Lesson 10-3: Fundamental Properties of Volume (Day 124)

Chapter 2 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is "Flatland" -- and this refers to a famous book written by Edwin Abbott. He was a 19th century British headmaster of the school he once attended as a student. City of London School.

The main character of Flatland is a square -- A Square, to be precise. Rucker jokes that since the author's full name was Edwin Abbott Abbott, he might have had the nickname "Abbott Squared," abbreviated to "A Squared." And so that's how he came up with the character.

English teachers often criticize writers when they make their characters too flat, condemning them as mere "two-dimensional" characters. Well, Abbott's A Square actually is two-dimensional! This is where the title Flatland comes from -- it refers to A Square's two-dimensional world.

Now you may ask, why do we keep going back to two dimensions when Rucker's book is supposed to be about four dimensions? Well, as Rucker writes, "The fourth dimension is to three-dimensional space as the third dimension is to two-dimensional space. 4-D : 3-D :: 3-D : 2-D." And so we learn about the fourth dimension by analogy -- by understanding how A. Square struggles to learn about the third dimension, we can learn better about the fourth dimension.

And for A Square, he learns about the third dimension via the analogy 3-D : 2-D :: 2-D : 1-D. So the story begins with A. Square having a strange dream -- he finds himself in a one-dimensional world known as Lineland. Rucker writes:

A Square tries to tell the king [of Lineland -- dw] about the second dimension. The king doesn't understand, and asks A Square to move in the direction of the mysterious second dimension. A Square complies, and moves right through the space of Lineland. Naturally enough, the king simply perceives this "motion" as a segment that appears out of nowhere, stays for a minute, and then disappears all at once. The king denies the reality of the second dimension, A Square loses his temper, the dream ends.

And just as the second dimension is a mystery to the king of Lineland, the third dimension is a mystery to A Square himself:

The next evening A Square and his wife are comfortably sealed up in the safety of their home, when suddenly a voice out of nowhere speaks to them. And then, a moment later, a circle appears in the confines of their tightly locked house. It is A Sphere, come to teach A Square about the third dimension.

So what would happen if a four-dimensional being were to visit our three-dimensional world? Well, Rucker continues:

Reasoning by analogy, you can see that a four-dimensional creature would be able to reach into any of our rooms or cubbyholes, no matter how well they are sealed up. A four-dimensional creature could empty out a safe without cracking it, for the safe has no walls against the fourth dimension. A four-dimensional surgeon could reach into your viscera without breaking your skin.

Five months ago -- back when we were discussing Mandelbrot and fractals -- I mentioned an episode of Futurama, "2-D Blacktop." The Professor actually wanted to reach the fourth dimension in order to cheat in a race. But instead of going up a dimension, he went down a dimension, and he and his crew found themselves in a two-dimensional world similar to Flatland. The fourth dimension itself appears in another Futurama episode, "Mobius Dick" (as in Moby Dick) -- the crew found itself hunting a four-dimensional whale that seemed to appear and disappear at will, just like A Sphere in Flatland and A Square in Lineland.

After being inspired by both Rucker's book and the Futurama episode, I did finally purchase a book of Abbott's classic novella. I recommend it to anyone who wants to read about a fictional world that revolves around math and Geometry.

Rucker includes two puzzles in this chapter:

Puzzle 2.1:
It would seem that Flatlanders cannot have a complete digestive system in the form of a tube running the length of their bodies, for such a tube cuts them in half. Is there any way around this problem?

Answer 2.1:
One way to keep poor A Square from going to pieces would be to have the type of gut depicted here. The projections on the upper half grip the knobs on the lower half, keeping A Square's body together. Food is passed down along the gut in the fashion of a barge moving through locks in a canal, with one after another of the barriers momentarily opening.

By the way, Futurama provides a different solution to this puzzle -- the two-dimensional creatures in "2-D Blacktop" simply absorb their food!

Puzzle 2.2:
Abbott's Flatland is not really a close analogue of our world. For although our space is 3-D, we cannot move around freely in a 3-D space. Instead we must walk around on the surface of a sphere. What would be an analogous design for a 2-D world?

Answer 2.2:
We would, in strict analogy to our world, expect two-dimensional creatures to crawl around on the rim of a disk: their planet.

Today I subbed in an English class. I usually don't write about my subbing when it's for classes other than math, but this English class is significant because it's the Expository Reading and Writing Course for seniors planning to enter the California State University system. In other words, it's the English equivalent of the math course that our governor is holding a contest to create (and in which I want to participate). So we can look at today's ERWC class for hints as to what the proposed math course should look like.

The ERWC course consists of 12 modules. Like many Integrated Math texts, the workbook is divided into two volumes, one for each semester. Naturally, only the second semester workbook is available in the classroom today:

7. Bring a Text You Like to Class
8. Juvenile Justice
9. Language, Gender, and Culture
10. 1984 by George Orwell
11. Brave New World
12. Bullying: A Research Project

Wow -- the students have to read 1984 and Brave New World. That's a lot of dystopia!

Within each module are several reading passages, followed by several numbered activities. Module 8 has 26 such activities, while Module 9 has 35 of them. There are additional numbered activities for Rhetorical Grammar. So my plan is to organize my "Algebra III" course the same way.

By the way, I've subbed in some English classes that are well below senior year -- including some at the middle-school level -- in which the students are assigned ERWC workbooks. So apparently, the
ERWC program is so successful that it has developed material for the lower grades.

Last year, when I wrote about today's lesson, I once again devoted much of what I wrote to the CAHSEE -- a test that no longer exists. I decided that I'll post some of what I wrote about the test again this year -- at least the questions relevant to surface area and volume -- and edit out the sentences that specifically mention the CAHSEE test, replacing them with Common Core Standards:

So with that being said, this is what I wrote last year about today's lesson:

128. One-inch cubes are stacked as shown below. What is the total surface area?

So clearly we have a surface area problem. Notice that in some ways, yesterday's Dan Meyer problem may prepare the students for this problem:

134. The short stairway shown below is made of solid concrete. The height and width of each step is 10 inches (in.). The length is 20 inches. What is the volume, in cubic inches, of the concrete used to create the stairway?

Once again, notice that these surface area and volume questions are all based on the seventh-grade standards, in theory.

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

CCSS.MATH.CONTENT.7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

CCSS.MATH.CONTENT.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Today I will be doing Lesson 10-3 of the U of Chicago text, on the fundamental properties of volume, rather than Lesson 10-2, on the surface areas of pyramids and cones. Lesson 10-3 more naturally flows from yesterday's Meyer project. Then tomorrow's lesson can cover Lesson 10-5, which then naturally flows from both Lessons 10-1 and 10-3 -- in 10-1 we have surface areas of prisms, and in 10-5 we have their volumes.

(Also, I might add that Lessons 10-1 and 10-3 also flow naturally from Monday's 8-8 and 8-9. Both the formulas for a circle appear in the surface area formula of a cylinder -- the circumference of a circle leads to the lateral area of a cylinder and the area of a circle leads to the full surface area including the bases.)But some people might point out that this would confuse the students even more. Instead of doing all of the surface area formulas at once (as the U of Chicago does) and all of the volume formulas at once, we'd keep going back and forth between surface area and volume. But another argument is that it's better to do all of the prism formulas at once, then all of the pyramid formulas, and finally all of the sphere formulas.

The cornerstone of Lesson 10-3 is a Volume Postulate. The text even points out the resemblance of the Volume Postulate of 10-3 to the Area Postulate of 8-3:

Volume Postulate:
a. Uniqueness Property: Given a unit cube, every polyhedral solid has a unique volume.
b. Box Volume Formula: The volume of a box with dimensions l, w, and h is lwh.
c. Congruence Property: Congruent figures have the same volume.
d. Additive Property: The volume of the union of two nonoverlapping solids is the sum of the volumes of the solids.

Just as we derived the area of a square from part b of the Area Postulate, we derive the volume of a cube from part b of the Volume Postulate:

Cube Volume Formula:
The volume of a cube with edge s is s^3.

And just as we can derive the area part of the Fundamental Theorem of Similarity from the Square Area Formula, we derive the volume part of the Fundamental Theorem of Similarity from the Cube Volume Formula:

Fundamental Theorem of Similarity:
If G ~ G' and k is the scale factor, then
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.