Lesson 9-8: The Four-Color Problem (Day 98)

Today I subbed in an eighth grade science class. It's in my first OC district -- I've subbed at this school before, but not in this particular classroom. I will do "A Day in the Life" today as it's middle school -- and also because it's a science class. Yes, I'm returning to my old habit of comparing any middle school science class to what I taught (or failed to teach) at my old charter school.

8:30 -- This is the district where at middle schools, three periods meet in numerical order. But as it turns out, first period is the teacher's conference period.

Near the end of this period, a fire drill begins. Fortunately, there is no evacuation needed -- compare this to last week, when an evacuation occurred during my fourth period conference and students had to report to their first period class, so I suddenly had students.

9:40 -- Second period arrives. This is the first of two honors eighth grade science classes.

The students have a Go Formative quiz today. The regular teacher was hoping to set me up in her Go Formative for the day, but she can't -- that requires Formative Premium, which costs money. The same thing happened during my long-term assignment -- the regular teacher ended up giving me his password, since he doesn't have a premium account either.

The quiz is on Newton's Laws of Motion. Since these are honors students, most of them appear to finish the assessment quickly. The students have an online Equal and Opposite Reactions game to play, as well as a Bill Nye video on motion.

As I've been doing all week, I sing the Palindrome Song today. And I tell them that it's from Square One TV, a show that ended just before Bill Nye the Science Guy premiered in 1993. (Indeed, I suspect that my local TV station aired Bill Nye in the spot vacated by Square One TV.) Today's episode is actually the series finale, which first aired in 1998.

10:40 -- Second period leaves for snack break.

10:55 -- Third period arrives. This is the second of two honors eighth grade science classes.

It's just the luck of the draw that I get a conference followed by two honors classes today. There was also a sub yesterday, and he drew the three non-honors classes.

11:55 -- Third period leaves for lunch.

12:45 -- As is typical for this district, academic support begins. Students log into Zoom from home if they need extra help.

One girl does try to show up for support, but I miss her -- I only notice her comment in the Zoom chat a few minutes after she leaves. Then again, there's no guarantee I could have helped her anyway. (Of course she's not in the honors classes, so it's possible she had a question about her quiz tomorrow.)

2:00 -- Academic support ends, thus completing my day.

OK, let's compare this class to science at the old charter school. Like most science classes in California these days, this class uses the Preferred Integrated Model of the Next Generation Science Standards. I was at the charter school during the transition to the new standards, so the correct thing for me to do was teach Physical Science to eighth grade my first two years there. By my third year, the first Integrated cohort makes it to eighth grade -- but by then there's a separate science teacher. Then again, Newton's Laws count as Physical Science, and so it's likely that I would have taught this to eighth grade each year anyway.

In the classroom, I find texts for the four units taught in Integrated Science 8, albeit in Spanish. I use Google to translate the names of these four units:

1. Change over Time

2. Energy and Movement

3. Understand the Waves

4. Human Beings and Their Place in the Universe

The texts are titled California Inspire Science, published my McGraw-Hill. Notice the irony here -- for the first few years of NGSS, the district had no science texts for the new standards, and so science teachers had to cover the whole curriculum online. And now the new science texts are finally here -- right when the pandemic begins.

The first unit is all about how our planet changes over time, as well as evolution of species. The second and third units are Physical Science -- the second unit was clearly taught to eighth graders both before and after the dawn of NGSS, but the unit on waves is fairly new. And as I've mentioned on the blog before, the last unit on astronomy was once taught to sixth graders as part of pre-NGSS Earth Science.

At my old charter school, if I followed the Illinois State text, then I would have taught the four main units of Physical Science as listed in the NGSS to my eighth graders. These four units are:

1. Matter

2. Motion

3. Energy

4. Waves

That's right -- energy and movement count as one unit in Inspire but two units in Illinois State. And with motion being the second unit for both Inspire and Illinois State, I would have reached it around the same time as this district -- just after Thanksgiving.

Here's what the motion unit would have looked like at the old charter school -- including the science projects that appeared in the Illinois State text. This schedule follows the four-week cycle that I suggested in previous science posts:

Week of November 28th, 2016 -- MS-PS2-1: Apply Newton's Third Law .. (Unit 4 Science Test covering standards MS-PS1-6 and 2-1)

Week of December 5th -- Science Projects (Water Bottle Rocket, Egg Crash Box)

Week of December 12th -- MS-PS2-2: Plan an investigation to .. forces ..

Week of January 10th, 2017  -- Science Projects (create your own forces project)

Week of January 17th -- MS-PS2-3: Ask questions about .. magnetic forces (Unit 5 Science Test covering standards MS-PS2-2 and 2-3)

Week of January 23rd: Science Projects (How can electricity cause magnetism?)

Week of January 30th: MS-PS2-4: Construct and present arguments .. gravitational ..

Week of February 6th: Science Projects (How do forces act on objects? Weight on Other Planets)

Week of February 13th: MS-PS2-5: Conduct and investigation and ... fields (Unit 6 Science Test covering standards MS-PS2-4 and 2-5)

Week of February 21st: Science Projects (sandpaper friction, magnet-levitated trains)

Recall that here, a "unit" is simply my own way of numbering the four-week chunks of time during which I give two science projects and one science test.

Some of the projects listed here might not work during the pandemic, but they should have been feasible back in 2016-17. Oh, and I could have shown the students some Bill Nye videos, just like this teacher -- except that I could have sung the Bill Nye songs just as I now sing Square One TV. (The song in today's Bill Nye episode is "All in Motion.")

Lecture 12 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Winning Ways -- It's Your Move." Here is a summary of the lecture:

• In our previous lecture, we focused on the game of Chess, but of course there are many other games of strategy that lead to interesting mathematical questions. And armed with the proper insights, these games and questions can be conquered by the mathematically inclined player.
• There are three types of games -- ones where the last player to move wins (Chess, Checkers, NIM, Cram, Domineering), ones where the winner is the first to create a structure (Tic Tac Tow, Hex, Connect Four), and ones where the winner accumulates the most (Scrabble, Go).
• In Cram, players take turns placing dominoes on a 4 * 8 chessboard. Northrop's Game is played on the full board, where each player has eight checkers on his first rank. Players can only move forward along each file and can't pass an opponent. The last player to move is the winner.
• NIM is a two-player game. There are piles of coins. You can take as many as you want from any one pile. The player who takes the last coin is the winner. For example, if there are three piles with 7, 5, 4 coins, you might take 4 coins from pile 1, then I take 2 coins from pile 2, and so on.
• Charles Bouton came up with a winning strategy in the early 20th century. If there are two unequal piles and it's your turn, then take enough to the larger pile to make them equal. Then you are guaranteed to win. From a good position, all opponent moves go to a bad position.
• If there are three or more piles, use the binary system. For example, suppose the three piles have sizes 13, 10, 6. In binary, 13 = 8 + 4 + 1, 10 = 8 + 2, 6 = 4 + 2. A good position contains an even number of each power of two -- since there's a single 1, you should take 1 from the 13 pile.
• As it turns out, Northrop's game is just NIM in disguise. The number of spaces between checkers corresponds to the number of coins in each pile. And even Cram is a version of NIM -- in both games, the players take turns, the last move wins, and both are finite impartial games.
• Sprangue and Grundy proved the following theorem: Every finite impartial game where whoever makes the last move wins can be transformed into a game of NIM. We can assign a NIM number, or "nimber," to each set of empty squares.
• Chomp is an impartial game that is not equivalent to NIM. Players take turns chomping off a square on a rectangular chessboard, which eliminates all squares above it and to its right. T he player that takes the last square loses.
• There is an existence proof that a winning strategy exists for the first player. Suppose taking the square in the upper right is a good move, then the first player should do so. If it's bad, then the second player has a good response to it -- which the first player preempts by doing so herself.
• In a variant of Hex, red goes first. Then blue can either make the next move, or decide this one time to switch places and play as red. So red shouldn't start too strongly (such as in the center) or blue will want to switch. This is called the pie rule.
• Indeed, I cut, you choose is a strategy to divide a cake into two pieces. If there are three or more people wanting cake, then the first player moves the knife until someone says stop. That person takes 1/3 of the cake, and then this is reduced to the two-person case.
• Connect Four is a more complex version of Tic Tac Toe. White and Black take turns dropping checkers into a vertical 7 * 7 board. The winner is the one who places four in a row. White should begin in the center to force a win. If White doesn't, Black should start one spot closer to center.
• Here are some Connect Four tips: appreciate the center column, create (and guard against) double threats, make forcing moves, and be patient and ponder parity.
• Computers can beat humans in so many games. One game that has yet to be cracked by a computer is Go.
• Games and mathematics are alike in many ways -- in both cases, we learn how to apply rules in the correct order in order to reach a goal. This is why some schools teach puzzles such as Rubik's Cube to young students.

This completes another highly enjoyable course. I've seen many other MTBoS teachers introduce puzzles to their math classes for the same reason. And today's U of Chicago lesson is, in many ways, just one big puzzle.

Lesson 9-8 of the U of Chicago text is called "The Four-Color Problem." This lesson doesn't appear anywhere in the modern Third Edition, because this is one of those "extra" lessons that we include mainly for fun.

In the past, I've mentioned several books and lectures which discuss the Four-Color Conjecture. One of these was David Kung's lectures. [2021 update: Let me snip out David Kung's lectures here, since we're now watching Arthur Benjamin's lectures.]

I wish to link to a member of MTBoS who actually teaches the Four-Color Theorem in class:

http://eatplaymath.blogspot.com/2015/10/the-four-color-theorem-and-pumpkin-time.html

Lisa Winer is the author of this post that is over five years old. She doesn't specify in what state she lives, nor does she make it easy for me to figure out what grade or class this is.

Anyway, in Winer's class, she uses the term "chromatic number" to describe the fewest number of colors required to fill in a map. The Four-Color Theorem, therefore, states that the chromatic number of any planar map is four. On a Mobius strip the maximum chromatic number is six, and of course on a torus the maximum is seven.

It's time to return to Euclid. Of course, he writes nothing about Four Colors or reflections across an axis, and so we proceed with the next proposition instead:

Proposition 11.

To draw a straight line perpendicular to a given plane from a given elevated point.

Propositions 11 and 12 are both constructions. Many of Euclid's propositions are constructions -- indeed, "The First Theorem in Euclid's Elements" (that is, Proposition I.1) featured in Lesson 4-4 is actually a construction.

Classical constructions are performed with a straightedge and compass, and David Joyce writes about the importance of actually proving constructions as theorems. But it's awkward to ask our students to perform a construction in three dimensions.

In this construction, we have a point A and a plane P, and we wish to construct the line perpendicular to P through A. How can our students do this? Is P the flat plane of the paper and A a point floating up in space?

It might be interesting to attempt Euclid's construction in the classroom. Here's how: We choose A to be a point on the ceiling and P is the plane of the floor. Thus our goal is to draw a point on the floor directly below A.

The key to this construction is to hang a rope from point A -- a rope that should be longer than the room is high. We can pull the rope at any angle and double-mark the points where the rope is touching the floor. I say "double-mark" because the point on the floor (where the rope touches) is marked (say with chalk), and then the point on the rope (where the floor touches) is marked (say with a piece of tape). The rope now can serve as a compass -- the point of the compass is at A, and the opening of the compass is set to the distance between A and the tape. The locus of all points on the floor that are the same distance from A as the point marked on the floor is a circle, and the locus of all points on a given line on the floor that are the same distance from A is a pair of points. So if we have a point (say B) drawn on a line on the floor, then we could find the unique point C on that line such that AB and AC are congruent.

All the lines on the floor can be drawn in chalk. There will be some plane constructions drawn on the floor as well, so we could use a large compass where the pencil has been replaced with chalk.

OK, so let's begin the construction. We start by drawing any line on the floor, and then we label any point on that line B. We now find C on this line exactly as given above -- we double-mark B on both the rope and floor, and then swing the rope to find C such that AB = AC.

Now we use the chalk compass to find the perpendicular bisector of BC. The midpoint is D.

Then we double-mark D with a second piece of tape, and then find the point on the last line we drew (that is, the perp. bisector of BC), to be labeled E, such that AD = AE. The second piece of tape must be higher up than the first since AD < AB, and so there's no danger of confusing which piece of tape is which.

Finally, we find the perpendicular bisector of DE. The midpoint is F. Euclid's G and H are any points on this last line -- their location doesn't matter. Only F is relevant here. AF is the desired line through A that is perpendicular to the plane of the floor, and F is directly below A.

Of course, this whole construction seems silly because of gravity. We can just hang a rope freely from A, label the point where the rope touches the ground F, and then we're done! The difference, of course, is that Euclid's three-dimensional space isn't physical space, and so there's no direction that's "favored" because of gravity or any physical force.

And so I'm not quite sure how David Joyce has in mind when he says he wants "the basics of solid geometry" to be taught better. Does he include Euclid's spatial constructions -- does he really want students to perform them? Or maybe he merely desires that students visualize the proofs in their minds while looking at the proof.

(Do you remember Euclid the Game, which is played on computers? Maybe in higher levels, players can make three-dimensional constructions that are difficult to perform in the real world!)

By the way, we can still modernize Euclid's proof:

Given: the segments and angles in the above construction.
Prove: AF perp. plane P

Proof:
Statements                              Reasons
1. bla, bla, bla                         1. Given
2. BC perp. plane (ED, DA)   2. Proposition 4 from last week
(Call it plane Q. In the classroom, Q is an invisible plane parallel to a wall.)
3. GH | | BC                            3. Two Perpendiculars Theorem (planar version)
4. GH perp. plane Q               4. Perpendicular to Parallels (spatial, Tuesday's Prop 8)
5. AF in plane Q                     5. Point-Line-Plane, part f (A, D, F all in plane Q)
6. GH perp. AF                       6. Definition of line perpendicular to plane
7. AF perp. plane (GH, DE)   7. Prop 4 (AF perp. DE is part of "Given")
8. AF perp. plane P                 8. From construction (both lines were drawn in plane P)

It might be tricky to reconcile this proof with the "rope" construction from above. In Euclid's construction, AD is designed to be perpendicular to BC, likewise AF is perp. to DE. Both of these perpendicular constructions technically occur in planes other than the floor -- yet earlier I direct you to perform perpendicular constructions on the floor -- which is the wrong plane.

But think about it -- given a point A and a line, how do we construct a line through A perpendicular to the given line? The answer is that, using the compass, we find points B and C on that line that are equidistant from A, and then find the perpendicular bisector (in that plane) of BC.

But technically, all we really need is D, the midpoint of BC. Then the line through points A and D is automatically the perpendicular bisector of BC in the correct plane. It doesn't matter how we obtain the midpoint D -- all that matters is that we find it. This includes finding the perpendicular bisector of BC in the wrong plane (that is P, the plane of the floor). This is why Euclid is able to assert and use statements like AD perp. BC in his proof, even though this isn't obvious from our ropes. (And as it happens, the perpendiculars in plane P appear later in the proof anyway, so we might as well construct these.) In the end, let's just stick to the Four-Color Theorem and two-dimensional reflections.

Today I'll post my second pandemic-friendly activity (especially after posting yesterday's making surfaces activity -- that's what the lesson was about, but it's hard to do in a pandemic). I'm not sure how to find a Four-Color activity online -- but when in doubt, just go to Desmos:

https://www.desmos.com/calculator/r1rwm2lymo

So students can create their own maps in Desmos and demonstrate how to four-color them.

I'll keep last year's worksheet on reflections -- but then again, I suspect that these could be "desmofied" as well.

Lesson 9-7: Making Surfaces (Day 97)

Today I subbed in a middle school math class. It's in my new district. Since I really do teach today (as opposed to watching the students see their teacher on Zoom), it's worth doing "A Day in the Life" today.

8:45 -- My day begins with -- fifth period? Actually, this school has a schedule similar to those at the other middle schools in the district (including my long-term school). But the periods are numbered differently -- Cohort A (early alphabet) has periods 0-4, while Cohort B (later alphabet) has periods 5-9. Zero and fifth periods correspond, as do first and sixth periods, second and seventh, and so on. (They could have easily numbered the periods 1-5 for both cohorts like most other schools.)

And fifth period isn't a math class, but an ASB elective. Some of the students create posters and hang them all around the school.

Taking attendance is confusing today -- but it's not because in-person students are in "fifth period" while at-home students are in "zero period." It's really because some students are enrolled in a second elective -- and there aren't enough periods in the day for students to attend both of them everyday. I believe that there are some rules that determine which one these double-elective students are supposed to attend each day, but I can't figure out what these rules are.

Moreover, zero and fifth periods together have a total of 45 students -- the largest class by far that I've taught in the pandemic era. Roughly a third of these students each are on Cohort A, Cohort B, and opting out of hybrid. These leaves about 15 students packed into the activity room. Fortunately, some students help create posters outside, and so the room isn't actually that crowded.

9:40 -- Fifth period leaves the activity room, and sixth period arrives at the math room. This is the first of three eighth grade classes, and the only class with an aide.

As it turns out, Math 8 is still in APEX Unit 4b, on solving systems of equations. The pacing guide always had this unit spanning all of January. I taught the first week of this unit while still at the long-term position, and now they are in the last week of the unit, preparing for the test on Friday. When the students finish the worksheet, they also have a practice test on Quizizz.

One thing I notice about today's review worksheet is that there are questions on solving systems by graphing or substitution, but not by elimination. And -- since you know that I'm still keeping up with what my long-term kids are learning now -- they aren't doing elimination either. I'm not sure whether APEX actually omits elimination, or the teachers reached the consensus to eliminate elimination.

In some ways, this makes me feel a little less guilty for not reaching elimination with the eighth graders at the old charter school four years ago. But then again, elimination appeared in the Illinois State text, so technically I was supposed to teach it.

As I've been doing all week, I sing the Palindrome Song to these students.

10:35 -- Sixth period leaves for snack break.

10:45 -- Seventh period arrives. This is the second of three eighth grade classes.

11:40 -- Seventh period leaves and eighth period arrives. This is the last of three eighth grade classes.

By the way, since I wrote how I'm still following my long-term class, let me reveal what they're doing this week. Instead of worksheet, they have a Google Slides activity on solving systems. They must create five slides -- introduction, what a system is, graphing, elimination, and a pizza problem. The students must complete this assignment before their test, which is also on Friday.

12:35 -- Eighth period leaves for lunch.

1:15 -- At this school, after lunch is tutorial. Just like at my long-term school, the students are assigned to go to a different period each day -- and today, it just happens to by fifth period ASB.

But this leads to confusion. First, I start to pack up my Chromebook and sub folder and head over to the activity room -- only to have some students show up outside my door. They tell me that they're not quite sure whether to go to the math room or ASB room on tutorial days. The last time fifth period was assigned to tutorial was last Tuesday, but that was a Cohort A day -- and the last time before that was before the schedule changed at all middle schools. Thus today was the very first day that these particular kids were on campus on a day that tutorial was tied to fifth period -- and so they genuinely don't know where to go. Well, since there are kids outside my door anyway, I let them in and hold tutorial in the math room.

But then there was a problem with double-electives again. Students are told that if they don't attend one of their electives during the day, then they should attend it during tutorial instead. And so I see several in-person students who are absent earlier in the day. And, as it turns out, some of these kids are on my zero period roster -- meaning that they're in Cohort A, which shouldn't even be on campus today!

So there might have been fifteen students in the classroom, maybe slightly more -- but there aren't 15 desks in the room, since some were removed during the pandemic. And so some students end up sitting on the floor.

And moreover, my last class of the day is Math Skills, which counts as an elective -- and for some students, it's their double-elective. And so some of these students try to come in during tutorial, since they won't be able to come during the last period! I tell them that my classroom is full, and so they should just attend their other elective. But I don't know how to mark this on attendance.

The students in this room can't work on posters, and so I just sing the Palindrome Song again. Then two girls put on their own dance performance. And with a few minutes left in tutorial, I have them choose an extra song for me to perform -- they choose "Count on It," also from Square One TV.

1:40 -- Tutorial ends and ninth period begins. This is a Math Skills class.

Just like at my long-term school, Math Skills is the last class of the day. It's also the smallest -- there are only three in-person students, all girls, today. Some students have opted out of hybrid and are attending class online, while I may have chased some double-elective students away during tutorial. Also, the ninth period cohort is much smaller than the corresponding Cohort A class (fourth period).

And before you ask, no, these students don't do 60 minutes of ST Math or Dreambox in Math Skills the way we did at my long-term school. Instead, the regular teacher directs me to go over the worksheet and Quizizz just as I do in the other classes. The three girls tell me that they already finished both assignments in their real Math 8 class, and so they work on English or other assignments.

2:35 -- Ninth period leaves, thus ending my day of subbing. (Just as the other middle schools, the students have one period of independent study PE.) Of course, I head directly to the attendance office to explain all the problems I had with attendance, electives, and double-electives.

OK, so today my schedule was very confusing. A few years ago, I subbed an entire week in a class that was very similar to today's -- the regular teacher also had five classes, including three eighth grade math classes (albeit Algebra I, not Math 8), a Math Skills class, and an ASB class. Of course, that was both in a different district and before the pandemic. (You can read more about that class on the blog -- I was there in mid-November 2018 and then again on Pi Day 2019.)

At that school there was also several double-elective students. But over there, the solution to the double-elective problem is to assign such students a zero period -- as in a real zero period that starts an hour before the rest of the students arrive, not today's so called zero period (that was really first period).

While yesterday's high school had a genuine zero period (and indeed, my long-term middle school also had a real zero period), today's middle school likely wanted to avoid having students attend an extra hour at school. So instead, they came up with this confusing schedule where students go to different electives on different days.

I'm not quite sure how the school could improve this schedule (other than offer the extra period before school starts, like most other schools). Perhaps the tutorial schedule could be improved -- notice that there are five possible periods for tutorial and five days of the week. Yet the tutorial periods don't correspond directly to days of the week -- instead, they rotate every fifth day. Thus holidays and monthly minimum days (like the one we had a few days ago on Monday) disrupt the correspondence.

Perhaps instead, tutorial should be tied to Periods 0 and 5 every Monday, Periods 1 and 6 every Tuesday, and so on. If there's a holiday, then that class simply doesn't have tutorial that week. Then ASB only has tutorial on Mondays -- and since Mondays are fully online, the problem of which room to go to for tutorial and having enough seats in that room rarely occurs.

Lecture 11 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Mathematics and Chess." Here is a summary of the lecture:

• In this lecture, the professor plans to talk about one of the world's oldest games, namely, the game of chess. Despite not requiring many calculations, the game is still very mathematical, because mathematics is the study of patterns.
• Two of the greatest grandmasters of all time -- Emanuel Lasker and Max Euwe -- also held doctorates in mathematics. And some of the greatest mathematicians -- Euler and Gauss -- were greatly interested in Chess puzzles.
• The king can move one square in any direction. The rook makes either horizontal or vertical moves of multiple squares. The bishop makes diagonal moves. The queen makes horizontal, vertical, or diagonal moves. The knight makes an L-shaped move. 
• The Knight's Tour is a special problem -- can we move a knight around a chessboard so that it visits every square in as many moves? One possible solution: divide the chessboard into four quadrants, and divide each quadrant into four diamonds (up, down, right wheel, left wheel).
• Each player starts with eight pawns. Pawns move only forward. The first move can be one or two squares, but thereafter it moves only one square. It captures diagonally. If it reaches the eighth rank, it can be promoted to a queen.
• If the king is threatened, it's said to be in check. The next move must be to get it out of check, either by moving the king or blocking/capturing the checking piece. If it's impossible to get out of check, it's called checkmate, and the checking player is the winner.
• Here's where math comes in: a pawn is worth 1 point, a knight or bishop is worth 3 points, a rook is worth 5 points, and a queen is worth 9 points. The king is infinite. This is used to determine whether to exchange pieces -- so don't sacrifice a rook (5 points) for a bishop/pawn (4 points).
• Nowadays, computers can analyze millions of moves in a split second. Since Deep Blue in 1996, computers have grown in power. No human has beaten the top computer program since 2005.

Once again, chess is an advanced game that I can't fully describe on the blog in a short summary. The professor discusses strategies for the opening, middle game, and endgame.

Lesson 9-7 of the U of Chicago text is called "Making Surfaces." In the modern Third Edition of the text, making surfaces appears in Lesson 9-8.

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.

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 about four years ago today:

https://rawsonmath.com/2016/01/26/how-do-we-know-that/
https://rawsonmath.com/2016/02/07/more-3d-geometry/

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 2021, let's go back to Euclid, who gives definitions of the Platonic solids:

Definition 25.

A cube is a solid figure contained by six equal squares.

Definition 26.

An octahedron is a solid figure contained by eight equal and equilateral triangles.

Definition 27.

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

Definition 28.

A dodecahedron is a solid figure contained by twelve equal, equilateral and equiangular pentagons.

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:

Proposition 10.

If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles.

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. AD | | BE, BE | | CF                      4. Definition of parallelogram
5. AD = BE, BE = CF                       5. Parallelogram Consequences, part b
                                                              (opposite sides of a pgram are congruent)
6. AD | | CF                                      6. Transitivity of Parallels (Prop 9 from yesterday)
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.