As we already know, reflections reverse orientation. Indeed, reflections are unlike translations and rotations in that it's easy to take a real object and translate it or rotate it. But we can't very easily reflect a real object -- transform a right shoe into a left show, for example.
The basic idea of this chapter is that a reflection in n-dimensional space is equivalent to a 180-degree rotation in (n+1)-dimensional space. So we convert a transformation that's physically impossible, a reflection, into one that's physically possible, a rotation.
Rucker describes a new dream of A Square in which our quadrilateral hero visits the king of Lineland again -- only this time he rotates the monarch 180 degrees, centered at his midpoint. This is equivalent to a 1-D reflection.
Then A Cube visits Flatland and rotates A Square 180 degrees, which A Square's midsegment as the axis of rotation. This is equivalent to a 2-D reflection.
And so Rucker explains that it's possible to reflect a 3-D object like A Cube by rotating him 180 degrees in the fourth dimension. The center of a 4-D rotation is a plane -- the same plane as the mirror of the desired 3-D reflection. Rotating a right shoe 180-degrees in 4-D results in a left shoe.
Rucker ends the chapter by describing a "Necker cube." A cube is drawn just as it is in Lesson 1-5 of the U of Chicago text, in non-perspective -- except that none of the sides are dotted. Now it is ambiguous which side of the cube is the front. Rucker points out that there are two ways of viewing the Necker cube as an actual cube -- with one of the cubes the reflection image of the other.
Here are the puzzles in this chapter:
Puzzle 4.1:
A cube that intersects a plane at right angles makes a square cross section. Would it be possible to place a cube so that it intersects a plane in a triangular cross section? How? What other cross-sectional shapes could a cube show?
Answer 4.1:
In cross section, a cube can appear to be a square, a triangle, a rectangle, or a hexagon. This is illustrated in a drawing by Claude Bragdon.
Commentary: Cross-sections of 3-D figures appear in Lesson 9-4 of the U of Chicago text (where they are called "plane sections"). In particular, Exploration Question 28 in Lesson 9-4 is identical to Rucker's Puzzle 4.1. Of course, we skipped Chapter 9 altogether here on the blog.
Puzzle 4.2:
Here is a very remarkable Necker-type illustration. Can the little man see the beetle or not?
Answer 4.2:
This illustration was sent to me [Rucker -- dw] by Orville L. Parrinello of Brazoria, Texas.
Commentary: I'll throw the rest of this puzzle out -- it doesn't really work without the visual.
Puzzle 4.3:
Looking at a Necker cube can help us think about the fourth dimension. Can you think of a "Flatland Necker Illusion" that A Square could use to help think of the third dimension?
Answer 4.3:
If A Square were transparent, then he could seem to reverse when seen edge on, just as the transparent A Cube reverses. The reversal would seem sort of as if the Square were "pulled through himself," just as a left glove can be turned into a right glove by turning it inside out.
Commentary: Here is the best link I could find about the Necker cube. Rucker's Necker Cube has a face -- to represent A Cube's face -- rather than a red dot as in the link. (Just pretend that the red dot is A Cube's nose!)
Today is another activity day -- because I have so many links from back in February about the volumes and surface areas of prisms and cylinders.
https://edtechenthusiast.wordpress.com/2016/02/05/same-lesson-5-years-apart/
Joel Speranza is an Australian secondary math teacher. He wrote two versions of this lesson -- the original lesson five years ago where the students are assigned a block letter and must calculate its surface area and volume, and a new version which incorporates the block letters into a sign.
This is what I wrote to Speranza in the comments:
It’s nice to see how you were able to differentiate five years ago just by using the letters of your name JOEL — but I’m wondering about the students who were stuck with the question mark — which looks the most difficult!
I enjoyed reading about how this began as a spur-of-the-moment lesson in 2011 and will now be a full-fledged project. I don’t see any ways to improve the lesson off the top of my head. It appears that the signs could be created by a 3D printer, but that’s expensive and I doubt anyone has access to one.
I look forward to reading about how this project turns out, even more so because I’m especially interested in Geometry lessons.
And this is how he responded:
The question mark was an afterthought, for a couple of the students who nailed the J. It was super difficult for them, they really enjoyed it.
Definitely thinking I’ll go the 3D printer route, I know our school has a couple and I’ve been looking for an excuse to try one out.
https://matheasyaspi.wordpress.com/2016/02/10/tetris-jenga-reviewing-nets-surface-area-and-volume/
The anonymous author of this website is a middle school teacher. Considering that this author writes:
I am an engineer turned math teacher on a mission to engage more girls in STEM.
therefore I will take a guess and use female pronouns to refer to the author. (This isn't to say that we males aren't interested in engaging more girls in STEM, but my guess is that the best person to inspire the girls is a female role model.)
Now the first project is what she created back in February, so naturally it referred to the holidays that took place during that month -- Chinese New Year and Valentine's Day. Because we really don't want to have a project celebrating February holidays in March, I provide a second link to her site -- but notice that this one requires the use of special "Tetris Jenga" blocks.
Both projects require students to find nets, surface areas, and volume. Nets appear in Lesson 9-7 of the U of Chicago text. Notice that even though we skipped Chapter 9 of the text, the ideas from that chapter keep coming up over and over again in Chapter 10, as we see in this post. By the way, the unfolded hypercube (eight cubes glued together) from yesterday's Rucker chapter is actually a 3-D net of a 4-D object, just as the crucifix is a 2-D net of a 3-D cube.
Indeed, I was considering posting Speranza's project today, but I decided to go with the one from the Math Easy as Pi site -- especially since it does incorporate a lesson we skipped. I scanned these pages from her site, but you may prefer to get them directly from the source linked above -- especially since she posted them as .pdf files that may be more convenient. I could have posted just a single worksheet, but the author intends these to be used as a foldable.
The foldable contains only nets and surface area -- but we taught volume yesterday, so you might want to have the students find the volume as well. I wanted to add volume in, but I didn't want to modify a project created by another person.
Some schools take the week before Easter (Holy Week) for spring break -- but the district whose calendar the blog follows takes the week after Easter (Easter Week or Bright Week) off. So we still have a week remaining in the Long March -- see you next week!
The anonymous author of this website is a middle school teacher. Considering that this author writes:
I am an engineer turned math teacher on a mission to engage more girls in STEM.
therefore I will take a guess and use female pronouns to refer to the author. (This isn't to say that we males aren't interested in engaging more girls in STEM, but my guess is that the best person to inspire the girls is a female role model.)
Now the first project is what she created back in February, so naturally it referred to the holidays that took place during that month -- Chinese New Year and Valentine's Day. Because we really don't want to have a project celebrating February holidays in March, I provide a second link to her site -- but notice that this one requires the use of special "Tetris Jenga" blocks.
Both projects require students to find nets, surface areas, and volume. Nets appear in Lesson 9-7 of the U of Chicago text. Notice that even though we skipped Chapter 9 of the text, the ideas from that chapter keep coming up over and over again in Chapter 10, as we see in this post. By the way, the unfolded hypercube (eight cubes glued together) from yesterday's Rucker chapter is actually a 3-D net of a 4-D object, just as the crucifix is a 2-D net of a 3-D cube.
Indeed, I was considering posting Speranza's project today, but I decided to go with the one from the Math Easy as Pi site -- especially since it does incorporate a lesson we skipped. I scanned these pages from her site, but you may prefer to get them directly from the source linked above -- especially since she posted them as .pdf files that may be more convenient. I could have posted just a single worksheet, but the author intends these to be used as a foldable.
The foldable contains only nets and surface area -- but we taught volume yesterday, so you might want to have the students find the volume as well. I wanted to add volume in, but I didn't want to modify a project created by another person.
Some schools take the week before Easter (Holy Week) for spring break -- but the district whose calendar the blog follows takes the week after Easter (Easter Week or Bright Week) off. So we still have a week remaining in the Long March -- see you next week!
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