# My New Math Project: Calculus

Now that I’ve let the whole world know this, I have to follow through on my plan. It is difficult to embarrass me, unless I deliberately set up a situation that uses embarrassment on a global scale, as a self-motivational tool, and that’s what I am doing right now. I fully intend to learn calculus in June, and this will help.

I already have books, and a plan of attack. I am not working for pay in June, nor taking any classes, so that gives me the time, and you can’t beat a tuition-price of zero.

The key was moving calculus from my mental “incomprehensible” Venn diagram bubble to my “I can do this” Venn diagram bubble. I never should have created that “incomprehensible” bubble in the first place, but it took a lot of time (30 years or so) to figure that out.

~~~

Update, 14June2016: I have decided to turn this from a project for the current month into an ongoing project . . . for the rest of my life.

# For me, geometry for breakfast is not unusual. This morning, though, I’m sprinkling calculus on top before eating it.

It’s important to explain, right up front, that Ronald Reagan was president when I last took calculus. However, I have a new determination to learn the subject. I have a hunch this may go better without the “help” of actually being enrolled in a calculus class, since the way I learn things, and the way most people learn things, aren’t much alike.

My current calculus puzzle started when I noticed that taking the derivative of the volume of a sphere, in terms of the radius, (4/3)πr³, yields the formula for the surface area of a sphere, 4πr². That was both unexpected and exciting, so I tried applying the same idea to another solid: the cube. With edge length e, the volume of a cube is e³, and the derivative of that is 3e² . . . but that’s only half of the surface area of a cube, which is 6e².

Half? What’s going on here? I mentioned this puzzle on Facebook, where I have many on my friends’-list whose mathematical knowledge exceeds my own. It was pointed out to me that I’d made an important and unhelpful change by going from using the radius, for the sphere, to the edge length, for the cube.

So I’ll try this again, but do it in terms of the radius of the cube, rather than the edge length. For a cube, the radius extends from the center to any of the cube’s eight vertices. Both the light and dark blue segments in the diagram below are cube radii.

This radius is sqrt(3)/2 times the cube’s edge length, as can be verified by applying the Pythagorean Theorem twice, first to triangle ABC (which shows that the green face-diagonal is sqrt(2) times the edge length), and then to triangle BCD (which yields sqrt(3) times the edge length for the interior diagonal DC, half of which is the radius).

It then follows that, if r = [sqrt(3)/2]e, that e = [2/sqrt(3)]r, which “cleans up” to e = (2/3)sqrt(3)r, when the denominator is rationalized.

If a cube’s volume is e³, and e = (2/3)sqrt(3)r, it then follows that V = [(2/3)sqrt(3)r]³ = (8/27)(3)sqrt(3)r³ = (24/27)sqrt(3)r³ = [8sqrt(3)/9]r³. If I take the derivative of the last expression, I get [8sqrt(3)/3]r² for the derivative of the volume, which I now need to compare to the surface area of a cube, in terms of its radius, rather than edge length.

So here goes . . . SA = 6e² = 6[(2/3)sqrt(3)r]² = [48(3)/9]r² = 16r², which isn’t what I got for the derivative of the volume, above.

Well, I was using, as the radius, the radius of the cube’s circumscribed sphere. Perhaps I should have used the inscribed sphere, instead? The radius of the cube’s inscribed sphere is the “invisible” segment FM in the diagram above, which I’m going to call “a” (for “apothem,” because this looks like the 3-d version of the apothem of a regular polygon). The length of a is exactly one-half that of e, the cube’s edge length, which means that e = 2a. Therefore, V = e³ = (2a)³ = 8a³, the derivative of which is 24a².

Now to check the surface area, in terms of a: SA = 6e² = 6(2a)² = 24a², and that’s what I got when I took the derivative of the volume, in terms of a.

So this trick works for the cube if you use the radius of the inscribed sphere, but not the circumscribed sphere. This leaves me with three questions to address later:

1. Will this also work for other polyhedra? This is something I intend to explore in future blog-posts, starting with the tetrahedron and the octahedron.
2. Why did this work at all?
3. Why was it necessary to use the radius of the cube’s inscribed sphere, rather than its circumscribed sphere?

If any reader of this post knows the answer(s) to #2 and/or #3, sharing your knowledge in a comment would be very much appreciated.

# A Torus and Its Dual, Part II

After I published the last post, which I did not originally intend to have two parts, this comment was left by one of my blog’s followers. My answer is also shown.

A torus can be viewed as a flexible rectangle rolled into a donut shape, and I had used 24 small rectangles by 24 small rectangles as the settings for Stella 4 for the torus, and its dual, in the last post — which, due to the nature of that program, are actually rendered as toroidal polyhedra. To investigate my new question, I increased 24×24 to 90×90, and these three images show the results. The first shows a 90×90 torus, the second shows its dual, and the third shows the compound of the two.

When I compare these images to those in the previous post, it is clear that these figures are approaching a limit as n, in the expression “nxn rectangle,” increases. What’s more, I recognize the dual now, of the true torus, at the limit, as n approaches infinity — it’s a cone. It’s not a finite-volume cone, but the infinite-volume cone one obtains by rotating a line around an axis which intersects that line. This figure, not a finite-volume cone, is the cone used to define the conic sections: the circle, ellipse, parabola, and hyperbola.

What’s more, I smell calculus afoot here. I do not yet know enough calculus.

“Learn a lot more about calculus” is definitely on my agenda for the coming Summer, for several reasons, not the least of which is that I plainly need it to make more headway in my understanding of geometry.

[Note: Stella 4d, the program used to make these images, may be found at http://www.software3d.com/Stella.php.]

# Circumparabolic Regions Inside a Unit Circle

A circumparabolic region is found between a circle and a parabola, with the circle being chosen to include the vertex and x-intercepts of the parabola used, with the circle, to define the two circumparabolic regions for a given parabola-circle pair. There are four such regions shown above, rather than only two, because two parabolas are used above. The formulae for the parabolas, as well as the circle, are shown.

A puzzle which I will not be solving, I suspect, until I learn more integral calculus: what fraction of the circle’s area is shown in yellow?

# How I Hit My Personal Mathematical Wall: Integral Calculus

To the best of my recollection, this is the first time I have written publicly on the subject of calculus. The fundamental reason for this, explained in detail below, is something I rarely experience: embarrassment.

Unless this is the first time you’ve read my blog, you already know I like mathematics. If you’re a regular follower, you know that I take this to certain extremes. My current conjecture is that my original motivation to learn how to speak, read, and write, before beginning formal schooling, was that I had a toddler-headful of mathematical ideas, no way to express them (yet), and learned to use English in order to change that. Once I could understand what others were saying, read what others had written, write things down, and speak in sentences, I noticed quickly that interaction with other people made it possible to bounce mathematical ideas around, using language — which helped me to develop and expand those mathematical ideas more quickly. Once I started talking about math, as anyone who knows me well can verify, I never learned how to shut up on the subject for longer than ten waking hours at a time.

A huge part of the appeal of mathematics was that I didn’t have to memorize anything to do it, or learn it. To me, it was simply one obvious concept at a time, with one exposure needed to “get it,” and remember it as an understood concept, rather than a memorized fact. (Those math teachers of mine who required lots of practice, over stuff I already knew, did not find me easy to deal with, for I hated being forced to do that unnecessary-for-me chore, and wasn’t shy about voicing that dislike to anyone and everyone within hearing range, regardless of the situation or setting. The worst of this, K-12, was long division, especially the third year in a row that efforts were made to “teach” me this procedure I had already learned, on one specific day, outside school, years earlier.) It might seem like I have memorized certain things, such as, say, the quadratic formula, but I never actually tried to — this formula just “stuck” in my mind, from doing lots of physics problems, of different types, which required it. Similarly, I learned the molar masses of many commonly-encountered elements by repeatedly using them to show students how to solve problems in chemistry, but at no time did I make a deliberate attempt to memorize any of them. If I don’t try to memorize something, but it ends up in memory anyway, that doesn’t count towards my extremely-low “I hate memorizing things” threshhold.

When I first studied calculus, this changed. Through repeated, forced exposure in A.P. Calculus class my senior year of high school, with a teacher I didn’t care for, I still learned a few things that stuck: how to find the derivative of a polynomial, the fact that a derivative gives you the slope a function, and the fact that its inverse function, integration, yields the area under the curve of a function. After I entered college, I then landed in Calculus I my freshman year. Unbeknownst to me, I was approaching a mental wall.

My college Cal I class met early in the morning, covered material I had already learned in high school, and was taught by an incomprehensible, but brilliant, Russian who was still learning English. Foreign languages were uninteresting to me then (due to the large amount of memorization required to learn them), and I very quickly devised a coping strategy for this. It involved attending class as infrequently as possible, but still earning the points needed for an “A,” by asking classmates when quizzes or tests had been announced, and only waking up for class on those mornings, to go collect the points needed for the grade I wanted.

This was in 1985-86, before attendance policies became common for college classes, and so this worked: I got my “A” for Cal I. “That was easy,” I thought, when I got my final grade, “so, on to the next class!”

I did a lot of stupid things my freshman year of college, as is typical for college freshmen around the world, ever since the invention of college. One of these stupid things was attempting to use the same approach to Calculus II, from another professor. About 60% of the way through that course, I found myself in a situation I was not used to: I realized I was failing the class.

Not wanting an “F,” I started to attend class, realizing I needed to do this in order to pass Cal II, which focuses on integral calculus. A test was coming up. In class, the professor handed out a sheet of integration formulas, and told us to memorize them.

Memorize them.

I read the sheet of integration formulas, hoping to find patterns that would let me learn them my way, rather than using brute-force memorization-by-drill. Since I had been skipping class, I saw no such patterns. All of a sudden, I realized I was in a new situation, for me: mathematics suddenly was not fun anymore. My “figure it out on the fly” method, which is based on understanding, rather than memorization, had stopped working.

A few weeks and a failed test later, I began to doubt I would pass, and tried to drop the class. This is how I learned of the existence of drop dates for college classes, but I learned it too late: I was already past the drop date.

I did not want an F, especially in a math class. Out of other options, I started drilling and memorizing, hated every minute of it, but did manage to bring my grade up — to the only “D” I have on any college transcript. Disgusted by this experience, I ended up dropping out of college, dropped back in later, dropped out again, re-dropped back in at a different university, and ended up changing my major to history, before finally completing my B.A. in “only” seven years. I didn’t take another math class until after attempting to do student teaching, post-graduation . . . in social studies, with my primary way of explaining anything being to reduce it to an equation, since equations make sense. This did not go well, so, while working on an M.A. (also in history) at a third college, I took lots of science and math classes, on the side, to add additional teaching-certification areas in subjects where using equations to explain things is far more appropriate, and effective. This required taking more classes full of stuff I already knew, such as College Algebra and Trigonometry, so I took them by correspondence (to avoid having to endure lectures over things I already knew), back in the days when this required the use of lots of postage stamps — but no memorization. To this day, I would rather pay for a hundred postage stamps than deliberately memorize something.

In case you’re wondering how a teacher can function like this, I will explain. Take, for example, the issue of knowing students’ names. Is this important? Yes! For teaching high school students, learning the names of every student is absolutely essential, as was quite evident from student teaching. However, I do this important task by learning something else about each student — how they prefer to learn, for example, or something they intensely like, or dislike — at which point memorization of the student’s name becomes automatic for me. It’s only conscious, deliberate memorization-by-drill that bothers me, not “auto-memorization,” also known as actually understanding something, or, in the case of any student, learning something about someone.

I don’t know exactly why my to-this-point “wall” in mathematics appeared before me at this point, but at least I know I am in good company. Archimedes knew nothing of integral calculus, nor did his contemporaries, for it took roughly two millennia longer before Isaac Newton and Gottfried Leibniz discovered this branch of mathematics, independently, at roughly the same time.

However, now, in my 21st year as a teacher, I have now hit another wall, and it’s in physics, another subject I find fascinating. Until I learn more calculus, I now realize I can’t learn much more physics . . . and I want to learn more physics, for the simple reason that it is the only way to understand the way the universe works, at a fundamental level — and, like all people, I am trapped in the universe for my entire life, so, naturally I want to understand it, to the extent that I can. (A mystery to me: why isn’t this true for everyone else? We’re all trapped here!) Therefore, I now have a new motivation to learn calculus. However, I want to do this with as much real understanding as possible, and as little deliberate memorization as possible, and that will require a different approach than my failed pre-20th-birthday attempt to learn calculus.

I think I need exactly one thing, to help me over this decades-old wall: a book I can read to help me teach myself calculus, but not a typical textbook. The typical mathematics textbook takes a drill-and-practice approach, and what I need is a book that, instead, will show me exactly how various calculus skills apply to physics, or, failing that, to geometry, my favorite branch of mathematics, by far. If any reader of this post knows of such a book, please leave its title and author in a comment. I’ll then buy the book, and take it from there.

One thing I do not know is the extent to which all of this is related to Asperger’s Syndrome, for I was in my 40s when I discovered I am an “Aspie,” and it is a subject I am still studying, along with the rest of the autism spectrum. One thing Aspies have in common is a strong tendency to develop what we, and those who study us, call “special interests,” such as my obsession with polyhedra, evident all over this blog. What Aspies do not share is the identity of these special interests. Poll a hundred random Aspies, and only a minority will have a strong interest in mathematics — the others have special interests in completely different fields. One thing we have in common, though, is that the way we think (and learn) is extremely different from the ways non-Aspies think and learn. The world’s Aspie-population is currently growing at a phenomenal rate, for reasons which have, so far, eluded explanation. The fact that this is a recent development explains why it remains, so far, an unsolved mystery. One of things which is known, however, is the fact that our status as a rapidly-growing population is making it more important, by the day, for these differences to be studied, and better understood, as quickly as the speed of research will allow, in at least two fields: medicine, and education.

Only one thing has fundamentally changed about me, regarding calculus, in nearly 30 years: I now want to get to the other side of this wall, which I now realize I created for myself, when I was much younger. I am also optimistic I will succeed, for nothing helps anyone learn anything more than actually wanting to learn it, no matter who the learner is, or what they are learning. In this one respect, I now realize, I am no different than anyone else, Aspie or non-Aspie. We are all, after all, human beings.