This lesson is all about making nets that can be folded to form polyhedra and other surfaces. Some figures have much simpler nets than others.
Today is an activity day -- and fortunately, today's lesson naturally leads to an activity. As I wrote in earlier posts, our activities will be based on the questions in the Exploration section. The first such question is the following:
25. A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices. There are only five regular polyhedra; they are pictured here.
a. Determine the number of vertices of each regular polyhedron.
b. Determine the number of edges of each regular polyhedron.
Ah -- we've seen these before. The five regular polyhedra are also called the Platonic solids. I've mentioned these in previous posts -- three summers ago we explained why there are only five of them, and two years ago we discovered that there are six regular polytopes in four dimensions. The Platonic solids are the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron.
The Exploration section continues thusly:
In 26-30, use cardboard and tape to construct a model of the regular polyhedron from the net provided. The patterns below should be enlarged. Cut on solid lines, fold on dotted lines.
Many teachers have given Platonic solid lessons in their classes. Since I don't want to try to create the nets myself, I link to previously made lessons. The first page, based on Question 25 (counting the vertices and edges) comes from the following link -- an elementary school in Washington State:
http://wilderptsa.ourschoolpages.com/Doc/Math_Adventures/Platonic_Solids.pdf
Question 26-30, the nets themselves, come from the following link:
https://www.math-drills.com/geometry/net_platonic_solids.pdf
The Math Drills link provides two nets for the dodecahedron. I chose the second one, since it more closely resembles the net in the U of Chicago text. On the other hand, their icosahedron net is very different from ours in the U of Chicago text.
Several members of the MTBoS have had Platonic solid activities in their own classes. Let's link to some of them:
Our first link is to Pamela Lawson, a Maine charter high school teacher. She taught her class about the Platonic solids exactly two years ago today:
https://rawsonmath.com/2016/01/26/how-do-we-know-that/
https://rawsonmath.com/2016/02/07/more-3d-geometry/
Notice that these posts were part of the 2016 MTBoS Blogging Initiative. (And no, there's still no sign of a 2018 Initiative.) Lawson begins:
I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?
(She's teaching a 12-week Geometry course? That's right -- hers is one of the rare high schools that uses trimesters!) Of course, I'd already give a full explanation here on the blog, just after Independence Day in 2015. Let me repeat parts of that post here:
Legendre's Proposition 357 states that the sum of the plane angles that make up a solid angle must be less than [360 degrees]. He proves this essentially by "flattening out" the solid angle -- he takes a plane that intersects all sides of the solid angle and uses the previous Proposition 356 (which we've already proved here on the blog) to show that each plane angle of the solid angle is less than the same angle projected onto the new plane. A good way to visualize this is to imagine that the solid angle is formed at the vertex S of a pyramid -- the points A, B, C, etc., mentioned Legendre can be the vertices of the base of the pyramid, and the point O can be any point in the plane of the base -- for example, the center of the polygonal base.
I won't take the time to show the full proof of Proposition 357, but I will mention an application of this theorem. Suppose we want to figure out how many Platonic solids there are. Recall that a Platonic solid is a completely regular polyhedron -- all of its faces are congruent regular polygons. As it turns out, we can use Proposition 357 to find all of the Platonic solids.
We start with the equilateral triangle, with each angle measuring 60 degrees. Now each vertex of our Platonic solid forms a solid angle. We need at least three plane angles to form a solid angle, but there is an upper limit to how many plane angles there can be. Proposition 357 tells us that the plane angles must add up to less than 360 degrees, and since each angle is 60 degrees, there must be fewer than six of them (since 6 times 60 is 360). So there can be three, four, or five 60-degree plane angles. The Platonic solid with three 60-degree plane angles is the tetrahedron, with four is the octahedron, and with five is the icosahedron.
If we move on to squares with their 90-degree angles, we can have three 90-degree plane angles, but not four (since 4 times 90 is 360). Three 90-degree plane angles gives us the cube. Regular pentagons have 108-degree angles. Again, we can't have four of them (since 4 times 108 is more than 360), and three 108-degree angles gives us the dodecahedron. Regular hexagons have 120-degree angles, but 3 times 120 is already 360. Since each solid angle must contain at least three plane angles, we are done, since increasing the number of sides in the polygon only increases the angle. Therefore, there are only five Platonic solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Returning to 2018, let's go back to Euclid, who gives definitions of the Platonic solids:
We notice that the tetrahedron is missing. According to David Joyce, Euclid refers to the tetrahedron merely as a triangular pyramid. In Book XIII, he also proves that these are the only five Platonic solids -- and there, he refers the tetrahedron simply as "pyramid."
Since I don't wish to jump to Book XIII of Euclid, let's look at the next proposition here in Book XI:
As usual, let's modernize the proof:
Given: l, m intersect at B, n, o intersect at E, l | | n, m | | o (lines not all coplanar)
Prove: The angle between l and m is congruent to the angle between n and o.
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Choose A, C, D, F on l, m, n, o 2. Point-Line-Plane, part b (Ruler Postulate)
so that AB = DE, BC = EF
3. ABED, BCFE are parallelograms 3. Parallelogram Tests, part d
(one pair of sides is parallel and congruent)
4.
5. AD = BE, BE = CF 5. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
6.
7. AD = CF 7. Transitivity of Congruence
8. ADFC is a parallelogram 8. Parallelogram Tests, part d
(one pair of sides is parallel and congruent)
9. AC = DF 9. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
10. Triangle ABC = Triangle DEF 10. SSS Congruence Theorem [steps 2,2,9]
11. Angle ABC = Angle DEF 11. CPCTC
We can't help but notice that the six points A, B, C, D, E, F are the vertices of a triangular prism. And indeed, we see that the translation that appears in the U of Chicago definition of prism is the same translation that maps Triangle ABC to Triangle DEF.
Of course, this requires us to show that if two lines are parallel, then a translation must map one line to the other. I've alluded to the proof of this in posts from previous years, but I no longer include it as part of our curriculum.
Meanwhile, in this post I wish to think ahead to Pi Day. One of my favorite pi websites is the old Sailor Pi website, created by "Bizzie Lizzie" -- indeed I linked to it in previous Pi Day posts.
But sadly, the Sailor Pi website no longer exists. Last year, I had been looking forward to singing some of her pi songs in my class, but of course I left my classroom before Pi Day. On Pi Day itself, recall that I delivered pizza to my old class, but I didn't stay and sing any songs.
But I did write some of Bizzie Lizzie's old songs in a notebook that I purchased that day. One problem in my old class was that I didn't use music break to its full potential. I recall one day when I sang a song about Mean, Median, and Mode when my seventh graders needed to recall measures of center during a Monday lesson with the coding teacher. This was about a month after I'd written the song for my sixth graders when they first learned about measures of center.
This is something that I should have done more often -- when my students need to recall an old lesson from a few months earlier, sing an old song from a few months earlier to jog their memory. But I rarely did so, because I'd written the songs on poster paper and then packed them away when it was time for the next song. A notebook full of songs would allow me to recall songs more quickly. And so on Pi Day, I bought the notebook and wrote in all my old songs and some of Bizzie Lizzie's pi songs.
And now I'm glad I did it, since the Sailor Pi website is now defunct. (I made this sad discovery back on Third Pi Day in November, but I didn't blog about it until today.) But now I'm curious as to the current whereabouts of this mysterious Bizzie Lizzie -- maybe she posted her songs at a new website.
A Google search for Sailor Pi I made back in November returned the following Instagram page:
https://www.instagram.com/sailorpi/
The owner of this Instagram page calls herself Liz. Hmm, is this Liz, as in Bizzie Lizzie? It's logical to assume that this Liz, with a user name of @sailorpi, is indeed Bizzie Lizzie, the author of the old Sailor Pi website. Nonetheless, I sought more evidence to prove that these are the same person.
I looked back to some of Liz's old pictures. One photo, dated January 26th, 2014, is of a cake, and on it is written "Happy Big 3-0, Liz!"
We can't conclude that Liz's birthday is exactly on the 26th -- she could have celebrated a few days before her actual birthday, or she could have a few days after the party to post the photo. But it's safe to say that she turned 30 in late January 2014. For simplicity, assume that her birthday is the 26th -- which, as you might notice, is today's date. That's the real reason that I chose today to blog about the Liz mystery -- it's her 34th birthday.
(Oh, and speaking of science-related birthdays, there is a Google Doodle today for Wilder Penfield, an American-Canadian neurosurgeon. According to the doodle, he used burnt toast to more learn about the brain.)
How does this help us tell whether Liz is Bizzie Lizzie? Well, on the old Sailor Pi site, Bizzie Lizzie wrote a "2003 Season of Sailor Pi," which details her days as a college student. If Liz's 30th birthday was indeed 1/26/14, then she would have celebrated her 19th birthday in 2003. This is consistent with Bizzie Lizzie attending college in 2003.
As we look at some of Liz's other Instagram photos, we see her wearing pi blouses and celebrating Pi Day on March 14th. This suggests that the "pi" in her username @sailorpi really does refer to the number pi and not, for example, "Sailor P.I." So far, the evidence that Liz and Bizzie Lizzie are the same person is compelling, but I wanted one more piece of solid evidence.
Liz often took photos of her workplace. In June, five months after her 30th birthday, she posted photos of her last day working at Cable News Network (CNN) and first day working at Jet Propulson Laboratories (JPL) right here in Southern California.
Hmm, CNN is a news source, and it's possible to search the CNN website for articles. If Liz indeed worked at CNN, perhaps some of her articles are still archived there. And since she's enjoys Pi Day, maybe she'd have been called upon to write a Pi Day article for CNN.
We now perform another Google search for pi day site:cnn.com. The first two results are for Pi Day 2017 and Pi Day 2016, which are no good since we know Liz left in 2014. But the third result is dated Pi Day 2014, so this is promising. Here's a link to the article:
http://www.cnn.com/2014/03/14/tech/innovation/pi-day-math-celebrations/index.html
The author of this article is Elizabeth Landau. Hmm, Elizabeth...Liz...Bizzie Lizzie -- could it be? We now perform one last Google search for the name Elizabeth Landau. The first result is LinkedIn:
https://www.linkedin.com/in/elizabethlandau/
And in this post, we confirm that Elizabeth attended Princeton from 2002-2006, which places her in college in 2003, just as Bizzie Lizzie wrote on the Sailor Pi website.
Oh, and there's one more thing to clinch it -- Liz's hair on both LinkedIn and Instagram is red. On the old Sailor Pi website, she drew herself as a redhead playing the guitar and dancing to her songs.
The evidence is now overwhelming. Elizabeth Landau is indeed Bizzie Lizzie, the author of the old Sailor Pi website.
By the way, all of this makes me seem like a stalker. After all, who else would search various social networking sites just to find one particular person?
The reason I searched for Liz is that I want to keep track of her songs. On her old website, she would claim her songs as falling under a "copyright." And so I respect this copyright by searching for the author and citing her name when writing about her songs. I'm surprised that I was able to find Landau so easily -- it helps that she ultimately became a journalist. Indeed, according to LinkedIn, one her roles at JPL is "writing long-form articles about space," so she is still technically a journalist.
By the way, Landau has written some earlier Pi Day articles for CNN:
http://lightyears.blogs.cnn.com/2012/03/13/pi-day-how-3-14-helps-find-other-planets-and-more/
http://www.cnn.com/2010/TECH/03/12/pi.day.math/index.html
http://scitech.blogs.cnn.com/2010/03/12/geek-out-my-life-with-pi/
http://scitech.blogs.cnn.com/2009/03/13/pi-day-and-american-pi/
In the last of these, Landau wrote her song "American Pi," a parody of Don McLean's song "American Pie." I'd copied the words of this song from the old Sailor Pi site into my notebook. This 2009 version is slightly different from the original, but it's still recognizably the same song.
I wrote the lyrics to one more songs from the Sailor Pi website. Since these words are, as far as I know, no longer posted on the Internet, I'll post them here. The song is "The Digit Connection," a parody of Kermit the Frog's song "The Rainbow Connection" from The Muppet Movie. Here is the song -- with proper attribution, of course.
THE DIGIT CONNECTION
Copyright 1998 by Elizabeth Landau, aka Bizzie Lizzie
1st Verse:
Why are there so many debates about pi?
And what's on the other side?
Pi is a ratio of random proportions.
Its digits have nothing to hide.
So we've been told and some choose to believe it,
But I know they're wrong, wait and see!
Someday we'll find it, the digit connection,
Mathematicians, logicians, and me.
2nd Verse:
Who said that everything has some sort of pattern,
Consisting of nothing but math.
Somebody thought of that, and someone believed it.
Now we're all caught in its wrath....
What's so hypnotic in something chaotic,
And what do we think we might see?
Someday we'll find it, the digit connection,
The optimists, the theorists, and me.
All of us under its spell,
We know it must be math-e-magic...
3rd Verse:
Have you been half asleep? And have you heard voices?
I've heard them calling my name.
Is this the sweet sound that calls the young sailors,
The voice might be one and the same....
I've heard it too many times to ignore it,
Irrational, random, and free.
Someday we'll find it, the digit connection,
The lovers, the dreamers, and me.
3.1415926535 dot, dot, dot!
People of my and Landau's generation -- late X'ers and early Millennials -- often created our own websites on the early Internet. I remember seeing the website of one of my former classmates. She, like Landau, was also a huge Sailor Moon fan, and both girls recast their friends as various characters from that particular anime. To me, it's sad that nowadays, AOL and Geocities are disappearing, because youngsters are simply creating profiles on social networking websites like Instagram and LinkedIn rather than making their own webpages. Our generation showed much more creativity.
Anyway, I wish Elizabeth Landau a happy 34th birthday, wherever she may be. (Sorry, it just seems so strange to keep writing out her real name -- to me, she'll always be Bizzie Lizzie.) And I'll spend the rest of her birthday reading all the wonderful articles she links to on her LinkedIn page.
Here are the worksheets for today's activity:
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