Course Homepage Preface On Learning
Build Your Intuition
1. 1-Minute Summary 2. X-Ray Vision 3. 3d Intuition
Learn The Lingo
4. Integrals, Derivatives 5. Computer Notation
Basic Understanding
6. Improved Algebra 7. Linear Changes 8. Squared Changes
Deeper Understanding
9. Infinity 10. Derivatives 11. Fundamental Theorem
Figure Out The Rules
12. Add, Multiply, Invert 13. Patterns In The Rules 14. Take Powers, Divide
Put It To Use
15. Archimedes' Formulas Summary
Welcome to an intuition-first calculus course.
Read online, or buy the complete edition for videos, PDF, and discussion.
15 min read

2. Practice Your X-Ray and Time-Lapse Vision

Calculus trains us to use X-Ray and Time-Lapse vision, such as re-arranging a circle into a "ring triangle" (diagram). This makes finding the area… well, if not exactly easy, much more manageable.

But we were a little presumptuous. Must every circle in the universe be made from rings?

Heck no! We're more creative than that. Here's a few more options for our X-Ray:

circle variations

Now we're talking. We can imagine a circle as a set of rings, pizza slices, or vertical boards. Each underlying "blueprint" is a different step-by-step strategy in action.

Imagine each strategy unfolding over time, using your time-lapse vision. Have any ideas about what each approach is good for?

Ring-by-ring Analysis

circle variations

Rings are the old standby. What's neat about a ring-by-ring progression?

Now let's get practical: why is it that trees have a ring pattern inside?

It's the first property: a big tree must grow from a complete smaller tree. With the ring-by-ring strategy, we're always adding to a complete, fully-formed circle. We aren't trying to grow the "left half" of the tree and then work on the right side.

In fact, many natural processes that grow (trees, bones, bubbles, etc.) take this inside-out approach.

Slice-by-slice Analysis

circle variations

Now think about a slice-by-slice progression. What do you notice?

Time to think about the real world. What follows this slice-by-slice pattern, and why?

Well food, for one. Cake, pizza, pie: we want everyone to have an equal share. Slices are simple to cut, we get nice speedups (like cutting across the cake), and it's easy to see how much is remaining. (Imagine cutting circular rings from a pie and trying to estimate how much area is left.)

Now think about radar scanners: they sweep out in a circular path, "clearing" a slice of sky before moving to another angle. This strategy does leave a blind spot in the angle you haven't yet covered, a tradeoff you're hopefully aware of.

Contrast this to sonar used by a submarine or bat, which sends a sound "ring" propagating in every direction. That works best for close targets (covering every direction at once). The drawback is that unfocused propagation gets much weaker the further out you go, as the initial energy is spread out over a larger ring. We use megaphones and antennas to focus our signals into beams (thin slices) to get the max range for our energy.

Operationally, if we're building a circular shape from a set of slices (like a paper fan), it helps to have every part be identical. Figure out the best way to make a single slice, then mass produce them. Even better: if one slice can collapse, the entire shape can fold up!

Board-by-board Analysis

circle variations

Getting the hang of X-Rays and Time-lapses? Great. Look at the progression above, and spend a few seconds thinking of the pros and cons. Don't worry, I'll wait.

Ready? Ok. Here's a few of my observations:

Ok, time to figure out where this pattern shows up in the real world.

Decks and wooden structures, for one. When putting down wooden planks, we don't want to retrace our steps, or return to a previous position (especially if there are other steps involved, like painting). Just like a tree needs a fully-formed circle at each step, a deck insists upon components found at Home Depot (i.e., rectangular boards).

In fact, any process with a linear "pipeline" might use this approach: finish a section and move onto the next. Think about a printer that has to spray a pattern top-to-bottom as the paper is fed through (or these days, a 3d printer). It doesn't have the luxury of a ring-by-ring or a slice-by-slice approach. It will see a horizontal position only once, so it better make it count!

From a human motivation perspective, it may be convenient to start small, work your way up, then ease back down. A pizza-slice approach could be tolerable (identical progress every day), but rings could be demoralizing: every step requires more than the one before, without yielding.

Getting Organized

So far, we've been using natural descriptions to explain our thought processing. "Take a bunch of rings" or "Cut the circle into pizza slices". This conveys a general notion, but it's a bit like describing a song as "Dum-de-dum-dum" -- you're pretty much the only one who knows what's happening. A little organization can make it perfectly clear what we mean.

The first thing we can do is keep track of how we're making our steps. I like to imagine a little arrow in the direction we move as we take slices:

circle variations

In my head, I'm moving along the yellow line, calling out the steps like Oprah giving away cars (you get a ring, you get a ring, you get a ring…). (Hey, it's my analogy, don't give me that look!)

The arrow is handy, but it's still tricky to see the exact progression of slices. Why don't we explicitly "line up" the changes? As we saw before, we can unroll the steps, put them side-by-side, and make them easier to compare:

circle variations

The black arrow shows the trend. Pretty nice, right? We can tell, at a glance, that the slices are increasing, and by the same amount each time (since the trend line is straight).

Math fans and neurotics alike enjoy these organized layouts; there is something soothing about it, I suppose. And since you're here, we might as well organize the other patterns too:

circle variations

Now it's much easier to compare each X-Ray strategy:

The charts made our comparisons easier, wouldn't you say? Sure. But wait, isn't that trendline looking like a dreaded x-y graph?

Yep. Remember, a graph is a visual explanation that should help us. If it's confusing, it needs to be fixed.

Many classes present graphs, divorced from the phenomena that made them, and hope you see an invisible sequence of steps buried inside. It's a recipe for pain -- just be explicit about what a graph represents!

Archimedes did fine without x-y graphs, finding the area of a circle using the "ring-to-triangle" method. In this primer we'll leave our level of graphing to what you see above (the details of graphs will be a nice follow-up, after our intuition is built).

So, are things starting to click a bit? Thinking better with X-Rays and Time-lapses?

PS. It may bother you that our steps create a "circle-like" shape, but not a real, smooth circle. We'll get to that :). But to be fair, it must also bother you that the square pixels on this screen make "letter-like" shapes, and not real, smooth letters. And somehow, the "letter-like pixels" convey the same meaning as the real thing!


1) What's your grandma-friendly version of what you've learned?

2) Let's expand our thinking into the 3rd dimension. Can you think of a few ways to build a sphere? (No formulas, just descriptions)

3) What clicked? Any questions?

I'll share a few approaches with you in the next lesson. Happy math.

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Class Discussion

Guided class discussions are available in the complete edition.