Before we get to similar figures, notice that today marks the release of A Wrinkle in Time. I've mentioned this movie several times on the blog, because it mentions the higher-dimensional figure known as a "tesseract."
I still haven't decided whether I'll watch the movie or not. Based on its Rotten Tomatoes score of 42%, the film might not be worth watching. Nonetheless, it's a movie which mentions Geometry, and so I do wish to revisit the movie in that light.
This is what I wrote back in December about A Wrinkle in Time and the tesseract:
"Let's look at the worlds which are created by the idea of dimensions. A mathematical world can exist on a single point, on a single line, on a plane, in space, in a hypercube (tesseract)."
This is the only page of the section "The Worlds of Dimensions." By the way, the name "tesseract" for the 4D hypercube appears in Madeleine L'Engle's Wrinkle in Time. (L'Engle refers to the tesseract as 5D, because she's including time as Einstein's fourth dimension.)
Here are some excerpts from this Pappas page:
"Imagine your world and your life on a flat plane. Three dimensional creatures can invade your world without you even knowing by simply entering your domain from above or below. Dimensions beyond the third have always been intriguing. The cube was one of the first 3D objects to be introduced into the fourth dimension by becoming a hypercube. Computer programs have even been devised to derive glimpses of the fourth dimension by picturing 3D perspectives of the various facets of the hypercube."
There is only one picture on this page. It shows all the figures, or worlds, mentioned in the second sentence -- from the point to the hypercube. Its caption simply reads: "The four dimensions."
OK, so a "tesseract" is just another name for a hypercube. What does this have to do with anything in the movie? Here's a link that explains it:
https://www.thewrap.com/wrinkle-time-tesseract-sound-familiar/
A tesseract is the literal “wrinkle in time” from the title, which is also a wrinkle in space. While “A Wrinkle in Time” keeps its tessering fairly simple, the idea is that you use your mind to fold the fabric of space together to bridge two faraway points. In other words, tessering creates a so-called Einstein-Rosen Bridge, a concept predicted by Albert Einstein as part of his theory of general relativity.
According to the link, other movies such as Captain America and Interstellar also mention tesseracts in their plots. In all three films, the idea is the same -- characters wish to travel to places that are hundreds or thousands of light years away, so they must use a tesseract to get there.
Here's another link, written by Ivy League mathematician Jessica Weare twenty years ago:
http://www.math.brown.edu/~banchoff/Yale/project12/
Here Weare quotes Rudy Rucker -- whose book The Fourth Dimension I've also discussed earlier on the blog:
"James Clerk Maxwell (1831-1879), founder of the modern theory of electromagnetism, wrote [that] 'Whatever difficulties we may have in forming a consistent idea of the constitution of the aether [proposed invisible 'glue' of the universe, which some mathematicians have purported to exist between objects in space, both on large and small scales], there can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform body of which we have any knowledge.'"
Jessica quotes L'Engle's first mention of math in "The Tesseract" chapter:
"That, of course, is the impractical, long way around. We have learned to take short cuts wherever possible."
"Sort of like in math?" Meg asked.
"Like in math." Mrs. Whatsit looked over at Mrs. Who.
Weare reminds us that a tesseract is a 4D spatial figure -- it's only 5D if we include time:
"That is because you think of space only in three dimensions," Mrs. Whatsit told her. "We travel in the fifth dimension. This issomething you can understand, Meg. Don't be afraid to try. Was your mother able to explain a tesseract to you?"This is where L'Engle's math gets a little confusing. Traditionally, the tesseract is a term used to describe the hypercube. The hypercube is an object in the fourth dimension, analogous to a point in the zero dimension, a line in the first dimension, a square polygon in the second dimension, and a cube in the third dimension. Thus, a "tesseract" is an object of the fourth dimension, not the fifth.
But if L'Engle had said, "We travel in the fourth dimension," some might count this as time travel, since she's counting time as the fourth dimension. Meanwhile, Jessica also criticizes the author's use of the phrase "square a dimension":
"Well, you'd square the second dimension. Then the square wouldn't be flat any more. It would have a bottom, and sides, and a top."
If we were to follow the pattern above, then we'd square the second dimension, or x^2. However, (x^2)^2 is x^4, not x^3. (This follows from the additive property of multiplying exponents) To go from one dimension (expressed in terms of x), you only need to multiply the previous dimension by x. This is not a drastic mistake, but it shows that the mathematics in A Wrinkle in Time were not L'Engle's primary concern. In many ways, the mathematical concepts in A Wrinkle in Time were used as a vehicle to provide an opportunity for L'Engle to express her religious beliefs.
And later on, she writes....
"That's right," Charles said. "Good girl. Okay, then, for the fifth dimension you'd square the fourth, wouldn't you?"
Well, no. You know by now that if we squared the fourth dimension we wouldn't get the fifth dimension. (x^4)^2 equals x^8, not x^5. To go to the fifth dimension from the fourth, you multiply the fourth dimension by x, as in (x^4) times x equals x^5. Is this getting more familiar now?
Weare mentions Edwin Abbott's Flatland, a two-dimensional world. Apparently Meg, in trying to get to the fourth (fifth) dimension, accidentally gets stuck in the second dimension instead. (An episode of Futurama has a similar plot.)
It's therefore interesting to think about what a wormhole would look like to a Flatlander. For beings in 3D space, a wormhole is a tesseract, but to 2D Flatlanders, a wormhole is a cube. We can imagine Flatland as a large sheet of paper. It takes a long time for a Flatlander (like A. Square) to get from one side to the other. But if the paper were folded up, a cube could connect one side to the other, so A. Square can travel through the cube to get to the other side faster. That's what L'Engle's tesseracts do for us in 3D space -- the idea is that the universe is folded (or "wrinkled") in such a way that a tesseract can take us from one side to the other.
The Brown link about contains some other interesting pages. "Educational Application" contains suggestions on how to incorporate ideas from the film into the classroom. Recall that this webpage is from 1998, back before the MTBoS, so the idea of finding lessons online was new. There's also a page called "Animation." The authors envisioned that a Wrinkle in Time movie would contain some animated scenes -- and indeed there are some CGI scenes in today's movie. I wonder whether the authors of this webpage are watching today's film -- and if so, whether they enjoy it or not.
Because of the timing of its release, A Wrinkle in Time is being compared to Black Panther. Both movies are considered to be of the Afrofuturist genre. (Oops -- well, I already mentioned religion in this post, and now here comes race. But race and religion are strong themes of the film.) Therefore, both movies are being used to promote math and science education. Disney is using proceeds from the Black Panther movie to fund STEM centers in several cities, including Oakland, California.
OK, that's enough about movies. Let's get back to similarity.
This is what I wrote last year about today's lesson:
Lesson 12-5 of the U of Chicago text is about similar figures. There is not much for us to change about this lesson from last year, except for the definition of similar itself. Recall the two definitions:
-- Two polygons are similar if corresponding angles are congruent and sides are proportional.
-- Two figures are similar if there exists a similarity transformation mapping one to the other.
The first definition is pre-Core, while the second is Common Core. The U of Chicago text, of course, uses the second definition. But that PARCC question I mentioned last week must be using the first definition, since it requires that we know what similar means before we can define dilations and ultimately similarity transformations.
In the U of Chicago text, the Similar Figures Theorem is essentially the statement that the second definition implies the first definition. We would actually need to prove the converse -- that the first definition (at least for polygons) implies the second. But the proof isn't that much different -- suppose we have two figures F and G satisfying the first definition of similar -- that is, corresponding angles are congruent and sides are proportional, say with scale factor k. Then use any dilation with scale factor k to map F to its image F'. Now F' and G have all corresponding parts congruent, so there must exist some isometry mapping F' to G. Therefore the composite of a dilation and an isometry -- that is, a similarity transformation -- maps F to G. QED
Today is an activity day. So I include the first part of last year's worksheet with a new worksheet based on the two Exploration Questions:
30. Scale models are objects that are similar to larger objects. Find a scale model. For the scale model you find:
a. What is the ratio of similitude?
b. A length of 1" on the model corresponds to what length on the larger object it models?
31. In Example 2, BC can be found using a theorem called the Law of Cosines. Look in some other book to find out what this "law" is.
For the first question, the teacher should bring some scale models to class. Even a cheap toy car, for example, might suffice.
For the second question, here's one book with the Law of Cosines -- the Glencoe Geometry text. In this text the Laws of Sines and Cosines are formally taught in Geometry. Of course, in practice students will research answers for both questions on the Internet.
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