Delta Pack the uncreative way (euclidea series)

Returning with a very brief installment of geometric constructions. Basically, I was just brute-forcing the constructions in this pack, so most of the cases where I managed the minimum move constructions were just because those were straightforward.  Reviewing the pack in preparation for this post, I did find a couple of ways to shave down some of the constructions, but I don't feel like there was a major breakthrough.

Here's a snapshot showing (almost) where the V stars are:

If you do some simple counting, you'll see that there is one more V-star missing. Since this somehow escaped me the first time, I'll leave its location as an exercise for you, too.

Favorites

I've mentioned the idea from 4.5 before and I still like that construction, even though it is very simple.

Constructing the two equilateral triangles also seemed nice. Constructing the inscribed and circumscribing circles from the triangle seems more like the "usual" direction of construction, so these reverse constructions appealed to me.

Compass only non-collapsing compass (Euclidea Series)

Since I've branched into this topic, I want to include some notes on additional references I've found and some results I've been able to get. First, an admission:

I'm still having trouble finding the intersection points of a circle and a line when the circle's center is on the line!

Another resource
James King (UW home page) has a nice session outline working through compass only constructions. Be warned, there are some spoilers for some of the Euclidea challenges in that material. Unfortunately, I can't tell from write-up when and where this was used. Maybe this NWMI meeting?

In Professor King's notes, he describes the construction I'm struggling with as "important and difficult," so I will have to redouble my efforts.

Non-collapsing compass from collapsing compass
As an aside, perhaps you have noticed that everyone else refers to the two types of compasses as collapsible and non-collapsible. Doesn't that terminology strike you as wrong, too? The point isn't that one type of compass is able to collapse, but rather that it always collapses when not drawing a given circle. Also, a common form of "non-collapsible" compass is able to collapse!

Just squeeze the legs together to collapse

Anyway, I was able to figure out the construction of a non-collapsing compass, just in time for Euclidea 13.2.

I'll give a construction below. For a more mild hint, the key is the ability to reflect points through a line we've already "constructed" (where we have two points on the line).

I've chosen to do this in GeoGebra so I can show a sequence, explain some of the reasoning, and grey out earlier parts of the construction that we don't need anymore. Check my work to make sure I didn't slip in any subtle straightedge moves!

Here, we are given points A, B, and C and want to construct the circle centered at B with radius AC.

First step is two circles centered at A and B with radii AB, giving us points D and E. The line DE will be the one we are reflecting through. Key is that A and B are the reflected images of each other through this line.

Next, we draw the circle at A with radius AC. This intersects our circle centered at B in point F. F is a special point, distance AC from A and AB from B, and distance DF from D.

That means the reflection of F through line DE is a point distance AC from B! We can find it using the fact that it will also be distance AB from A and distance DF from D:

It almost looks like C and F are the same distance from D. That isn't true and is entirely coincidental. Try it out yourself and use the move tool to push C around and see that this is the case. In particular, you might notice that we could have made another choice for F (it is a point of intersection of two circles, we could have taken the other) and that other choice is on the opposite side of AB from C.

I'll leave to you the satisfaction of completing the construction and drawing our target circle.

Also, the Euclidea challenge is formulated very slightly differently: instead of point C, we are given the starting circle centered at A. That means the construction is one move shorter than what I've shown here.

Beginning Restricted Constructions (Euclidea Series)

Note: This is a little bit out of order, but reflects the challenge on which we're currently working.

Choices of Construction Rules
The classic construction problems involve specific, restricted tools:

1. unmarked straight-edge: this tool is only able to determine a line that contains two existing points (previously given or already constructed). It has no length markings and only one side.
2. collapsing compass: this tool is only able to determine a circle given the center and a point on the circle. Again, both center and an included point must already be given earlier in the construction. Like the straight-edge, the compass has no way to indicate the length of the radius. (side note: when I was in HS geometry, we didn't require the compasses to be collapsing).
3. A plane as a writing surface and infinitely fine/precise pens: these allow us to identify the intersection points of the constructed objects (circles and lines).

I, and maybe you, for years considered these rules to be natural. However, there are several other construction rules that are at least as natural:

• Origami: moves based on folding paper. This is really a fascinating topic. Take a look at this Numberphile video for a taste.
• Straight-edge only constructions: for those unfortunate to have left their compass at home.
• Compass-only constructions: easy to imagine how ancients made a good compass, but how would they have gotten a really straight edge anyway?!

Several of the Euclidea challenges impose either straight-edge only or compass-only restrictions, which got me interested in this general topic again.

Examples in Euclidea
Unless I've missed some, the restriction challenges start officially in Theta pack:

• Drop a perpendicular (8.4): we are only given the straight-edge, but we are also given a circle.
• Mid-point (8.5): find the midpoint of a segment with the straight-edge and a parallel line.
• Segment trisection (9.7): same restrictions as 8.5
• Midpoint (13.1): this time, we've only got the compass, no straight-edge
• Some I've not yet unlocked: Tangent to Circle (13.5), Drop a Perpendicular (13.7), Line-Circle Intersection (13.8)

In addition, Zeta pack has a some challenges that, while not marked as restrictions, are good warm-ups: Point Reflection (6.1), Line Reflection (6.2), and Translate Segment (6.6).

Humans can forget things!

If you've made it to Zeta pack, you know that the collapsing compass is able to replicate the function of a non-collapsing compass, so that particular restriction doesn't change what is theoretically constructible (but it sure reduces a lot of extra steps and auxiliary objects in real constructions.)

It turns out that the straight-edge is also unnecessary! This result is the Mohr-Mascheroni Theorem. One really fascinating/disturbing fact about this result is that it was first proven, as far as we know, in 1672 by Mohr, but his proof was lost for over 250 years. Mascheroni independently discovered and proved the result about 120 years after Mohr.

I think this is an important lesson that, with search-empowered internet, is easy to forget: human knowledge accumulation isn't always steady or certain.

Compass-only program

Cut-the-Knot (a generally excellent resource) has a good discussion of restricted constructions and outlines a program for proving the sufficiency of compass-only constructions.

I have started to work through this list. Generally, I won't post solutions since Cut the Knot already has solutions. However, I was concerned about whether the compass they were using was collapsing or not. The solution for 6 (given three points, find a fourth point that makes them vertices of a parallelogram) clearly uses a non-collapsing compass. So, that leaves an open challenge: how to build a non-collapsing compass from just a collapsing compass.

I'm currently stuck on this. There are two ways I could break through:
(a) find the point of intersection of two lines. If I could do this, I would use the ability to find perpendicular lines to move distances around.
(b) Find the point of intersection of a line and a circle with the center on the line.

Note, this second one is very similar to Cut the Knot's third challenge, but that assumes the center of the circle is not on the line segment. Amazingly, this slight change makes the challenge much harder, at least for me.

Some Reverses in Gamma (Euclidea series)

I found several of the puzzles in Gamma pack to be cute, even though they aren't necessarily hard. In especially liked the "reverse" constructions, finding the triangle given the orthocenter (3.2) or given the circumcenter (3.3).

Note: toward the latter half of this pack, I was getting anxious to see what Delta pack had in store, so I bashed through constructions for 3.5-3.8 without always finding the minimal moves solutions. Among those four challenges, I still have 5 missing stars.

Triangle from orthocenter (3.2)
Obtaining the E star gave me trouble on this one. I didn't originally get one, but figured I would try harder for these notes.

The key idea is to use the E-optimal perpendicular construction from 2.6 and the vertex of the angle we're given as one of the center points. That allows us to pick up two perpendiculars for the cost of 5 E moves, leaving one last move to connect the two new vertices.

Triangle from intersection of perpendicular bisectors (3.3)

Well, I actually already gave a spoiler above when I shortened the name of this challenge. If you've got the intersection of the perpendicular bisectors, then you have the center of the orthocircle, the circle that contains all the vertices of the triangle.

Since we already have one vertex and rays where the other edges are....

Three equal distances (3.4)
When going back to write up these notes, I didn't remember how this construction worked and was concerned I'd have trouble working through a tricky challenge. Fortunately, ....

The key insight is the relationship between points B, D, and M. Just think about which of our favorite construction tricks relates them and you are done. Finding E from D and M is straightforward, but keep your eyes opened for a nice surprise!

Those V stars

Three equal distances (3.4), Forty-five degree angle (3.7), and lozenge (aka rhombus 3.8) all have V stars. In fact, 3.8 needs 4 versions to collect the V!

A construction mess (Euclidea Series)

Part of my motivation for writing this series is to make a confession: some of my constructions are just a mess built of geometrically calculating a length that I've determined algebraically. Typically, my approach is to create a coordinate system, then set up a couple of equations that determine a key length for the construction, solve those equations, then use geometric operations to construct that value.

Here is an example. Theoretically, it is a spoiler for a construction in mu pack (circle tangent to two circles 12.5) but I doubt anyone will really be able to see what is going on here:

Another example is construction of the regular pentagon. I know the golden ratio figures prominently, so one approach I often use is just to build that ratio between two lengths and then impose it on the basic construction template.

Other than massively overrunning the move targets for E and L stars, the weakness of this approach is that it doesn't link with any geometric insight about the construction. In a sense, this is the power of analytic geometry: you can get results without having to find a new insight.

Over the course of the series, I'll try to work on constructions that are geometrically insightful, but won't shy away from letting you know where I've had to push through with a brute force attack.

Perpendiculars (Euclidea series)

Beta pack is the only one for which we have received all of the stars. Even so, a couple of the puzzles are worth discussing because they illustrate some interesting ideas. In particular, we like Drop a Perpendicular (2.6) and Erect a Perpendicular (2.7).

Perpendicular bisectors
One of the ideas lurking is around the perpendicular bisector, which we saw way back in alpha pack (1.2). One of the things I always found interesting about the perpendicular bisector was that it was easier to construct an object with more conditions than either constructing a perpendicular alone or finding a midpoint alone.

The other great thing about the perpendicular bisector is that it can be defined in an entirely different way: it is the locus of all points that are equidistant from our two starting points. In case that isn't clear, assume we start with two distinct points A and B. The perpendicular bisector of the segment AB is also the locus of all points C such that distance AC is equal to distance BC.

Of course, the fact that this is a line means that we only have to find two such points to construct the perpendicular bisector (which is how you solved 1.2, right?)

Drop a perpendicular
For the 2 move solution using tools, I'll let you find a solution on your own. In case you need a hint: how many combinations of 2 moves are there anyway? You could just try them all and see what you find.

Way back in my HS geometry class, I learned a construction for dropping a perpendicular that uses 4 elementary moves. For the E start, though, that's not good enough.  The cool idea that breaks through here is to choose two totally arbitrary points on the line as centers of circles that we draw. For some reason, I get a kick out of the idea that arbitrary points can be helpful ("if the point we choose doesn't matter, how can it help to choose a point anyway?")

In this case, while the points on the line we choose don't matter, the circle we draw with those points as centers need to have the right radius. Click the button below if you want to see how it is done and a bit more explanation.

The key here is that the circle radii are equidistant from our target point and the new intersection point of the two circles. That means they are on the perpendicular bisector of the segment between those points. Another observation is that the new intersection of the two circles is the reflection of our target point through the line. Kind of cool that, no matter which two center points we choose for the two circles, the new intersection point will always be the same!

Erect a perpendicular
Erecting a perpendicular also uses the idea of choosing an arbitrary point, but goes a step farther. This time, we choose any point we want that is not on the line already! Well, it also fails if we happen to choose a point that is already on the perpendicular, but that should be impossible.... In any case, just choose a point somewhere off to the side.

This construction uses Thales' Theorem. I don't know exactly why I find this result to be so cool, since the proof isn't hard. For me, it transforms a circle into a family of right triangles. Given a length for the hypotenuse, all the right triangles with that hypotenuse are living right there on the arc of the circle with that length as the diameter.

Incidentally, Thales' Theorem and the two defining properties of perpendicular bisectors will come up a lot in other Euclidea Constructions.

V stars
If you want to know where the V stars are, click below:

Angle of 30 degrees (2.3) and Double Angle (2.4) both have two solutions.

Castle overrun and storm the boat (creating games)

Playing Santorini with a make-shift set inspired the older two Js to create their own games. We test played J1's game, but haven't yet had a chance to test play J2's game.

Castle Overrun

Equipment: 
TRIO set. In particular, we used the 7x14 top, 5 short and 1 long magenta and 5 short and 1 long green straight connectors, 50+ 1x1x1 cubes. The long connectors are kings, the short connectors are soldiers. Key feature is that the cubes can stack on top of each other and on top of the soldier/king pieces.

Set-up:
Kings start in diagonally opposite corners with their soldiers one square away. Pre-placed cubes also start on positions (6, 3), (6, 5), (9, 3) and (9, 5).
You can see these details in the picture below (the extra chains of cubes were just on the board to give a tidy "resting" set-up when we weren't playing):

Turns
Turns alternate. On a player's turn, they move one of their pieces (the king or a soldier) one grid space (pieces can move diagonally). Then, they have an option of one of 3 possible actions:

• build a cube on an adjacent space (including diagonally adjacent). There is no limit to how high they can build. Think of this as building part of a wall or tower. A king can build two bricks on one turn.
• destroy the top cube on an adjacent space. This is tearing down a part of a wall.
• Attack an opponent's piece on an adjacent square. This involves putting a cube on top of that piece. The J's called this "pig-heading" and a piece with a cube on top is "pig-headed." If a soldier has 2 cubes on top, they get removed from the board. If a king gets 3 cubes on top, then it gets removed from the board.

Winning
First player to remove the opponent's king from the board is the winner.

Playability
There are some elements that suggest this game has possibilities, but the current rules quickly become a stalemate. First, we noticed that any attempt to pig-head an opponent is a half-suicide mission, since they can pig-head right back on their turn. There is one exception, but it involves the opponent accidentally walking into an obvious trap (and there's no reason why they would do that).

Also, the king can quickly build a fortress around himself before the enemy soldiers get close. Because the king can build two bricks on a turn, it would be impossible for the enemy soldiers to break through. In fact, this strategy is what gave us the idea for the name of the game.

Possible modifications
The main problem is that wall building is too fast. Perhaps we should allow the soldiers to break down two bricks each turn, but restrict all building to one brick at a time. This means that any fortress can, eventually, be broken down, if the siege army is unmolested.

Storm the Boat

Equipment
Again, this is based on the TRIO set, but also requires 2 dice (choose your favorite sizes/shapes/markings).

Set-up
The straight connectors start in the same corner configuration as Castle Overrun:

Again, place one cube on each space (6, 3), (6, 5), (9, 3), (9, 5). Further, add bricks in a line from (1,2) to (4, 2) and (11, 6) to (14, 6).

There is another special piece in the center of the board, the boat:

Turns
Player rolls two dice and then moves a piece according to the values shown on the dice. They can move one piece for both values, or move two different pieces. Restrictions: movement is orthogonal (not diagonal) and pieces can not pass through a space occupied by another piece.

Pieces can move up one level on the boat. For example, if they are next to an open bottom magenta space, they can move onto the boat. If they are on the magenta level, they can move onto the blue seats.

After moving, the player pushes the boat one space toward their starting corner. This means that only one of the boat entry points is available at a time. This is like a moored boat rocking back and forth with the waves.

Winning
The first player to get three pieces onto the boat seats is the winner. They get a double win if their "king" is one of the pieces to get onto the boat.

Playability
This seems to be a solid racing game, similar to many other games for children. The boat mechanic adds a small twist. It seems playable and fun for kids, but lacks any deep strategy.

Possible modifications suggested have been:
(a) some way for the two teams to attack each other
(b) a way to release the boat from its anchor so that the movement is more extreme and/or erratic
(c) possibility for a team to attack the boat and sink it (perhaps to prevent an opponent from reaching the winning condition).

Summary thoughts

Putting aside how good their games were, I was delighted with the ideas and enthusiasm both J1 and J2 had for creating their own games. While I had been hoping they would have been more pro-active about designing a DIY Santorini set the day before, it seems that exposing them to this gave them a boost of inspiration to try their own.

Start of vacation math

Here in Thailand, the kids are now on summer vacation, so I've been exceptionally busy and found it hard to write blog posts. We've still been doing a lot of math, so I wanted to at least note down some of the activities.

Sonobe Modular Origami

Credit for this one goes to another parent at J1's school. One day when school was still in session for J1, but J2 and J3 were already on vacation, we happened to see a group making sonobe modules at the elementary school. This is something my mother tried to introduce to the kids last year, but it didn't catch them at the time. Not sure what was different this time, but J2 was really intrigued.

Most of the work has been J2, but J3 also got into the act and folded about 9 units herself. I helped to sharpen the folds for 6 of them, which J2 and J3, working together, made into a cube.

Here are two of J2's stellated octahedra, made in between swimming sessions at the beach:

Video references we found useful for guiding our sonobe exploration:

Sonobe Cube: this is where we started.

Stellated Octahedron: shows how to make the basic module and how to assemble 12 into a stellated octahedron. This proved to be easier than I had anticipated, so J2 was able to do this all on his own.

Doubled tetrahedron: unfortunately, this doesn't give a sonobe module tetrahedron, but rather a 6 sided shape. We had actually made one of these ourselves when experimenting with 3 sonobe modules created from post-it notes. These are delightfully strong and feel really nice to toss from hand to hand.

Some advanced observations
Euler Characteristics
The model for the cube gives a triangulation for which it is particularly easy to count vertices, edges, and triangles. I talked about the Euler characteristic and we then did the counts for other sonobe models as well as other platonic solids to check. We also talked about the Euler characteristic for a cylinder (without end caps) and a torus.

Finally, I liked to describe our calculation as "0 dimensional things" - "1 dimensional things" + "2 dimensional things." This led J2 to ask what we would get if we included the space inside the shapes (extending to a "3 dimensional thing") and to get curious about 4 dimensional objects (which led to a lot of discussion, below).

Chirality
Sonobe modules come in two flavours, depending on which corners you fold into the center. All three of us (J2, me, grandpa) accidentally made the two kinds of sonobe modules without realizing it, only to struggle when we went to assemble a larger model. This started a brief conversation about chirality. While we didn't explore in depth, it did spark a later moment of recognition when Matt Parker mentioned the concept for mobius strips (see below).

Robustness
I was really surprised that the sonobe models we built are so robust. The completed shapes don't fall apart very easily (especially the doubled tetrahedron). Even better, for the models we have built so far, the sonobe modules themselves don't need to be perfect. This was crucial for the beginning origamists in our gang as it allows them to do almost all the work from start to finish.

It also made me think about my old idea of error in construction recipes (straight-edge and compass constructions as well as origami folding constructions). To illustrate this for my father (and J2 unintentionally) I showed them David Eisenbud/Numberphile's construction of the 17-gon. This, in turn, prompted J2 to get excited about straight-edge and compass constructions (below).

Constructions

As mentioned above, J2 was excited about investigating classical construction problems. This is something he has seen a couple of times in the Abacus Math curriculum that P does with the kids. However, in that, we haven't generally been strict about the tools they use and the collection includes some standard triangles (45-45-90, 30-60-90) and measuring tools (rulers, protractors). Now, J2 was curious to see what could be done with just a straight-edge and compass.

For physical construction, our tool of choice has been the Safe-T compass:

Obviously, this doesn't quite correspond to the ancient limitations, but we ignore the ruled markings and the various holes are always within our margin of error on radius lengths anyway. Other than the obvious (for a parent) advantage, I think this compass style makes it easier for 2 people to help each other when drawing the circular arcs.

J2 has also been playing Euclid the Game. The virtual constructions have added two interesting points:

• strict restriction to what is constructed; he eyeballed a pair of equal lengths at one point, then wanted to talk about why he hadn't gotten credit for the construction (for angle bisection). This actually helped him identify that he needed the lengths to be equal; once he articulated that, he knew to use the compass to draw a circle.
• Chunking into advanced tools. For example, the game gives you a mid-point tool once you demonstrate the construction of a midpoint of a given line segment. This has helped him start to work on the coding concepts of functions/procedures and also gives him a new visual reference for the problem solving strategy "try to use what you already know."

Fourth dimension

The big insight here is about the relationship between understanding four dimensional shapes and our own ability to perceive 2 and 3 dimensional shapes. Other than that tip, I'd direct you to these great resources:

Flatland: we don't actually have a copy, but I know it well and it is a foundation behind a lot of the discussions we had and videos we watched.

Matt Parker's Royal Institution talk: He covers a lot that isn't about four dimensional shapes, but I'm not going to point you to the relevant starting time because the whole talk is great!

XKCDhatGuy: This was actually one of the first videos J2 found and, hey, the kid does a pretty good job explaining. Don't, by any means, stop with this explanation, but it is a fine place to start.

Carl Sagan's Flatland: J2 liked it, but I still remember the old cartoon from the BU library that we would watch at PROMYS and wasn't so impressed.

4th dimension explained: [no comment]

Drawing 4th, 5th.. dimension:  Somehow, the simple points in this video really helped demystify higher dimensional shapes for J2. I think part of the point is that it encourages us to attend to aspects of the shapes that we can understand easily (for example, the number of vertices in an n-dimensional cube) rather than worrying about the macro shape.

Rotations of 4 dimensional shapes:  Similar message to the previous video

There is no 4th dimension: Nice, quick explanation living up to the high standards of other One Minute Physics videos. A very worthwhile series, in its own right.

Cricket

This isn't mathematical, but I wanted to record this for posterity as well. Through random chance, J2 asked, "How do you play cricket?" At this point, he is still 2/3 English, having spent most of his life in London, so I guess he has a right to some information about this crazy pastime. For better or worse, though, that's not what he got. Our house rules:

1. batsman/batsmen pretend to stand in a traditional cricket batting pose, mime holding a bat
2. I stand a couple feet away, mime delivery of a cricket pitch (straight arm overhead throw)
3. we all (including spectators) run together into a giant heap, tickling each other and saying vaguely English-y, vaugely cricket-y things: "good show old chap," "ooh, looks like rain," "how do  you take your tea?" "you were almost to a century, young chap" etc
4. I break off the tickling and everyone stands in line, waiting for their points and an explanation of why points were awarded. Not spilling the tea is the most common reason for someone getting points, though boundary hits are also frequent. Note that points can be awarded in non-integer amounts (1/3, sqrt(2), and pi have all been awarded at some point).
5. Finally, I declare the match over with a result of "no result." As you can tell, we take our cricket seriously and are committed to the "test match" format.

Some important summary points:

• J2 is the player who has managed to accumulate the largest score in a single match. He was awarded "one hundred million one thousand" points by J3 for not spilling the tea.
• We are coining a new English-y phrase: "young old chaps." Usage example: when J3 is setting up a chasing game outside, she will point to J1 and J2 saying "these young old chaps are the runners," then point to me "and that old young chap is the monster."