Chapter 3Expanding Our Intuition
I hope you thought about the question from last time: how do we take our X-Ray strategies into the 3rd dimension?
Here’s my take:
- Rings becomeshells, a thick candy coating on a delicious gobstopper.每一层都比前一层略大。
- Slices becomewedges, identical sections like slices of an orange.
- Boards becomeplates, thick discs which can be stacked together.(我有时会幻想开一家只提供球形煎饼堆的床上早餐店。)
The 3d segments can be seen as being made from their 2d counterparts.例如，我们可以像硬币一样旋转一个环来创造一个壳。A wedge looks like a bunch of pizza slices (of different sizes) stacked on top of each other.Lastly, we can imagine spinning a board to make a plate, like carving a wooden sphere with a lathe (video).
The tradeoffs in 3d are similar to the 2d versions:
- Organic processes grow in shell-by-shell layers (pearls in an oyster).
- Fair divisions require wedges (cutting an apple for friends).
- The robotic plate approach seems easy to manufacture (weightlifting plates).
An orange is an interesting hybrid: from the outside, it appears to be made from shells, growing over time.But on the inside, it forms a symmetric internal structure with wedges – good for evenly distributing seeds, right?We could analyze it both ways.
3.1Exploring The 3d Perspective
In the first lesson we had the vague notion that the circle/sphere formulas were related:
With our X-Ray and Time-Lapse skills, we have a specific idea for how:
- Circumference:从一个戒指开始。
- Area:用一个环一个环的时间推移制作填充光盘。
- Volume:把圆做成一个盘子，然后逐盘延时制作一个球体。
- Surface area:X-Ray the sphere into a bunch of shells; the outer shell is the surface area.
Wow!We now have detailed descriptions of how one formula is related to the other.We know, intuitively, how to morph shapes into alternate versions by thinking “Time-Lapse this” or “X-Ray that”.We could even work backwards: starting with a sphere, we can X-Ray it into plates, and then take one plate and X-Ray it into rings.
3.2The Need For Math Notation
You might have noticed it’s getting harder to describe your ideas.We’re reaching for physical analogies (rings, boards, wedges) to explain our plans: “Ok, take that circular area, and try to make some discs out of it.是的,像这样。Now line those discs up into the shape of a sphere…”.
I love diagrams and analogies, but should they berequiredto explain an idea?可能不会。
看看数字是如何发展起来的。At first, we used very literal symbols for counting: I, II, III, and so on.最后，我们意识到像V这样的符号可以代替iiii3，更棒的是，每个数字都可以有自己的符号。(The number “1” reminds us of our line-based history.)
Math notation helped in a few ways:
- 符号比单词短。Isn’t “2 + 3 = 5” better than “two added to three is equal to five”?Fun fact: In 1557, Robert Recorde invented the equals sign, written with two parallel lines (=), because “noe 2 thynges, can be moare equalle”.(我agrye !)
- The rules do the work for us.With Roman numerals, we’re essentially recreating numbers by hand (why should VIII take so much effort to write compared to I? Just because 8 is larger than 1?Not a good reason!).Decimals help us “do the work” of expressing numbers, and make them easy to manipulate.So far, we’ve been doing the work of calculus ourselves: cutting a circle into rings, realizing we can unroll them, looking up the equation for area and measuring the resulting triangle.Couldn’t the rules help us here?You bet.我们只需要把它们弄清楚。
- 我们把我们的思想普遍化。“2 + 3 = 5” is really “twoness + threeness = fiveness”.It sounds weird, but we have an abstract quantity (not people, or money, or cows… just “twoness”) and we see how it’s related to other quantities.算术规则是通用的，我们的工作是将它们应用到特定的场景中。
This last point is important.在学习加法的时候，你的老师可能会用字面上的苹果来表示二加三等于五。With enough practice, you started using abstract symbols without needing a physical example, and “2 + 3 = 5” made sense.
Calculus is similar: it works on abstract equations like\( f(x) = x^2 \), but physical examples are a nice starting point.When we see a shape like this:
we can actuallyseewhat Calculus does as we apply a technique, instead of pushing symbols around.Eventually, we can convert the shape into its corresponding equation and work with symbols directly.
所以，不要认为微积分需要一个真实的对象，就像加法需要苹果一样。它可以分析任何形状或公式(物理方程、业务场景、函数图)——形状更容易上手。