paper: tactile functions

Download paper (PDF)

Abstract: 

This article discusses the construction of tactile functions in the dynamic geometry environment.  Technology now allows us to visually/graphically define functions responding to informal cosmetic directives while inheriting the analytical properties of a function object in a software environment.  The specific topic explored is the Fundamental Theorem of Calculus.

tactile functions - piecewise cubic

Download: TactileFunction_PiecewiseCubic.gsp

This sketch follows an earlier experiment with piecewise linear tactile functions.  These functions are defined using a small number of user-positioned independent points which use a simple convention to define a function.  The linear one offered what I thought to be an interesting look at the antiderivative.  This sketch shows a piecewise cubic function, meaning that consecutive independent points are joined with a segment of a cubic function.  

Let me know what you think. 

There is something called a Bezier curve that involves cubics and addresses some similar applications, but I am really not sure if my construction is different. 

I decided that I wanted a manipulable function that was smooth on its domain.  I could not do that with lines, and quadratics offer only the most unwieldy options if we insist on differentiability.  I did make one with sinusoidal curves, but the first construction I tried demanded in horizontal tangents at each of the independent points, which was a bit limiting.  In the end, I realized it needed to be cubic and I was pretty sure that a unique curve existed where endpoints and slopes (at the endpoints) were defined. 

I'm not sure if it is good for anything. 

In this sketch, the derivative (red) and antiderivative (blue) can be shown and hidden easily.  The curve in the construction is really seven distinct objects, but it would be possible to create it as a single function object.

waiting for cab 1729

Korean license plates have a few small Korean characters followed by four large digits.  I have not devoted myself to the study of Korean, so the first few characters do little to distract me.  Mostly I enjoy the numbers.  A few weeks ago, it occurred to me that there may be a taxicab in this city with the plate "1729", which would be supercool.  So now I can't stop looking for it. 

The significance of 1729 as a taxi number is a legend of mathematics history.  G.H. Hardy visited Ramanujan at a hospital and noted that his taxi was number 1729, an (unfortunately) uninteresting number.  Ramanujan replied that 1729 is in fact interesting because it is the smallest number that can be written as the sum of two cubes in two different ways.  1729=1³ + 12³ = 9³ + 10³. 

It would be most satisfactory to see it on a taxi, but I find myself checking other license plates just in case.  I saw 1728 once, which is very close and is 12³, which is in fact part of why 1729 is special.  It seems that this would be a disappointing game with occasional near misses.  If I ever found cab 1729 it would surely be anticlimactic at this point. 

Unexpectedly, it is the near misses that have proved to be the most stimulating part of the game.  I was surprised to find myself coming up with arguments in favor of each number being somehow "close" to 1729.  For example 1279, 1698, 3729, and 3458 would all be close to 1729 for different reasons.  What if a license plate has two or three of the right digits?  (2175, 9243).  It's better if they are in the correct places (1754, 1889), or if the wrong digits somehow resemble the correct missing digit (1724, 1759, 1429).  Given the variety, I started to wonder, what is the probability that a given four digit integer--at least a little bit--resembles 1729?  After all, there is about an 87% chance that a license plate contains at least one of the digits {1, 7, 2, 9}.

I can hardly walk around Seoul anymore without getting sucked into this game, which is far from a voluntary obsession.  It is really more like humming the Pororo song that plays on a loop at E-mart.  Kind of catchy.  

When I am in the US, I always try to find acronyms for the letters on license plates.  I'm just throwing that out there in case you need any traffic obsessions of your own.  Mercifully, I do not understand Korean enough for that one to take hold of me.

favorite number?

I encourage my students to start thinking about a favorite number.  Sometimes I ask if anyone has a favorite number, and I get a few students tossing up a 'random' number to make their point that any number is as good as the next.  I get a few students staring blankly, like numerical favoritism is for nerds.  And it is.  I get some students who see that a number could be pleasing in some sense, and they are able to choose a favorite, or a few that stand out as candidates. 

A number can be fun to write.  Its numerical representation can possess some visual aesthetic quality.  It can be just the right size.  It can occur in interesting contexts.  It can play a role in elegant statements describing a complexity that has chosen to highlight a particular number. 

My own favorite number is 17.  It is small, but not too small.  It is prime, a Fermat prime in fact.  There are 17 distinct wallpaper patterns.  Gauss showed that you can construct a regular 17-sided polygon with compass and straightedge.  The smallest number of clues necessary for a uniquely solvable Sudoku puzzle is believed to be 17.  But I digress.  I was just giving an example of a favorite number (existential instantiation), I am not trying to make you feel bad about your own favorite number. 

slope field swarm

Download (for GSP v4 or v5):  Calculus_SlopeFieldSwarm_v4.gsp
This amused me for awhile today.  I wanted to see a slope field constructed from a single perspective, not delivered as a completed product.  I made a point with a segment whose slope was determined by a differential equation, and I enjoyed painting the slope field into being.  This was time consuming, however, and I got the idea of animating and duplicating the point.

calculus - simpson's rule

Download (GSP 5 file): calculus_SimpsonsRule.gsp

This sketch shows the area measured by Simpson's Rule using parabolic arcs. The area beneath a parabolic arc was known to the Greeks ages before calculus was used, and Simpson's Rule is a slick application of this fact. The interval is divided into an even number of sub-intervals and a parabolic arc is chosen for each sub-interval.  The application of the rule does not of course involve actually finding the parabolas, but I found this graphical exploration illuminating.  

word spelling

It has occasionally bothered me that the standard Microsoft Word spelling dictionary knows about "dodecahedron" but not "icosahedron".  It does acknowledge "semi-duality",  which I suppose explains the deeper issue.