagorean triples, allowing for negative values for

*** QuickLaTeX cannot compile formula:
a

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



or

*** QuickLaTeX cannot compile formula:
b

*** Error message:
Fatal Package fontspec Error: The fontspec package requires either XeTeX or
Emergency stop.



.

# What the Golden Ratio has to do with The Thomas Crown Affair

There is a Geogebra app written by Ben Sparks which can run on your browser. There are many facets to this app, such as an animation of “circles within circles”. When this app runs (see the screenshot on your left), the dark points appear to move together in circles, but are really moving in straight line segments along the green paths. The red point A is the only point actually moving in a circle. The number of dark points can be controlled. When reduced to two points, the positions of them resemble the sine and cosine of angles if the red point was on the terminal arm of an angle in standard position. A circle of dark points can be formed from the points of intersection between a perpendicular passing through A and any line.

There was another one to do with a Mandlebrot set and a complex coordinate plane, even offering a “painter” which colors the canvas from black to blue when the iteration forming the spiral goes from stable to unstable, or from a cohesive spiral to one whose points are scattered as you drag a point C with your mouse over the coordinate plane. The points converge on a single point at the origin.

The last and my favourite, is one called a “sunflower spiral”. Apart from checkboxes, a text box for numeric input, and a “start/stop” button to control the animation, there is another checkbox which says “show lyrics”. They are the lyrics to the 1968 hit song “The Windmills of Your Mind”, which first appeared as the theme to the movie The Thomas Crown Affair, and sung by Noel Harrisson, and won an Academy Award for best original song. I won’t bore you with the lengthy list of vocalists who have covered it since, but I can provide a link if you really want to know that and other details.

The idea here is, imagine a flower with many, many seeds, like a sunflower. As it distributes more and more seeds, it must do so from the centre, pushing the seeds it has already deployed, more and more to the outside. Seeds thrive when they are further apart, so that they may take advantage of greater food availability.

This app appears to mimic a flower which distributes its “seeds”, symbolized as small circles, based on the distance from the centre and the amount of rotation based on the desired amount of seeds per turn. The number of turns has to be a number between 0 and 1. So, if you want 5 seeds per turn, seeds are distributed every 1/5 of a turn, and so you enter r = 0.2 in the text box. However, this means that, rather than spirals forming, you get spokes, as in the illustration above. However if you let it run for a bit, it does a fine iteration to six decimals, and you begin to see spirals bending, disappearing and re-forming, as in the illustration to the left, with a number like 0.206270. This is still a rational number, since the decimal terminates.

Notice that the centre of the “windmill” still has “spokes”. That might still be because 0.206270 is still close to 1/5, or 0.2. The outer “seeds” seem to arrange themselves in such a way that any one of them are members of at least two kinds of spirals, criss-crossing each other in opposing directions.

$\pi$ is another number thought to be highly irrational. Not irrational enough to scatter the seeds, as they fall in an orderly fashion and too close together for both $\pi - 3$ (above left) and $1/\pi$ (above right). Strangely, the spiral on the right has 7 “spokes”, and the one on the left has 22 “spokes”. If you recall your high school teacher telling you that an approximation for $\pi$ is $\frac{22}{7}$, I am not sure if that has anything to do with it.

Another candidate is Euler’s constant, e. I tried $e - 2$ (right) and $1/e$ (left), and both looked somewhat more satisfying. They still had distinct spirals somewhere in the seed distribution. When I say that the spirals are distinct/not distinct, I mean that the eye is drawn less and less to one particular spiral. But the seeds are overlapping a lot less, though there is still some overlap.

The irrational number $\sqrt{2}$ can be applied, and here, it is applied as $1/\sqrt{2}$ (left), and $\sqrt{2} - 1$ (right). This seems to be the best so far, as the spirals get less distinct, and the seeds appear to be spread further apart still, not seeming to touch anywhere.

For the golden ratio, this time the decimal was simply stripped from the number. Their button says $\frac{\sqrt{5}-1}{2}$, but this is the same as $\Phi - 1 = \frac{\sqrt{5}+1}{2}-1 = \frac{1}{\Phi}=0.61803398875$, oddly enough. The “seeds” appear to have the best distribution. I had the feeling this is true of any number $\frac{\sqrt{n}+1}{2}$, where n is a prime number, or at least a number that is not a perfect square. I had no luck, however, after trying 2, 3, 6, 7, and 11, and using the decimal or taking the reciprocal. Of course, the metallic ratios have numerical properties similar to the Golden Ratio, and two others were tried.

The reciprocal of the Silver Ratio, $\delta_S=\frac{1}{\sqrt{2}+1}=\sqrt{2}-1$ was already tried, and worked well (see above). The Bronze Ratio, $\frac{3+\sqrt{13}}{2}-3=0.302775638$, led to less satisfactory results.  Metallic ratios follow the formula ${\large\frac{n+\sqrt{n^2+4}}{2}}$, for any positive whole number n.

# Whatever happened to the conic sections: Parabolas

I don’t know about anywhere else, but in our school district, any parabola that is not solvable by the quadratic formula is an endangered species. Specifically, I am mourning the loss of parabolas whose directrix is not parallel to the x-axis. The ones that are parallel to the x-axis are such a mundane part of the curriculum that we never discuss the existence of a focus or a directrix.

Parabolas can open up in any direction, not just up or down. And they all possess a focus and a directrix. As an example, suppose the focus of the parabola was at the origin, and the directrix was the line $-x - y = 2$, or $y= -x-2$ if you prefer slope-intercept form. This ought to lead to a single parabola as a result: $x^2 - 2xy + y^2 -4x - 4y = 4$. Parabolas like these cannot be resolved to familiar equations of the form of $y= ax^2 + bx + c$, largely due to the difficulty in separating $x$ from $y$.

I have gotten used to saying to my students that all polynomial functions of $x$ have a domain in all real numbers. This works because I am only talking about functions. But for relations like $x^2 - 2xy + y^2 -4x - 4y = 4$, both domain and range have a restriction. The parabola looks like the illustration to the right. This relation is not a function, because  1) $x$ and $y$ can’t be separated, and 2) you can pass a vertical line through more than one point on its graph – that is, one value of $x$ generates two values of $y$. Other patterns we take for granted are also broken: the vertex is not a local extrema anymore. We require implicit differentiation to obtain the local extrema: $\frac{dy}{dx} = \frac{-x+y+2}{-x+y-2} = 0$ , which boils down to $y = x-2$. From the point of intersection between this line and the parabola, we find it will have an absolute minimum at $(1, -1)$. The range, then, is $y \ge -1$.

The vertex is located at $(-0.5, -0.5)$.

The domain is, similarly, $x \ge -1$.

# Happy π Day 2018!

Today, we mourn the passing of Stephen Hawking (age 76). He joins Howard Aiken, a computer designer from the early days, who died on $\pi$ day back in 1973. For the record, Albert Einstein was born this day back in 1879.

From 2014: A Mile of π, from Numberphile: