**Ellipse: You’ll never see me bad-mouthing a circle. But, a circle only has TWO parts: The middle and the perimeter. I can live with that. That’s why they wait until analytic geometry to introduce you to the HORROR that is the ellipse. It has more parts than a shortwave radio. Besides a center and a perimeter, which all God-fearing circles have, it has two focii, a semi-major axis, a semi-minor axis, two vertices and two co-vertices. And, I’m not even going to talk about the semi-latus rectum. I’ve dissected things simpler than that.**

**“But”, you warn, “Without ellipses, we’d never be able to predict what a comet is going to do”. I’ll tell you EXACTLY what a comet is going to do: It’s going to circle the Sun until all of its gases are used up and then it will be a STUPID ASTEROID. All that time, it will be alternately raising and dashing the hopes of amateur astronomers everywhere.**

**Arc: Arcs have their uses, I suppose. I hear they are used for welding. Also, I think they are mentioned in Genesis somewhere. However…**

**If they had hands, arcs would have a lot of blood on theirs. Ballistics and artillery are all about arcs. You can calculate where the cannonball is going with an arc. You can kill people soundly with arcs. Whether those arcs be from a saintly circle or a God-forsaken ellipse. Like it or not, the arc is part of the military industrial complex. That makes it as guilty as Hitler…**

**Heart: The heart-shape is used in Valentines cards and to dot the ‘i’s of twelve year old girls. Now, I don’t mind the latter, due to the fact that I have very little contact with twelve year old girls and I think everyone is happy with THAT arrangement. My issue is, every time I look at a Valentines card, I am reminded of my own mortality. The heart is the organ that will decide when I die. You might say that the brain has something to do with it and you can’t live without a brain. Well, if I had a brain, would I be wasting my time complaining about SHAPES? I should say not!**

**Pentagon: First off, the pentagon joins the arc in complicity with all the wartime deaths since the invention of the sling. Is it evil? Well, at the actual Pentagon in Arlington, the very center is a little outdoor food court, shaped like a pentagon. And, food courts are only evil when they are vegetarian food courts. “More bulgar wheat paste?”… “Sure. I’ve lost the will to live, so it is only fitting that I lose the will to enjoy myself as well”…**

**When you are summoning a demon, it is advised that you stand inside a pentagon for protection. This is important to remember when asking an imaginary thing to help you out with your problems. The all-powerful being is INCAPABLE of stepping into a pentagram because of reasons that wouldn’t make any sense even if I replaced them with other reasons…**

**Star: It was bad enough when airlines offered kosher and non-kosher meals; but, stars come in five points and six points, which is kind of the same thing. A six-pointed star is the intersection of two triangles, which are, of course, the very WORST of shapes, so multiply that times two and then explain to me how you quantified it in the first place. It is called the “Star of David” and I’ll bet that David was glad to get rid of it.**

**Stars represent the heavenly bodies that we call, ironically enough, “stars”. Stars are spheres, of course, so their two-dimensional representation should be a circle; but, they didn’t want you to confuse Betelgeuse with, say, a cantaloupe…**

**Spiral: You ever know a spiral to be a good thing? Things spiral out of control all the time but do they ever spiral back into control? No! Never! Thanks, entropy, you little bitch. A larger version of a spiral would be a hurricane, which is always a welcome sight. Even larger spiral? The galaxy Andromeda, which is scheduled to crash into our galaxy in a few billion years. I’d tell you more but I THINK IT CAN HEAR ME…**

**In addition, spirals remind me of the Spiro-graph, which I believe was invented by Nixon’s vice-president. It was a kit with cogs and a pen that didn’t work, designed to be played with once and then put in the closet until it is either thrown away or consumed in a house fire…**

**Ray: A 2-dimensional entity that has the same name as the guy who bags your groceries. Difference is, the guy who bags your groceries is doing something remotely important. A ray has no use at all. Theoretically, it is a one-dimensional entity, with a starting point, that goes an infinite distance in one direction. In reality, it only goes to the edge of your page or blackboard, then it gets high and plays video games…**

Now this was excellent – a fun rant at things we all hate. Destroy the ellipse – a cathcry we can all relate to

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Some shapes have had it way to easy, way too long!

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Here’s the thing I discovered about an ellipse just last week–and I’m a high school math teacher. A student asked me about the area of an ellipse. Well, I said, think of it as a deformed circle–the area is pi*a*b, where a and b are half the lengths of the two axes; it makes sense because when a=b as in a circle, the area is then pi*r squared, as in a circle.

Yes says the student, but what about the circumference of an ellipse? So I thought well all I have to do is again think of an ellipse as a deformed circle, but soon realized that 2*pi*r would cause me problems. What combination of a and b should I use as r? The mean of a and b? After considering “a” very large and “b” very small, it soon became apparent that was impossible, because the circumference starts looking like two lines, each with length 2a. So I broke down, and looked it up on Wikipedia.

It wasn’t a pretty sight. Lots of approximation series, but no actual formula. I was very surprised. But more surprised that I had never run across this before.

Yes, ellipses are very nasty.

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I ended up getting my Masters Degree in Mathematics, but I never lost my frustration for the ellipse. I think because it ends up being the first shape you learn about that isn’t mathematically simple. That’s when you lose your math innocence for the first time. (The second is when you learn that you cannot explicitly solve fifth order or higher equations).

It’s rough having to learn that stuff, but I’d imagine it’s a LOT tougher trying to help others learn that stuff…

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Many of the formulas for calculating the approximate perimeter of ellipse are long and tedious. And many of them aren’t even that good (they have high percentage errors). I can only think of 4 formulas that are good enough off the top of my head. One of which by Ramanujan, the other one by Peano and two others I found in math journals before.

As for exact solutions, all of them involve summing infinite series. There is one that is pretty good since after only a few terms, it quickly converges to the right answer. However, it involves binomial coefficients so the calculation can be rather complicated if done manually.

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One of the many great things about Newton was his method of approximating solutions. If not just for “Newton’s Method” but also for the concept of a non-explicit solution. If Everest Galois taught us anything it is that SOME things cannot be accomplished explicitly… oh, and NOT to get into duels with soldiers…

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Very good 😀

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Thank you very much.

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