Throwing A Curve

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Where we learn that a curve is a bent line and a line is a straight curve.

I’ve never quite understood the idiom, ‘life throws you a curveball’. You’d think that with my love of math as well as my love of baseball this would be a pretty clear statement, and yah yah sure I think we can all agree that if you’re metaphorically thrown a curveball, that generally means something unexpected and/or potentially problematic is coming at you super quick and super immediate. And it’s likely that you’ll need to deal with it immediately, too.

It’s just, why call those moments in life a curveball? Why not ohIdunno calling that moment a slider (if we’re sticking with baseball speak)? A slider is as unexpected as a curveball. How about a dropshot 🎾🏸 (what, you have a problem with tennis)? A dropshot is as unexpected in tennis as a curveball or slider is in baseball. What about pressing the Push-to-Pass button on your Indycar during those final few road course laps, where you get a handful of extra power to pass your competitor? Fine, that last one is a bit involved and a bit too much of a 21st century reference. But hey, those all seem like moves that are made to be unexpected and will cause a problem to the other person in the mix, right? So whys it gotta be a curveball?

What’s so special about a curve, anyway?

Writing about the word ‘point’ for last week’s Math WOW led me to the mathematical explanation for a line as two points that are connected by a series of points. So it follows for me that when a line becomes a curve, then it’s a series of points that has become a curve.

A line is a straight curve and a curve is a bent line. Yeah?

A line is one kind of curve, but not the only kind, and lines enjoy all kinds of special properties that curves in general may not.

“How Not To Be Wrong: The Power of Mathematical Thinking” by Jordan Ellenberg
(pg. 23)

But what is it about the curve, itself? What makes a curve, curve?

CURVE
To have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity.

Merriam-Webster

Here’s a visual representation of a continuous function.

I started focusing on the formation of a shape of a curve, and sweet Mother Earth, there are sooooooo many types of curves. As my Curve List grows, it’s like I’m hawking counterfeit watches on the street
_{(All curve definitions courtesy of Wolfram Mathworld.)}

Whatcha need whatcha need?
Something basic, like a Simple Curve?

Or how about something super confusing?
Oooh, I’d bet you’d like an Elliptic Curve, then.

Or, hey, perhaps something that transcends?
Then you’ll want this Transcendental Curve.

On and on and on it goes, so many different types of curves – oh wait, a parabola, now that looks like a familiar one… yet, none of these seem applicable to what I’m looking for, which means I might need to get a lil more mathy. So folks, we gotta do a lil math history massage – yes, I said massage, let’s go into this with a positive mindset, this won’t take long, I promise, and let’s go into this with an outlook of massaging our brains with calming and invigorating info, ‘kay? I’m not any good at memorizing stuff and it’s part of the reason I didn’t excel at math in school, so, I’m not going to get carried away with this history lesson.

SO!
When I say Cartesian, you say Descartes.

If something is labeled as Cartesian, then it hails from the early 16th century and is named for the work that mathematician René Descartes produced; this is a guy who had quite a philosophical bend to his teachings. He had a big ol focus on connecting algebra and geometry together aka analytic geometry aka let’s name it after ourself, let’s call it Cartesian Geometry. Most notably, there is the Cartesian Coordinate System which begat the Cartesian Plane, which is quite simply a two-dimensional flat surface that the Cartesian Coordinate System is overlaid on – cuz it’s algebra and geometry, together! With a Cartesian Plane, you too can plot out the points on a graph that are the arc of a curve.

Really, the heart of the whole Cartesian mindset is this, this very thing of curves,
this connection between the geometry of the shaped dimensionality
and the algebra of the equations that determine their arc.

I got to understand there’s a thing called a Family of Curves. The familial part references that all those curves are of the same form (again, the geometry) but they have an algebraic quantity whose value is selected based on their particular circumstance and its relation to those other variables that are present.

When we’re talking about a line, we’re talking about just one-dimension, whether that line is straight or it’s curving. The Cartesian Plane is, by definition of a plane, two-dimensions. Then when we get to three-dimensions we need three axes (x-axis, y-axis, and also a z-axis) and they need to be “mutually perpendicular” to each other. Oh my. Deep breaths.

In general, curves in space can be conceptualized as being traced by motions of points. Those motions can in turn be conceptualized as having components of motion along the x- and y- axis of the Cartesian Plane. Descarte observed that such motions in the plane correspond to continuous variations of numbers in algebraic equations. Continuous changes of numerical variables in algebraic equations thus become correlated with the motion of points along the x- and y- axes. In the Cartesian Plane blend, continuous functions are correlated with curves formed by the motion of a point over the plane.
The Cartesian Plane blend is, of course, generalizable to spaces of any dimension.

Functions in the Cartesian Plane are often conceptualized in terms of motion along a path – as when a function is described as “going up”, reaching”, a “maximum” and “going down” again.

“Where Mathematics Comes From: How The Embodied Mind Brings Mathematics Into Being”
by George Lakoff & Rafael E. Núñez
(pg. 385-386 & pg. 8)

“…motion along a path,” you say? Ooooh, I do believe we are talking about a specific curve, let me go thru my flash cards, oh there it is, it’s a curve that is imaginatively called – a Path Curve! And lookee there, it’s those continuous functions, the curve that can be drawn without lifting the hand, without any sharp breaks.

Path Curve
A continuous mapping from an initial point to a final point, including a defined space for continuous functions.

Wolfram MathWorld

Here’s the info I’m gonna hold on to with these dang curves: You’ve got your beginning point of x and your ending point of y – those two points connected by a series of points, and the curve of the curve itself, that’s the parameter that you want / need to give value to. And you’re determining that value, the parameter, based on the particular circumstance at hand as well as its relationship to the other variables at hand, in that moment.

Oh, geometry. You’re so much more than a bunch of shapes. I knew that you have dimensions, but these equations that are now thrown on top of ya, that are dictating direction. Well.

The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony.

“The New York Times Book of Mathematics” by Gina Bari Kolata (editor)
(pg. 224)

Hey, harmony, what’s up, you cutie you? 😘

(y’all this is Platonic as in it’s associated with the Greek philosopher Plato,
not platonic as in “just friends.” 😉 )

All right. Back to this curve IRL and it being about the unexpected problem that you must immediately react to and solve.

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I do still like the thought of a group of people meeting and agreeing to the trajectory of their work together towards a common goal — and then, just as everyone splits up to head off in their own direction but with their group plan in hand, someone – is it you?!? – someone says, “This’ll work, as long as no one throws us a slider.”

Thanks math, you’re the best.