2.1Ring-by-ring Analysis

Using your Time-Lapse vision, imagine how the ring-by-ring strategy accumulates over time:

一个环接一个环的进程有什么有趣之处?

• 每个中间阶段本身就是一个完整的“迷你圈”。i.e., when we’re halfway done, we still have a circle, just one with half the regular radius.
• 每一步的工作量都在增加。Just imagine plowing a circular field and spreading the work over several days.第一天，你从中心开始，甚至不动。接下来，你要做一个尽可能紧的转弯。Then you start doing laps, larger and larger, until you are circling the entire yard on the last day.
• 这项工作具有合理的可预见性，这有助于制定计划。If we know it’s an extra minute for each lap, then the 20th ring will take 20 minutes.
• Most of the work happens in the final laps.In the first 25% of the timelapse, we’ve barely grown: we’re adding tiny rings.Near the end, we start to pick up steam by adding long rings, each nearly the final size.

Now let’s get practical: why do trees follow a ring pattern?

A big tree must grow from a complete smaller tree.With the ring-by-ring strategy, we’re always building on 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.)采用这种由内而外的方法。

2.2Slice-by-slice Analysis

现在考虑一个切片的进程。What do you notice?

• We contribute the same amount with each step.Even better, the parts are identical.这可能对数学无关紧要，但在现实世界中(例如切蛋糕)，我们希望在切每一片时使用相同的运动。
• Because the slices are symmetrical, we can use shortcuts like making cuts across the entire shape.These “assembly line” speedups work well when generating identical components.
• Progress is extremely easy to measure.如果我们有10片，那么在第6片我们正好完成了60%(无论是面积还是周长)。
• We follow a sweeping circular path, never retracing our steps from an “angular” point of view.当雕刻出圆环时，每一步都涉及到完整的360度。

Let’s think about the real world: what objects use the slice-by-slice pattern, and why?

Well food, for one.蛋糕、披萨、派:我们希望每个人都能平等分享。Slices are simple to cut, we get nice speedups (cutting across the cake), and it’s easy to see how much is remaining.Imagine cutting circular rings from a pie and estimating how much is left.

现在想想雷达扫描器:它们在一个圆圈内扫一束光，在移动到另一个角度之前“清理”一片天空。This strategy does leave a blind spot in the angle you haven’t yet covered, a tradeoff you’re hopefully aware of.

相比之下，潜艇或蝙蝠使用的声纳会向各个方向发送声音“环”。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 most range for our energy.

Logistically, if we’re building a circular shape from a set of slices (like the folded sections of a paper fan), it helps to have every part be identical.找出制作单一切片的最佳方法，然后批量生产。更妙的是:如果一个部分可以折叠，那么整个形状就可以折叠起来!

2.3Board-by-board Analysis

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.别担心，我会等的。

准备好了吗?Ok.Here’s a few of my observations:

• 这是一种非常机械的模式，板子从左到右整齐排列。
• 每一步的贡献开始很小，逐渐变大，在中间达到最大，然后开始再次缩小。
• 我们的进展多少有些不可预测。Sure, at the halfway mark we’ve finished half the circle, but the pattern rises and falls which makes it difficult to analyze.相比之下，每个环的图案每次变化相同的数量，总是增加。It was clear that the later rings would add the most work.Here, it’s the middle section which seems to be doing the heavy lifting.

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 (especially if there are other steps involved, like painting).就像树每一步都需要一个完整的圆圈一样，桥牌也坚持使用我们在家得宝(Home Depot)可以找到的线性板。

事实上，任何具有严格“管道”的流程都可以使用这种方法:完成一个部分，然后进入下一个部分。想想一台需要在纸张通过时喷射出图案的打印机(或者现在的3d打印机)。The printer sees a position only once, so it better make it count!

圆不需要是字面上的形状。It could represent a goal you’re trying to accomplish, whether an exercise plan or topics to cover in a counseling session.

The board approach suggests you start small, work your way up, then ease back down.披萨片的方法可以忍受(每天相同的进度)，但戒指可能会让人泄气:每一步都需要比前一步更多的努力。

2.4Getting Organized

So far, we’ve been using natural descriptions to explain our thoughts: “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 with “Dum-de-dum-dum” – you’re probably the only one who knows what you mean.但是一个小小的组织可以让你的意图非常清晰。

To start, we can explain how we’re splitting the shape into steps.I like to imagine a little arrow in the direction we move as we cut out each piece:

在我的脑海中，我沿着黄线移动，一边走一边喊着我们要走的台阶(向前走，发出响声，向前走，发出响声……)

虽然箭头显示了环是如何制成的，但其步骤很难想象，因为它们被卡在圆圈内。As we saw in the first lesson, we can pull out the individual steps and line them up:

We draw a black arrow to show the trend in the size of each step.很好,对吧?我们一眼就能看出，这些环在增加，而且每次增加的幅度相同(趋势线是直线，就像一组楼梯)。

Math fans and neurotics both enjoying lining up the pieces.There is something soothing about it, I suppose: who doesn’t want to line up all the pencils that are scattered on the floor?

And since we’re on the topic, we might as well organize the other patterns too:

Ah!Now it’s much easier to compare each X-Ray strategy:

• With circular rings, steps increase steadily (upward sloping line)
• With triangular slices, steps stay the same size (flat, horizontal line)
• 用矩形板，台阶会变大，峰值，然后变小(上下)。The trend line looks elongated because the boards have been pushed down to line up at the bottom.

这些图表使策略比较更容易，不是吗?Sure.But wait, isn’t that trendline looking like a dreaded x-y graph?

Yep.不幸的是，许多类只是呈现黑色趋势线，而没有向您展示它所代表的部分。That’s a recipe for pain: be explicit about what the graph means!

The black trend line is the super-summarized description of an X-Ray strategy.我们展示了每一块的大小(图高度)以及它们的大小是如何变化的(趋势方向)。

The distinction between a ring, slice and board isn’t important: in Calculus, they’re all pieces of the whole pattern.Words like “slice”, “ring” or “board”, are just descriptive versions of “piece of the whole”, and are otherwise interchangeable.

在本指南中，我们将保持上述水平的图表:趋势线，并显示各个部分。这是以后基于性能的类的基础，在这些类中，您可以直接使用图形。但仅供参考，阿基米德在没有x-y图的情况下奠定了微积分的基础，他的结果也没有x-y图。

2.5Questions

事情开始有进展了吗?Thinking better with X-Rays and Time-lapses?

1. Can you describe a grandma-friendly version of what you’ve learned?
2. 让我们把思维扩展到第三维度。你能想出一些制作球体的方法吗?(No formulas, plain-English descriptions are fine.)

PS. It may bother you that our steps create a “circle-like” shape, but not areal, smoothcircle.Great question, and we’ll get to that.But to be fair, it must also bother you that the square pixels on a computer screen make “letter-like” patterns, but notreal, smoothletters.不知何故，“字母样模式”传达的信息与真正的字母相同!