Did I ever tell you about the time I got dragged out of a cathedral?

It was when I was about 17. My aunt and uncle were visiting from Ohio. This was the trip where they rented a minivan so the five of us could all travel in one vehicle. The glove compartment had a three digit combination lock on it (by which I mean a permutation lock) and I spent the week trying to guess the code. After some days of this, I said “right ha ha I’ll try this” and scrolled in 666. It opened. Ha ha ha! True story. This minivan also came with a cassette of Achtung! Baby in the tape deck, which some previous renters had left behind and to which I gave a loving home.

So one day we drove up to Spokane to visit St. John the Evangelist, which is, for out west, a pretty swank cathedral. We were nosing around, and my aunt said to one of the vergers, “Oh you have a pipe organ! My niece plays the piano” and before I knew it, I found myself sitting on the organ stool. “Play something!” they encouraged.

Now, I had taken a few years of piano lessons when I was younger, but I’d never built a repertoire, and I certainly couldn’t remember anything. But pause to consider the situation laid out before me: I’m seventeen. I’m sitting in front of a beautiful keyboard. The keyboard of a pipe organ. In a big, vaulted building. In a church! What an opportunity.

So off I go, with gusto:

Screen Shot 2017-03-13 at 17.18.39.png

I made it through one pass of the next section before the verger made me stop.

We weren’t thrown out, but I did get quite a telling off. I can’t remember now the particulars of what was said. Something something “inappropriate”, something something “disrespectful”, I don’t know. He was pretty angry.

In hindsight, I wish I’d been able to Michigan J. Frog it with a solid round of Hello! Ma Baby. But hey, that’s the nature of impromptu situations. You have to work with what you’ve got at the time.

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ps to previous post (3:13am)

Say … check this out (and pardon the inability to superscript):

Divide 4r^4 + 12r^3 + 9r^2 – 486r – 729 by (r + 1.5)

Go on, it’s fun! It all shakes out in whole numbers. I’ll wait.

You get 4r^3 + 6r^2 – 486. Which is interesting for his lack of an r term (yes yes coefficient of zero, okay). But it’s also interesting because you can factor a 2 out of that sucker.

How come I can’t factor a 2 out of the original quartic?

But hey, that’s interesting: the factor (r + 1.5) really is (2r + 3). There we go. Yays!

4r^4 + 12r^3 + 9r^2 – 486r – 729 = (r + 1.5) (4r^3 + 6r^2 – 486)

= 2 (r + 1.5) (2r^3 + 3r^2 – 243)

= (2r + 3) (2r^3 + 3r^2 – 243)

Perhaps fractions should never. If you get a non-integer yet rational root, de-fractionate that guy before you polynomially divide. Or hey — don’t! Is it easier to deal with the r than 2r? I don’t know; would you rather double or halve?

(If you’re really clever, you’ll end up with four, and that’s when you know you need to go to bed.) (If four makes sense because quartic, you should go ahead and turn off your alarm clock.)

(Correct answer: door number 3. Yes yes okay I’m tucking myself in now.)

Unedited maths at 2am: rrrr, circles

Thoughts in response to Colin’s post about a circle that won’t behave

Caveat / explanation of self: I started poking at this problem around midnight-thirty and now it’s 2:31am. I wrote up my thoughts in an email and then thought “ooh or should I just comment directly on Colin’s post” and then “no no this is way waffley, don’t do that to Colin” and then “ooh maybe I could blog it myself.” Then “omg this is way untidy.” However. If I save the below till morning for a proper edit, I’ll either never get around to tidying it up, or I’ll decide it’s all rubbish, and post nothing. (One reason this blog is so empty.) So here are my scribbles, wholesale. … You’re welcome.

Hey! I’ve had a play with your Ill-Behaved Circle post.

Tangent theta, yes certainly; but I can generate a quartic.* 😀 It’s teh awesomes. Same solution of 4.5. (and one negative solution of 1.5 which I presume is interesting but I’m too tired to try to think of how it might be interpreted) (or it could be one of those “square it and generate erroneous answers” things) I got this via looking at the big triangle (A B and O) and the little triangle at the top (C B and the point where the tangent touches the circle). Pythagoras twice: once on the little triangle to get its slopey leg; again on the big triangle. Or, to reword that, the distance between A and B can be found using the coordinates of A and B; and also by saying “it’s 9 plus √(6r + 9)”. I solved the quartic via Desmos (thank you Desmos xx).

4r^4 + 12r^3 + 9r^2 – 486r – 729 = 0 (thank you so much, blog, for apparently not supporting superscripts)

* take that, polite society!

Or, for another cubic, you can think about incircles. If you mirror the tangent line around the y-axis, you get a triangle with vertices A, B, and -A. Isosceles, with base 18 and sides however-you-want-to-write-it (big distance formula or “9 plus √(6r + 9)”). Our circle in question is the incircle of this triangle, and a quick trip to Wikipedia shows you how the radius of this relates to the triangle’s sides. It’s a tidy little proof, pretty much just “area = 1/2 base x height”. You can do the whole shebang. Or you might have gotten yourself halfway to an incircle by the same ideas, thinking about the A B O triangle area splitting into two littler ones and etc. In any case, you get a cubic again … wouldn’t you know it! it’s the same as your tangent-theta cubic. And solution 4.5.

Thinking about it now, after typing out the above, my quartic, if divided by (x + 1.5), should produce your cubic. It better! else trouble. … How interesting … families of polynomials generated by dividing out roots … hmmm … *strokes chin* (if I’m going ooh at something that’s less than profound, again I tell you that I am tired). And again I wonder about the -1.5 business.

ps: I bet your tangent approach and my incircle approach amount to the same thing … tangent is just a triggy way to say “area of a triangle”, because legs. I mean, not just that we get the same cubic, but the very process is the same thing. You’re going from side to side via angles, whereas I crawled over the credenza. (Sanity Reference Marker: this made sense in my head at the time)

pps: Yes yes the long division works (hooray!) except for a factor of 2 because apparently quartics are twice as nice.

ppps: I wrote my grin in ASCII, because ASCII, but WordPress auto-emojified it. Grrrrrrrrrrrrrr. I’ll figure out how to turn that off later.

Look For The Axis Label

For Colin @icecolbeveridge

Look for the axis label
When you are sketching relation or curve,
Abscissa ‘cross there, the ordinate’s growing,
Our numbers showing by ordered pairs the graphs they serve,
We maths hard, but who’s abstaining?
Thanks to M. RDC, we’re plotting away,
So always look for the axis label,
It says we’re able to function in the proper way!

with apologies to the International Ladies’ Garment Workers’ Union

Everybody!

Once more with feeling (you in the back: pipe up)